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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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Why is the Frob in elliptic curve not called an automorphism

Please apologize, if that's a stupid question. Why is the Frobenius Endomorphism of an elliptic curve over a finite field not regarded as an automorphism? Since it is an Isogeny, it is surjective Since it has trivial kernel, it is injective Why…
weissnix
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Isogenous elliptic curve.

I'm studying elliptic curves and I have a question Take two $k$-isogenous elliptic curves defined over a number field $k$ and fix a place $v$ of good reduction. Are the reduced curves $\mathrm{mod} \:v$ isogenous?
user262440
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Reduction of an elliptic curve defined over $\mathbb{Q}$

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $\ell$ be a prime number such that the reduced curve $\tilde E_{\ell}$ is non singular. Assume that $\tilde E_{\ell}$ admits a subspace $E'_{\ell}$ and $E''_{\ell}=\tilde E_{\ell}/E'_{\ell}$ such…
user261123
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Showing $P=(o,m) \not\in 2E_m(\mathbb{Q})$ for an elliptic curve $E_m$

I have been having trouble with this question. Let $m\in \mathbb{Z}$ with $m > 0$ and define $E_m : y^2 = x^3 −x+m^2$ Then $E_m$ is an elliptic curve Determine the group sturcture of $E_m(\mathbb{Z}_5)$ and hence show that when $m\nmid 5$, $P=(0,m)…
Shevak
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Elliptic Curve $E/\mathbb{Q}$ with $\Delta_E^{1/3}$ a root of defining cubic

Consider the elliptic curve $$E'\colon y^2 = g(x) = x^3 + \frac{1}{432} .$$ One can check that the discriminant of $E'$ is $-1/432$, that $E'$ has complex multiplication, and that $(-1/432)^{1/3} = (-1/6 \sqrt[3]{2})$ is a root of $g(x)$. Hence…
Jackson Morrow
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Faster scalar multiplication over an elliptic curve by hand?

For an elliptic curve $y^2=x^3+ax+b$, I have $a=1, b=1, G=(3,10)$ private key of User $B$ as $4$. To calculate his public key, I have the formula: $Pb=nb \times G = 4(3,10)$. This makes my calculation$=4G= (3,10)+(3,10)+(3,10)+(3,10)$ I got $(7,10)$…
user2816215
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Does the Hasse inequality fail for supersingular elliptic curves?

For supersingular elliptic curve $E: y^2+y=x^3 + 2x$ over $\mathbb{F}_{27}$, $\#E\left(\mathbb{F}_{27}\right) = 55$ but $|\#E(\mathbb{F}_{q}) - (q+1)| \leq 2\sqrt{q} \iff 18 \leq \#E(\mathbb{F}_{27}) \leq 38 $
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How long does it take to double and add points on an elliptic curve?

I have some problems understanding how many multiplications it takes to add or double points on an elliptic curve in Weierstrass form. This link tells that it's 11 and 14, but I don't quite understand why. Can someone walk me through it? The…
MBrown
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Multiple points on an elliptic curve

I have given the following elliptic curve $E:F(x,y) = 0$: (Where $F(X,Y) := Y^2 + a_1XY + a_3Y - X^3 - a_2X^2 - a_4X - a_6$ with $a_1 = -1.5, a_2 = 3, a_3 = 1, a_4 = 0.5, a_6 = -1.5$.) The curve $F$ is the black graph, $F_X$ is the red one and…
ProfMeowsworth
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What is the argument used to dsitinguish the cases (a) and (b)

We know from [B. Mazur, Modular curves and the Eisenstein ideal, Publ. math. IHES 47 (1977), 33-186] that if $C$ is an elliptic curve of the form ($C:y²=x³+ax+b$ with $a,b∈ℤ$), then $C(ℚ)^{tors}$ (the subgroup of elements of finite order in $C(ℚ)$…
DER
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An isogeny of elliptic curves induces a $\mathbb{Z}_l$-linear map

On page 89 of The Arithmetic of Elliptic Curves (second edition), Silverman says: Let $\phi:E_1\rightarrow E_2$ be an isogeny of elliptic curves. Then $\phi$ induces maps $\phi:E_1[l^n]\rightarrow E_2[l^n]$, and hence induces a $\mathbb{Z}_l$-linear…
Haikal Yeo
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Why does an isogeny not ramify?

The following argument is, I believe, based on the premise that an isogeny (or a morphism of curves that is a group homomorphism) doesn't ramify: Considering the multiplication by $n$ map $[n]$ on a elliptic curve $E/K$, where char$(K)\not|\,\,n$ so…
Rodrigo
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Odd torsion of elliptic curves are isomorphic

$C: Y^2=X(X^2+aX+b)$ $D: Y^2=X(X^2+a_1X+b_1)$ where $a,b,\in\mathbb Z a_1=-2a,b_1=a^2-4b,b(a^2-4b)\neq0$ Let $C_{oddtors}(\mathbb Q)$ denote the set of torsion elements of $C(\mathbb Q)$ which have odd order and $D_{oddtors}(\mathbb Q)$ denote the…
Haikal Yeo
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How can I check I have a birational transformation

Updated this question, just focusing on the relevant part $$f_2(u,v) = \tfrac{27}{64} u^3 - \tfrac{81}{64} u^2 v + \tfrac{189}{20} u^2 + \tfrac{81}{64} u v^2 - \tfrac{189}{10} u v + \tfrac{1764}{25} u + \tfrac{165}{64} v^3 - \tfrac{99}{4} v^2 +…
user16697
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Elliptic curves, inflection points and divisors

I'm studying basics of elliptic curves. I'm reading An Elementary Introduction to Elliptic Curves by Leonard Charlap and David Robbins. It is stated there that the divisor of a line (i.e. a polynomial of the form $ax + by + c$) can have only few…
Jasiu
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