The elliptic curve group of an elliptic curve is a group in which the elements of the group are points on an elliptic curve.
The group operation is called elliptic curve point addition.
Not every belongs to the elliptic curve over a non quadratically closed field by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16One major difference between the elliptic curve over a finite field or the elliptic curve over the rational numbers the elliptic curve over the real numbers is that not every possible generates a member of the curve.
This is because on the Equation "Definition of the elliptic curves" we see that given an , we calculate , which always produces an element .
But then we are not necessarily able to find an for the , because not all fields are not quadratically closed fields.
For example: with and , taking gives:and therefore there is no that satisfies the equation. So is not on the curve if we consider this elliptic curve over the rational numbers.
That would also not belong to Elliptic curve over the finite field , because doing everything we have:Therefore, there is no element such that or , i.e. and don't have a multiplicative inverse.
For the real numbers, it would work however, because the real numbers are a quadratically closed field, and .
For this reason, it is not necessarily trivial to determine the number of elements of an elliptic curve.
Number of elements of an elliptic curve over the rational numbers by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16The elliptic curve group of all elliptic curve over the rational numbers is always a finitely generated group.
The number of points may be either finite or infinite. But when infinite, it is still a finitely generated group.
For this reason, the rank of an elliptic curve over the rational numbers is always defined.
TODO example.
Largest known ranks of an elliptic curve over the rational numbers by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16web.math.pmf.unizg.hr/~duje/tors/rankhist.html gives a list with Elkies (2006) on top with:TODO why this non standard formulation?
The BSD conjecture states that if your name is long enough, it will always count as two letters on a famous conjecture.
Maybe also insert a joke about BSD Operating Systems if you're into that kind of stuff.
The conjecture states that Equation 1. "BSD conjecture" holds for every elliptic curve over the rational numbers (which is defined by its constants and )
Equation 1. . Where the following numbers are defined for the elliptic curve we are currently considering, defined by its constants and :
- : number of elements of the elliptic curve over the finite field, where the finite field comes from the reduction of an elliptic curve from to . can be calculated algorithmically with Schoof's algorithm in polynomial time
- : rank of the elliptic curve over the rational numbers. We don't really have a good general way to calculate this besides this conjecture (?).
- : a constant
The conjecture, if true, provides a (possibly inefficient) way to calculate the rank of an elliptic curve over the rational numbers, since we can calculate the number of elements of an elliptic curve over a finite field by Schoof's algorithm in polynomial time. So it is just a matter of calculating like that up to some point at which we are quite certain about .
The Wikipedia page of the this conecture is the perfect example of why it is not possible to teach natural sciences on Wikipedia. A million dollar problem, and the page is thoroughly incomprehensible unless you already know everything!
Birch and Swinnerton-Dyer conjecture by Kinertia (2020)
Source. The Equation "Definition of the elliptic curves" and definitions on elliptic curve point addition both hold directly.
Something that is very not continuous.
Notably studied in discrete mathematics.
Chuck Norris counted to infinity. Twice.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
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Intro to OurBigBook
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Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
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Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
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