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Questions tagged [elliptic-curves]

For questions about elliptic curves.

An elliptic curve is a smooth, nonsingular projective curve of genus 1 with a specified point $\mathcal{O}$, defined over any field $K$. They form abelian groups under point addition. They are much studied in number theory, for example in cryptography and integer factorization.

An elliptic curve can be defined by an equation of the form: $$E:y^2=x^3+ax+b$$ with the discriminant $\triangle_E=-16(4a^3+27b^2)\ne 0$ so the curve is nonsingular, i.e. its graph has no cusps or intersections.

The elliptic curves with $a=0$ are .

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A question on level structures on elliptic curves

I have a question on $\mathbb{H}/\Gamma(N)$, which parametrizes level $N$ structures on elliptic curves. Let $Y(N)$ be the set of isomorphism classes of such objects, then, according to Fact 2 on page 2 on this note, parametrization is given by…
Pooya
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Minimal Weierstrass equation and different reduction types (good and bad) of an elliptic curve

Why do we need the 'minimal' Weierstrass equation of an elliptic curve in order to study it's different reduction types (good and bad) ? What happens if we don't start with a minimal Weierstrass equation ?
Andrew
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Proving Fermat's Last Theorem (easily) using "assumed" conjectures

It can easily be proven assuming Szpiro's conjecture that Fermat's Last Theorem is true for sufficiently large $n$. The proof consists of extremely straightforward computations. My question is, is there a refinement of that proof that prove's FLT…
Anurag
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On the relationship between Fermats Last Theorem and Elliptic Curves

I have to give a presentation on elliptic curves in general. It does not have to be very in depth. I have a very basic understanding of elliptic curves (The most I understand is the concept of ranks). I was wondering if anyone could explain to me…
Raj
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Isogenous elliptic curves over finite fields have the same number of points

I'm stuck in this question, it is the first part of exercise 5.4 from Silverman - The arithmetic of elliptic curves. Let $C,D$ be two isogenous elliptic curves over a finite field $\mathbb{F}_q$. Then $$\#C(\mathbb{F}_q)=\#D(\mathbb{F}_q)$$ Any idea…
Toni Calvo
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Elliptic curve with prescribed lattice

It's well known that there is a connection between elliptic curves and lattices. To establish such a connection one needs to use Eisenstein series. How one can one write down the explicit equation of an elliptic curve knowing its lattice? For…
user44636
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Hasse's Theorem for Elliptic Curves over Finite Fields + proof clarification

I need a little help understanding Hasse's theorem for elliptic curves over finite fields, as well as the proof of this theorem. (Sorry about my editing) Hasse’s Theorem: Let $E$ be an elliptic curve defined over $F_q$. Then…
Math Major
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Elliptic curves with twists of zero rank

I am new to field of Elliptic curves. When I was seeing some papers related to this area I have come across elliptic curves having quadratic(some times cubic ) twists with zero rank. What is the consequence by this? Any help is great. Thank you in…
MKJ
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Why is reduction modulo $p$ a group homomorphism on Elliptic Curves?

I am reading A. Knapp's book on elliptic curves right now. In Proposition 5.6 the author wants to prove that the reduction map (modulo $p$, where $p$ does not divide the discriminant) of an elliptic curve over $\Bbb Q$ preserves the group…
Lucas Mann
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Does the conductor of an elliptic curve always divide the minimal discriminant?

Of course, the primes dividing the conductor are precisely those dividing the minimal discriminant. But I cannot find any source that addresses the possibility of a prime appearing to the first power in the minimal discriminant but appearing to the…
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There are no elliptic curves over $\mathbb{F}_8$ with $7$ or $11$ points

This is taken from The Arithmetic of Elliptic Curves by Silverman on page 154, Q5.10(f). One way of directly solving this problem is to work out on sage all 8^5 possibilities of elliptic curves and show that no such curve with the required number of…
Haikal Yeo
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Hyperelliptic curve order

How to compute order of a hyperelliptic curve ($y^2=f(x)$, $deg(f)=2 \cdot g+1$, $g=4$), over $F_p$ for small $p$ ($p$ prime)? Are there any efficient algorithms to do so? Is it possible with Pari/Magma/Sage? Are there any efficient algorithms for…
ted.k
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Birationally equivalent elliptic curves

I encountered a question about showing that the curve $$ y^2 = x^4 + a_3 x^3 + a_2x^2 + a_1x + a_0, \qquad\qquad(1) $$ where $a_i \in \mathbb{Q}$, can be birationally transformed over $\mathbb{Q}$ to a curve of the form $$ y^2 = x^3 + Ax + B…
Sputnik
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Doubling a point on an elliptic curve

I've a programming background and am just about to get into a project where Elliptic Curve Cryptography (ECC) is used. Although our libraries deal with the details I still like to do background reading so started with the ECC chapter of…
Peanut
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Example of an elliptic curve with trivial torsion subgroup and rank 0

What is an example of an elliptic curve over $\mathbb{Q}$ with trivial torsion subgroup and rank 0?
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