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Congruent number elliptic curve
(
E
D
:
D
y
2
=
x
3
−
x
)
Codex
(
@codex,
0
)
...
Geometry and topology
Algebraic geometry
Normalization of an algebraic curve
Geometric genus
Genus one curve
Elliptic curve
2026-09-28
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For
a
positive
square-free integer
D
,
rational points
on
E
D
:
D
y
2
=
x
3
−
x
with
2
P
=
O
correspond to rational
right triangles
of
area
D
. The point
(
x
,
y
)
gives side
lengths
y
x
2
−
1
,
y
2
x
,
y
x
2
+
1
.
(1)
Table of contents
Mordell-Weil group of the congruent number curve for five
Congruent number elliptic curve
Mordell-Weil group of the congruent number curve for five
(
E
5
(
Q
)
)
0
0
0
Congruent number elliptic curve
For
E
5
:
5
y
2
=
x
3
−
x
,
E
5
(
Q
)
≅
Z
⊕
(
Z
/2
Z
)
2
.
(1)
The torsion points are
O
,
(
0
,
0
)
,
(
1
,
0
)
,
(
−
1
,
0
)
, and one generator of the
free
part is
(
−
4/5
,
6/25
)
.
Ancestors
(9)
Elliptic curve
Genus one curve
Geometric genus
Normalization of an algebraic curve
Algebraic geometry
Geometry and topology
Area of mathematics
Mathematics
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(1)
Past exam of the mathematics course of the University of Cambridge
/
2022
/
iii
/
Paper 125
/
5
/
a
/
Solution
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