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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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Question on avoiding complex numbers in quartic to Jacobian elliptic curve

I am working with a quartic (1) $ \quad y^2 = -x^4 + 1504271405904482x^2 - 16053296232241^2 $ A known solution is $ [x,y] = [1853^2, 131674446875520] .$ The ellfromeqn() function in GP-Pari takes the affine definition (very close to 1) and finds the…
Randall
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Write an elliptic curve with coefficients only depending on its j-invariant

Let $$E:y^2 = 4x^3-g_2 x - g_3$$ be an elliptic curve and $$j=\frac{g_2^3}{g_2^3-27 g_3^2}$$ denote to its $j$-invariant. I want to transform $E$ to find $f$ and $g$ s.t. $$E:y^2=4x^3-f(j)x-g(j).$$ I have no clue what ansatz I should take for $f$…
robertmiles
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Point Order on Elliptic Curve

i got stuck doing the exercise 4.5 d) in the Book Elliptic Curves: Number Theory and Cryptography by L. Washington and would be grateful for hints. Let $ p \equiv 1 \text{ (mod 4)}$ be prime and let $E$ be given by $y^2 = x^3 -kx$, where $k…
ASP
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Elliptic Curve over $\mathbb Z_p$ isomorphic to a subgroup of that Elliptic Curve over reals?

Is every Elliptic Curve group over finite field $\mathbb F_p$ isomorphic to some subgroup of the Elliptic Curve group over the reals having the same equation? Update: this is at last sometime the case. Take the equation $y^2=x^3+7$. On the field…
fgrieu
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$j$-invariants for elliptic curves over $\mathbb{F}_p$

I'm reading an article about elliptic curve volcanos. I know how to compute the $j$-invariant given a curve in Weierstrass form, but i don't have any idea on how to compute every possible $j$-invariant possible for curves defined over…
José
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Rewriting elliptic curve with point of order 3

I have a question regarding a change of coordinates with elliptic curves. I tried to show that an elliptic curve $E: y^2 = x^3 + ax^2 + bx +c$ with coefficients in $\mathbb{F}_q[t]$ and where $P$ is a rational point of order 3 can be rewritten…
Math4Life
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Do points on elliptic curves exist where the denominators of point multiples grows more slowly than normal?

Looking at prime multiples of $P=[1,1]$ on the curve $y^2=x^3+x-1$ the size of the denominator grows quite rapidly. So $5P=[\frac{685}{11^2},\frac{-18157}{11^3}],7P=[\frac{[154513}{443^2},\frac{-45623219}{443^3}]$ Looking purely at the x coordinate…
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Given a real number, how do I produce an elliptic curve with j-invariant equal to that number?

I have formula for computing the j-invariant but I was wondering if given number $j$, is there a formula for getting a curve $y^2=x^3+a_2x^2+a_4x+a_6$ with j-invariant j?
user54358
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Compute Kernel Polynomial Given Isogeneous Curve?

So I was trying to implement a CRS key exchange using modular polynomial. I start with a curve $E_1$, for Elkies prime $\ell$, I solve for the roots of classical modular polynomial $\Phi_\ell(X,j(E_1))$. Then I could obtain the j-invariant $j(E_2)$…
Taylor Huang
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In what contexts outside elliptic curves do any of the three rational elliptic curves of minimal conductor arise?

Particular (rational) elliptic curves arise in many contexts outside the study of elliptic curves themselves. For example, this solution to this question asking which squares of triangular numbers $T(k)$ are themselves triangular numbers proceeds by…
Travis Willse
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Can I find an endomorphism of an elliptic curve with a specific kernel size?

Say I want to find an endomorphism of an ordinary elliptic curve $E$ with kernel size of a prime $l$ that divides the cardinality of $E$. Is this possible in its endomorphism ring and what is the proof?
edlothia
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Do we have a criterion for $j$-invariants that gives $CM$ elliptic curves?

It is well-known that every CM elliptic curve has $j$-invariants which are algebraic integers. Is there any criterion to classify all the $j$-invariant that corresponds to $CM$ elliptic curves? For small degrees, such as over $\mathbb{Q}$, we can do…
Seewoo Lee
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Distribution of torsion subgroups of elliptic curve

By Mazur's theorem, we know the list of possible torsion subgroups of elliptic curves over $\mathbb{Q}$. Now, if we order them by height, can we compute the distribution of each possible groups? According to wikipedia, it is known that each group…
Seewoo Lee
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Is there an elliptic curve mod n with exactly one point?

I have tried many elliptic curves $y^2 = x^3 + ax +b$ with no success. I know that for prime modules there exists a minimum number of points the elliptic curve has to have, and I couldn't satisfy this for the smallest primes. So I decided to try…
SlowerPhoton
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Change of variables for elliptic curve

Say I have an elliptic curve $y^{2} + 5xy + y = x^{3}$ with the $(0, 0)$ being a rational 3-torsion point at $(x, y) = (0, 0)$. What change of variables would I need to get it into the form $y^{2} = x^{3} + a(x - b)^{2}$?
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