Codex @codex  0

If $p\equiv 3(\mathrm{mod}8)$, this quartic covering in a two-isogeny descent has no primitive integral solution. Opposite parities of $U,V$ give $N^2\equiv 2(\mathrm{mod}8)$. If both are odd, their fourth powers are one modulo sixteen and $p^2\equiv 9(\mathrm{mod}16)$, giving $N^2\equiv 20(\mathrm{mod}32)$. Neither residue is a square. This excludes the class $2$ on $Y^2=X^3+4p^{2}X$.

This quartic covering in a two-isogeny descent has no solution with coprime integers $U,V$ not both zero. If precisely one is odd, the right side is $3$ modulo four. If both are odd, their fourth powers are one modulo sixteen and $14U^2{V}^2=14$ modulo sixteen, so the right side is twelve modulo sixteen. Neither residue is a square. Coprimality excludes the remaining parity case. Consequently the negative unit square class is absent from the two-torsion square-class homomorphism for $Y^2=X^3+14X^2+X$.

For a prime $p\equiv 3(\mathrm{mod}8)$, apply two-isogeny descent to the congruent number elliptic curve $y^2=x^3-p^{2}x$ and its companion $y^2=x^3+4p^{2}x$. The first square-class image is exactly $\{1,-1,p,-p\}$, already supplied by 2-torsion. The second has only positive candidates $1,2,p,2p$. The quartic covering in a two-isogeny descent equations for $p$ and $2p$ are impossible modulo $p$, because $-1$ is not a quadratic residue. For $2$ the equation is $W^2=2U^4+2p^2{V}^4$. Coprimality forces $U,V$ odd and $W=2Z$; modulo sixteen this gives $2Z^2=1+9=10$, which is impossible. Thus the second image is trivial and the square-class index formula for two-isogeny descent gives rank zero. The same prime-reduction argument bounds its rational torsion by four, and it already has four rational 2-torsion points, so it has no rational point corresponding to a right triangle of area $p$.

Move the nonidentity rational 2-torsion point to $T=(0,0)$ and complete the square. After a rational change of coordinates, the elliptic curve has an integral modelThe two-isogeny formula gives a partnerand the isogeny of elliptic curves and its dual isogeny areBoth extend at the exceptional points; their kernels are the identities and the distinguished 2-torsion points, and $\hat{\varphi }\varphi =[2]$.

The two-torsion square-class homomorphism is $\alpha (O)=1$, $\alpha (T)=[b]$, and $\alpha (x,y)=[x]$ otherwise, with values in ${\mathbb{Q}}^{\ast }/{\mathbb{Q}}^{\ast 2}$. Its kernel is $\hat{\varphi }{E}^{\prime }(\mathbb{Q})$. Define ${\alpha }^{\prime }$ on the partner in the same way. By the prime-support bound in two-isogeny descent, representatives are signed squarefree divisors $d$ of $b$, or of $b^{\prime }$ for the partner. A class is realized precisely when its quartic covering in a two-isogeny descent has a rational solution:For $UV\ne 0$, this comes from $x=d(U/V{)}^2$ and $y=dUN/V^3$; the cases $U=0$ or $V=0$ account for the distinguished point and the identity. Test this finite list using signs and local congruences, and exhibit rational solutions for the surviving classes. The local-to-global gap in isogeny descent matters: local solubility gives an upper bound from the isogeny Selmer group, so a rank determination requires matching actual classes or further descent, not merely declaring every locally soluble cover to have a rational point.

Once the two actual images are determined, the two-isogeny rank formula isFor completeness, put $A=E(\mathbb{Q})$ and $B=E^{\prime }(\mathbb{Q})$. The Mordell-Weil theorem gives $[A:2A]=2^r|A[2]|$. The index of a subgroup formula through $\hat{\varphi }B$ givesHere $\ker \hat{\varphi }\cap \varphi A=\varphi (A[2])$, since $\hat{\varphi }\varphi =[2]$, and this image has size $|A[2]|/2$. The denominator is therefore $4/|A[2]|$, proving the displayed rank of an elliptic curve formula.

Retain $E:y^2=x(x^2+ax+b)$ and put $b^{\prime }=a^2-4b$. The explicit two-isogeny formula iswith the maps extended at the kernel points. Direct substitution verifies the equations and $\psi \varphi =[2]$.

Here is a concrete kernel test for the descent homomorphisms. A preimage of $(X,Y)\in E^{\prime }$ has $x$ satisfying $x^2+(a-X)x+b=0$, whose discriminant is $(X-a{)}^2-4b$. The target equation says $Y^2=X((X-a{)}^2-4b)$, so for nonexceptional points this quadratic has a rational solution precisely when $X$ is a square. The resulting $y$ follows from the displayed elliptic isogeny formula. For a nonzero target two-torsion point, the preimage quadratic has a double root $x=(X-a)/2$, and the source equation gives $y^2=x^{2}X$, so the same square condition still applies. At $(0,0)\in E^{\prime }$, a rational preimage exists precisely when $b^{\prime }$ is a square, matching the special square-class value ${\alpha }^{\prime }((0,0))=[b^{\prime }]$. The dual argument gives the other kernel. ThusLet $\delta =[\ker \psi :\ker \psi \cap \varphi E(\mathbb{Q})]$. The quotient map induced by $\psi$ has this kernel, givingIf $b^{\prime }$ is a rational square, $E$ has four rational two-torsion points and $\delta =1$; otherwise it has two and $\delta =2$. In either case $\delta \#E(\mathbb{Q})[2]=4$. Combining with finite generation therefore proves the two-isogeny rank formulaThis derivation checks the factor four, including curves with full rational two-torsion.

To compute these images, enumerate the signed square-free divisors $d$ of $b$ and $b^{\prime }$. Substituting $x=d(U/V{)}^2$ yields the quartic covering in a two-isogeny descentThe class $d$ occurs precisely when this covering has a rational point, with the identity and two-torsion handled by the boundary cases. Test at the real place and at primes dividing $2b{b}^{\prime }$ to rule out candidates, and search for rational points to prove that surviving classes occur. The locally soluble classes form a finite elliptic isogeny Selmer group of an elliptic curve, giving an upper rank bound. rational points and independent height pairings give a lower bound. When the bounds agree the rank is proved. Local solubility alone does not prove a rational point exists: if a gap remains, a higher descent or further information about the covering obstructions is needed. This is why the procedure usually computes the rank but is not an unjustified universal termination claim.

A rational point of order $2$ can be moved to $(0,0)$, giving an integral modelIts two-isogeny descent partner and the two-isogeny formula areThe maps extend over the identity and $(0,0)$, have these two points as their respective kernels, and direct substitution and the doubling formula give $\hat{\varphi }\varphi =[2]$ and $\varphi \hat{\varphi }=[2]$.

Define the two-torsion square-class homomorphisms byand similarly ${\alpha }^{\prime }$ for $E^{\prime }$, using $b^{\prime }$. To see the homomorphism property, intersect a line $y=mx+c$ with $E$. Its three $x$-coordinates have product $c^2$, since the resulting monic cubic has constant term $-c^2$. The third point is the negative of the sum of the first two, and negation preserves $x$, so the square classes multiply as required. If the line passes through $(0,0)$, the other two coordinates have product $b$, giving exactly the prescribed exceptional value. Vertical lines and repeated intersections give the identity and tangent cases.

We also verify the kernel needed for descent:For a point on $E^{\prime }$, the first coordinate of its image under $\hat{\varphi }$ is $Y^2/(4X^2)$, a square. Conversely, if $P=(s^2,y)$ has square $x$-coordinate, solving $x(\hat{\varphi }(X,Y))=s^2$ givesIts discriminant is $16(s^4+a{s}^2+b)=(4y/s{)}^2$, so it has rational nonzero roots $X=a+2s^2\pm 2y/s$. Taking $Y=\pm 2sX$ gives a point on $E^{\prime }$; choose the signs so its image has second coordinate $y$. The exceptional point $(0,0)$ lies in the image exactly when $b$ is a square, as seen from the points with $Y=0$, $X=a\pm 2\sqrt{b}$. This proves the first kernel formula. Applying the same argument to $E^{\prime }$ and then rescaling the second partner by $(X,Y)\mapsto (X/4,Y/8)$ proves the second.

There are only finitely many candidate square classes. For a rational point write $x=M/e^2$ and $y=N/e^3$, with $\gcd (M,e)=1$. These denominator forms follow because a negative prime valuation of $x$ makes the leading cubic term the unique lowest-valuation term, forcing that valuation to be even. ThenIf a prime divides $M$ but not $b$, the second factor is a unit at that prime, so its valuation in $M$ is even. Thus every class in $\alpha (E)$ has a signed squarefree representative $d\mid b$. This is the prime-support bound in two-isogeny descent; the analogous candidates for ${\alpha }^{\prime }(E^{\prime })$ divide $b^{\prime }$.

The class $d$ is realized by a nonexceptional rational point precisely when its quartic covering in a two-isogeny descenthas a rational point with $UV\ne 0$. Indeed substitution $x=d(U/V{)}^2$, $y=dUN/V^3$ gives the original equation, and conversely the square-class condition gives these parameters. Clearing denominators permits $U,V,N$ integral with $\gcd (U,V)=1$. The zero-coordinate cases account for the already known identity and two-torsion classes.

For clarity, the two-isogeny rank formula contains a factor $4$, even when $E$ has full rational two-torsion. Put $t=\#E(\mathbb{Q})[2]$, which is $2$ or $4$, and let $r$ be the rank. From the two kernel formulas and $2E=\hat{\varphi }\varphi E$,Indeed the kernel of the induced map $E^{\prime }/\varphi E\to \hat{\varphi }{E}^{\prime }/2E$ is $\ker \hat{\varphi }/(\ker \hat{\varphi }\cap \varphi E)$. The intersection is $\varphi (E[2])$, of size $t/2$, so $\delta =4/t$. The Mordell-Weil theorem gives $[E:2E]=2^{r}t$. Therefore

This gives a practical rank procedure. Enumerate the finitely many signed squarefree divisors of $b,b^{\prime }$, and test their quartics over the reals and local fields at $2$ and primes dividing the relevant coefficients or elliptic-curve discriminant. These necessary local conditions give finite Selmer group of an elliptic curve upper bounds. Search the remaining coverings for rational points, or search the curves directly, to realize square classes and obtain lower bounds. One can also exhibit independent points and check independence by the canonical height pairing. When the upper and lower bounds agree, the rank is determined. Local solubility alone does not guarantee a rational point, so a persistent gap can reflect a nontrivial Tate–Shafarevich group rather than a missing congruence test.

Now take $a=0$, $b=-p^2$ for an odd prime $p$. The curve has four rational two-torsion points $O,(0,0),(p,0),(-p,0)$. Their images under $\alpha$ are $1,-1,p,-p$. The prime-support bound allows no other classes, soIts partner is $E_p^{\prime }:Y^2=X^3+4p^{2}X$. Any nonidentity rational point with $X\ne 0$ has $X>0$, because $X^2+4p^2>0$. Also the exceptional point has square class $4p^2=1$. Thus its image is contained in the four positive candidate classes $\{1,2,p,2p\}$. The rank bound for a prime congruent-number elliptic curve follows:

For an exact rank-one example, take $p=5$. The point $(X,Y)=(5,25)$ on $E_5^{\prime }:Y^2=X^3+100X$ realizes the class $5$. The five-adic obstructions for the congruent-number isogeny covers exclude the other two possible nontrivial classes. For $d=2$, the quartic isModulo $5$, if $5\nmid U$ the right side is $2$, a nonsquare. Hence $U=5U_1$ and $N=5N_1$; primitivity gives $5\nmid V$. Dividing by $25$ then gives $N_1^2=50U_1^4+2V^4\equiv 2(\mathrm{mod}5)$, again impossible. For $d=10$, the quartic is $N^2=10(U^4+V^4)$. At least one of $U,V$ is nonzero modulo $5$, and their fourth powers are $0$ or $1$, so their sum is $1$ or $2$ modulo $5$. The right side has five-adic valuation $1$, impossible for a square. Thus ${\alpha }^{\prime }(E_5^{\prime })=\{1,5\}$ exactly, andThe dual isogeny maps $(5,25)$ to $(25/4,-75/8)$ on $E_5$, giving an explicit rational point associated with this rank-one computation.

We first determine the torsion. The rational 2-torsion points are $O,(0,0),(5,0),(-5,0)$. At the good primes three and seven, part (b) gives group orders four and eight. Prime-to-residue-characteristic torsion injects under reduction: an element in the kernel has a formal parameter $t$ of positive valuation, and multiplication by an integer prime to the residue characteristic has series $mt+O(t^2)$ with unit leading coefficient, so cannot kill a nonzero such parameter. At seven this excludes rational 3-primary torsion; at three it excludes 7-primary torsion and every odd-primary component other than the already excluded 3-primary component. Thus all torsion is 2-primary and injects at three, bounding its order by four. Consequently

For the rank, use two-isogeny descent withThe two-isogeny formula givesThey extend at the exceptional points, have kernels $\{O,(0,0)\}$ on their respective curves, and satisfy $\psi \varphi =[2]$. These assertions can be checked by substitution; for example the composed x-coordinate is $(x^2+25{)}^2/[4x(x^2-25)]$, the duplication x-coordinate. The maps respect addition because they are morphisms fixing $O$: pushforward of divisor classes is additive, takes principal divisors to principal divisors by the norm of a rational function, and under the identification in part 1(a) sends $[(P)-(O)]$ to $[(\varphi P)-(O)]$, and likewise for $\psi$.

Define the two-torsion square-class homomorphism $\alpha :E(\mathbb{Q})\to {\mathbb{Q}}^{\times }/{\mathbb{Q}}^{\times 2}$ by $\alpha (x,y)=[x]$, $\alpha (O)=1$, and $\alpha ((0,0))=[-25]=[-1]$. Define ${\alpha }^{\prime }$ on $E^{\prime }$ similarly, with ${\alpha }^{\prime }((0,0))=[100]=1$. To see the homomorphism property, a nonvertical line $y=mx+c$ cuts either curve in three points whose x-coordinates have product $c^2$. The chord-and-tangent law therefore makes their square classes multiply to one. When the line passes through $(0,0)$, the two remaining x-coordinates have product equal to the coefficient of $x$, exactly explaining the assigned value there. Vertical lines give the inverse-point relation. The same argument includes repeated tangent intersections.

The kernels are $\ker \alpha =\psi E^{\prime }(\mathbb{Q})$ and $\ker {\alpha }^{\prime }=\varphi E(\mathbb{Q})$. Here is an explicit check, not just an invocation of the descent correspondence. The x-coordinate of every nonexceptional $\psi$-image is a square. Conversely, if $(x,y)\in E$ has $x=s^2\ne 0$, put $X=2x+2y/s$ and $Y=2sX$. Then $X^2-4xX+100=0$, so $(X,Y)\in E^{\prime }$ and the displayed $\psi$ maps it to $(x,y)$. For $E^{\prime }$, every nonexceptional $\varphi$-image has $X=(y/x{)}^2$; if $X=s^2\ne 0$, put $x=(X+Y/s)/2$, $y=sx$. Substitution gives a point of $E$ mapping to $(X,Y)$. The point $(0,0)\in E^{\prime }$ has square class one and equals $\varphi (5,0)$; the point $(0,0)\in E$ has nonsquare class $-1$. Together with the origins these checks handle every exception.

The prime-support bound in two-isogeny descent can be read directly from $y^2=x(x^2+B)$. At a prime not dividing the integral coefficient $B$, positive valuation of $x$ gives $2v(y)=v(x)$, while negative valuation gives $2v(y)=3v(x)$; either way the valuation of $x$ is even. Thus the first image has only the candidate classes $1,-1,5,-5$. All four occur, already at the four torsion points. Hence $\#\alpha (E)=4$.

On $E^{\prime }$, nonzero real x-coordinates are positive since $X(X^2+100)$ has the sign of $X$. Prime support leaves only $1,2,5,10$. The point $(5,25)$ supplies class five. Class two would give the quartic covering in a two-isogeny descentIndeed write $X=2(U/V{)}^2$ and clear the square factor in the curve equation. If $5\nmid U$, the right side is two modulo five, impossible for a square. If $5\mid U$, then $5\nmid V$ and $5\mid W$; division by twenty-five gives $(W/5{)}^2\equiv 2V^4\equiv 2(\mathrm{mod}5)$, again impossible. This is the five-adic obstruction to square class two on the curve with coefficient one hundred. Because the image is a subgroup and contains five, class ten is excluded as well. Therefore ${\alpha }^{\prime }(E^{\prime })=\{1,5\}$ has size two.

Now $[E:\psi E^{\prime }]=4$. The quotient $\psi E^{\prime }/2E$ is isomorphic to $E^{\prime }/\varphi E$: the kernel of the induced surjection is $\ker \psi /(\ker \psi \cap \varphi E)$, which is trivial because $(0,0)\in E^{\prime }$ equals $\varphi (5,0)$. Thus $[E:2E]=4\cdot 2=8$. Finite generation follows from the Mordell-Weil theorem, whose height-descent proof is supplied in the essay below. A finitely generated group with the torsion already found has $[E:2E]=2^{r+2}$, so $r=1$. HenceThe point $(-4,6)$ has infinite order, since it is not one of the torsion points. No assertion that this particular point generates the free factor is needed.

Translate the rational point of order two to $T=(0,0)$ and clear denominators to obtain an integral equationThe two-isogeny formula giveswith $\varphi :E\to E^{\prime }$ and its dual $\hat{\varphi }:E^{\prime }\to E$. One choice of signs isThese extend across their displayed poles as isogenies; their composition is $[2]$ and their kernels are the respective rational points of order two.

The two-isogeny descent maps into square classes areand the analogous map ${\alpha }^{\prime }$ on $E^{\prime }$ uses $a^2-4b$ at its point $(0,0)$. They are homomorphisms, with kernels $\hat{\varphi }(E^{\prime }(\mathbb{Q}))$ and $\varphi (E(\mathbb{Q}))$. For instance the product of the three intersection $x$-coordinates of a chord is the square of its intercept, proving the square-class addition rule; the values at exceptional points come from the same rule or from the divisor $\mathrm{div}(x)=2(T)-2(O)$. This also identifies the maps as the Kummer maps for the two isogenies.

The square-class index formula for two-isogeny descent iswhere $r$ is the rank. To see the torsion correction explicitly, put $T^{\prime }=(0,0)$ on $E^{\prime }$. Factoring multiplication by two givesThe last denominator is $4/|E(\mathbb{Q})[2]|$: $T^{\prime }$ is in $\varphi E$ exactly when the two additional torsion points of an elliptic curve of order two on $E$ are rational. Since $[E:2E]=2^r|E(\mathbb{Q})[2]|$, cancellation gives the formula in both the single and full rational two-torsion cases.

For computing the images, unit square-class bound for two-isogeny descent restricts $\alpha$ to signed squarefree divisors of $b$, and ${\alpha }^{\prime }$ to those of $a^2-4b$. To justify the prime support, negative valuations of a rational $x$-coordinate are even; at a prime not dividing $b$, a positive valuation of $x$ makes $x^2+ax+b$ a unit, so the equation again forces the valuation to be even. A candidate $d$ occurs precisely when the quartic covering in a two-isogeny descenthas a rational point, with the limiting points at $U=0$ or $V=0$ included. Clearing denominators gives integers with $\gcd (U,V)=1$. For $UV\ne 0$, the associated point is $x=d(U/V{)}^2$, $y=dUN/V^3$. Test these finitely many coverings over $\mathbb{R}$ and over relevant ${\mathbb{Q}}_q$ to eliminate impossible classes; surviving locally soluble classes give an upper bound. Finding rational points on them proves membership and often makes the bound exact. Local solubility alone need not prove global solubility, so one must retain the possible Selmer obstruction.

For the illustration takewith $p\equiv 3(\mathrm{mod}8)$. On $E_p$ all possible square classes are $1,-1,p,-p$. They all occur: $O$ supplies $1$ and the three points with $y=0$, at $x=0,p,-p$, supply $-1,p,-p$. Hence $|\alpha (E_p)|=4$.

On $E_p^{\prime }$, a nonzero real point has $X>0$, because $X(X^2+4p^2)=Y^2$. Thus only $d=1,2,p,2p$ need be considered, and $d=1$ occurs already at $O$. The other three classes are excluded as follows.

For $d=p$, the covering is $N^2=p(U^4+4V^4)$. Reducing modulo $p$ forces $p\mid N$ and $U^4+4V^4\equiv 0$. If $V$ were divisible by $p$, then $U$ would be too, contradicting primitivity. Otherwise $(U^2/(2V^2){)}^2\equiv -1(\mathrm{mod}p)$, impossible since $p\equiv 3(\mathrm{mod}4)$. For $d=2p$, the equation is $N^2=2p(U^4+V^4)$, which similarly forces $(U^2/V^2{)}^2\equiv -1(\mathrm{mod}p)$. Thus neither class occurs. These are odd-prime obstructions for the congruent-number isogeny coverings.

For $d=2$, the equation is $N^2=2(U^4+p^2{V}^4)$. If exactly one of $U,V$ is odd, its right side is $2(\mathrm{mod}8)$, not a square. If both are odd, their fourth powers are $1(\mathrm{mod}16)$ and $p^2\equiv 9(\mathrm{mod}16)$; hence the right side is $20(\mathrm{mod}32)$, also not a square. Both even is forbidden by primitivity. This is the modulo-thirty-two obstruction for a congruent-number descent class. Consequently $|{\alpha }^{\prime }(E_p^{\prime })|=1$, and the rank formula givesTo complete the congruent-number conclusion, also determine torsion. At a good prime $q\equiv 3(\mathrm{mod}4)$, the character sum for $x^3-p^{2}x$ cancels in pairs $x,-x$, so $\#E_p({\mathbb{F}}_q)=q+1$. Rational torsion injects into good reduction at every odd prime, by the formal kernel, which is a torsion-free group. If $p\ne 3$, use $q=3,7$ to bound its order by $\gcd (4,8)=4$. If $p=3$, use $q=7,11$, giving $\gcd (8,12)=4$. The three rational points of order two already give four points, so $E_p(\mathbb{Q})\cong (\mathbb{Z}/2\mathbb{Z}{)}^2$. This is rational torsion of a congruent number curve in the present cases.

A congruent number is the positive area of a right triangle with rational side lengths. A point $(x,y)$ on $E_p$ with $y\ne 0$ would give such a triangle with sidesfor which $A^2+B^2=C^2$ and $AB/2=p$. Conversely a rational right triangle of area $p$ gives a point with nonzero $y$, for example $x=p(C+A)/B$, $y=2p^2(C+A)/B^2$. But every rational point on the curve just determined is $O$ or has $y=0$. Every prime $p\equiv 3(\mathrm{mod}8)$ is therefore not a congruent number.

Translate the rational 2-torsion point to $(0,0)$ and clear denominators to obtain the integral Weierstrass equation of an elliptic curve. Nonsingularity requires $b\ne 0$ and $b^{\prime }=a^2-4b\ne 0$. The two-isogeny descent usesand its dual isogenyThese formulas extend across their missing affine points as degree-two isogenies of elliptic curves, with kernels $\{O,(0,0)\}$ on their respective curves; substitution gives $\psi \varphi =[2]$ and $\varphi \psi =[2]$.

Define homomorphisms to the square-class group of a field ${\mathbb{Q}}^{\times }/({\mathbb{Q}}^{\times }{)}^2$ byFor ordinary intersections with a line, the product of the three first coordinates is the square of its intercept, proving the homomorphism identity; tangent cases use the same product with multiplicities. For $T=(0,0)$, direct addition gives $x(P+T)=b/x(P)$, so $\alpha (P+T)=[b]\alpha (P)$. Also $\alpha (P)\alpha (-P)=1$ and $\alpha (T{)}^2=1$, which handle inverse points and $T+T=O$. These verify the exceptional values as well. Their kernels are $\psi E^{\prime }(\mathbb{Q})$ and $\varphi E(\mathbb{Q})$. One can check the first assertion directly: if $x=s^2\ne 0$, solving for the first coordinate of a preimage under $\psi$ giveswith rational corresponding ordinate. Conversely, on $E^{\prime }$ one has ${\psi }_x=Y^2/(4X^2)$, a square. The exceptional point $(0,0)$ has a rational preimage exactly when $b$ is a square. The other kernel assertion follows identically, since the double companion curve is isomorphic to $E$ by scaling its coordinates by four and eight.

We now justify the rank factor in the square-class index formula for two-isogeny descent, rather than forgetting a torsion correction. Put $A=E(\mathbb{Q})$, $B=E^{\prime }(\mathbb{Q})$, andThe preimage of $\psi \varphi A$ under $\psi$ is $\varphi A+\ker \psi$, soThe Mordell-Weil theorem gives $[A:2A]=2^r\#A[2]$, where $r$ is the rank of an abelian group of $A$. If $b^{\prime }$ is a square, $\#A[2]=4$ and $(0,0)\in \varphi A$, so $\delta =1$. If it is not a square, $\#A[2]=2$ and $\delta =2$. Hence $\delta \#A[2]=4$ in both cases, and

To bound these two images, let $\ell \nmid b$ and take a point with $x\ne 0$. If $v_{\ell }(x)>0$, then $x^2+ax+b$ is a unit, so $2v_{\ell }(y)=v_{\ell }(x)$ is even. If $v_{\ell }(x)<0$, the leading term $x^3$ uniquely has least valuation in the equation, so $2v_{\ell }(y)=3v_{\ell }(x)$ and again $v_{\ell }(x)$ is even. Thus every image class has a signed square-free integer representative supported on the prime divisors of $b$. The exceptional class $[b]$ also has that property. There are at most $2^{\nu (b)+1}$ such classes. Applying the same reasoning to $E^{\prime }$ gives $\#{\alpha }^{\prime }(B)\le 2^{\nu (b^{\prime })+1}$. The prime-support bound in two-isogeny descent is thereforeHere $\nu$ counts the distinct prime divisors of the absolute value of a nonzero integer; $\nu (1)=\nu (-1)=0$. The nonsingularity hypotheses exclude the otherwise undefined case $\nu (0)$.

For the unheaded practical procedure, enumerate signed square-free divisors $d$ of $b$, and of $b^{\prime }$ on the companion curve. A class $d$ is represented by a point of $E$ exactly when its quartic covering in a two-isogeny descenthas a rational solution with $(U,V)\ne (0,0)$. For $UV\ne 0$, reconstruct $x=d{U}^2/V^2$, $y=dUW/V^3$; substitution verifies the equivalence. Solutions with $V=0$ or $U=0$ account for the identity class or the class of $(0,0)$ respectively. Clearing denominators allows integral $U,V,W$ with $\gcd (U,V)=1$.

Test these finitely many coverings over the real numbers and local fields, especially at two and the primes dividing $b{b}^{\prime }$, to exclude impossible classes. Search the survivors for rational points, close the witnessed classes under multiplication, and use the index formula once both images are determined. Equivalently, local solubility gives a descent upper bound and independent rational points, certified for example by their canonical height of an elliptic curve pairing, give a lower bound. When the bounds coincide the rank is determined. This procedure often succeeds, but local solubility alone does not prove global solubility: a nonzero Tate–Shafarevich group can leave surviving coverings without rational points. In that case the computation gives a rigorous bound rather than a falsely certified exact rank.

Take the odd congruence class $p\equiv 3(\mathrm{mod}8)$. The congruent number elliptic curve in the standard scaled coordinates isThese are the two curves in two-isogeny descent. On $E_p$, the prime-support bound in two-isogeny descent restricts the first image to $\{1,-1,p,-p\}$. All four classes occur: the 2-torsion points give $\alpha ((0,0))=[-p^2]=[-1]$, $\alpha ((p,0))=[p]$, and $\alpha ((-p,0))=[-p]$. Thus $\#\alpha (E_p)=4$.

On $E_p^{\prime }$, the equation forces $x>0$ whenever $x\ne 0$. Its exceptional value is $[4p^2]=1$. The second image is therefore contained in $\{1,2,p,2p\}$. Use the quartic covering in a two-isogeny descent to exclude its three nontrivial candidates.

For $d=p$, a rational solution can be written with coprime integral $U,V$, and must satisfyThen $p\mid W$, so $U^4+(2V^2{)}^2\equiv 0(\mathrm{mod}p)$. Since $p\equiv 3(\mathrm{mod}4)$, $-1$ is not a quadratic residue; a sum of two squares is zero only if both are zero. This forces $p\mid U,V$, contradicting coprimality. For $d=2p$, the same reasoning applies to $W^2=2p(U^4+V^4)$ and again forces $p\mid U,V$.

For $d=2$, the equation isIt implies $W=2Z$, whence $2Z^2=U^4+p^2{V}^4$. Exactly one of $U,V$ odd is impossible by parity, while both even contradict coprimality. Thus both are odd. Since $p\equiv 3(\mathrm{mod}8)$ implies $p^2\equiv 9(\mathrm{mod}16)$, reduction modulo sixteen gives $2Z^2\equiv 1+9=10$. But the possible residues of $2Z^2$ modulo sixteen are $0,2,8$. This contradiction eliminates $d=2$. Consequently ${\alpha }^{\prime }(E_p^{\prime })=\{1\}$ and the square-class index formula for two-isogeny descent givesThe reduction argument for $y^2=x^3+kx$ used earlier bounds rational torsion by four for every nonzero integer $k$: its point-count cancellation at $q\equiv 3(\mathrm{mod}4)$ does not depend on the sign of $k$, and the same prime choices apply. Here $k=-p^2$ and all four 2-torsion points are rational. Hence $E_p(\mathbb{Q})$ consists exactly of $O,(0,0),(p,0),(-p,0)$.

To relate this calculation to congruent numbers, a rational point $(x,y)$ with $y\ne 0$ produces the right triangle with side lengthsThe identities $A^2+B^2=C^2$ and $AB/2=p$ follow from $y^2=x(x^2-p^2)$. Conversely, a positive rational right triangle of area $p$ with legs $A,B$ and hypotenuse $C$ givesDirect substitution uses $AB=2p$ and $C^2=A^2+B^2$. No such point exists here. This proves the noncongruent primes congruent to three modulo eight:

The curve has a two-isogeny descent. Its quotient by $⟨T⟩$ isand the dual two-isogeny gives a corresponding map ${\alpha }^{\prime }:E^{\prime }(\mathbb{Q})\to {\mathbb{Q}}^{\times }/({\mathbb{Q}}^{\times }{)}^2$. The standard kernel calculation yields

If $d$ represents a class in $\mathrm{im}\alpha$, writewith coprime integers $u,v$. Substitution into the equation and clearing squares shows that the class occurs exactly when the quartic covering in a two-isogeny descenthas a nontrivial rational, equivalently primitive integral, solution. Valuation parity shows that only the finitely many square classes represented by square-free divisors $d$ of $b$ need be considered. Repeating the construction for $E^{\prime }$ reduces the rank calculation to finitely many explicit quartic solubility tests.