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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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Duplication formula for the general case: $x$-coordinate of $n(x, y)$

The duplication formula for an elliptic curve over rationals: $$y^2 = x^3 + ax^2 + bx + c$$ for the $x$-coordinate is given by: $x$-coordinate of $2(x, y)=(x^4-2bx^2-8cx+b^2-4ac)/(4x^3+4ax^2+4bx+4c)$ Here $2(x, y)=(x, y)+(x,y)$ via the group law of…
Safwane
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The definition height of a rational point on an elliptic curve which is not in Weierstrass normal form

Assume that $$E:y^2 = x^3 + Ax + B$$ is an elliptic curve that is defined over ${\bf{Q}}$ and is expressed in Weierstrass form. Then, the height $H(P)$ of a rational point $P=({\frac{a}{b}},{\frac{c}{d}})\in{E({\bf{Q}})}$ is defined to be $$H(P) =…
student
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Is it possible to say that there is a curve $C$ such that its rank exactly $r$?

The motivation of this question can be found in Is it possible to say that every point $P$ in $C(ℚ)$ other than the 'basis' is of finite order? Given the elliptic curve: $$C:y²=x³+ax+b$$ for $a,b∈ℤ$. We know that $C(ℚ)≠∅$, so the rank is $r≥0$. From…
Safwane
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How often a number can be conductor of an elliptic curve

There are several upper bounds for number of elliptic curves (over $\mathbb{Q}$, say) upto-isomorphism with a given conductor $N$. Probably the best one is given by Helfgott-Venkatesh of order $N^{0.22}$ (or may be some improvement is possible…
dragoboy
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Is it possible to say that every point $P$ in $C(ℚ)$ other than the 'basis' is of finite order?

Let $C$ an elliptic curve over $\mathbb Q$. Assume that the rank of $C(ℚ)$ is equal to $r$. Then the cardinality of a maximal independent set in $C(ℚ)$ is $r$, thus there exists $r$ independent points ${P_1,P_2,\dots,P_r}$ of infinite order in…
Safwane
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Calculating 2P Elliptic Curve y^2 = x^3 + 3x + 1 mod 7 with P = (0,1)

Elliptic Curve: $y^2 = x^3 + 3x + 1 \bmod 7$ Point : $P(0,1)$ First I calculate lambda for when $P1=P2$ $\lambda = (3x_1^2 +a)/2y_1 = (0 + 3)/2(1) = 3/2$ now calculating $X_3 = \lambda^2 - x_1 - x_2 = (3/2)^2 = 9/4$ Finally Calculating $Y_3 =…
user3371137
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Image points over extensions fields

While trying to understand isogeny-based cryptography I stumbled over a passage in a paper by Galbraith. On p.4 he explains how to calculate the image points of an isogeny $\phi:E\to E^{\prime}$: What about the other 4 points in…
jvdh
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Elliptic curve scalar extraction

For any point $P$ on an elliptic curve such that $P = X*Q$ where $X$ is an unknown integer and $Q$ is another point on the curve, what information can we extract about $X$ from $P$? I understand we cannot get the complete value of $X$, but is there…
user306666
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Can the base point of an elliptic curve be any point?

An elliptic curve is defined as: "Formally, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O." From this it sort of sounds like you can just specify whichever point you like and it counts…
vacant
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Elliptic curve division

If a generator point G is used to create a prime point and this generated point is then divided by any integer, the result gives a non-integer multiple of G. Example: 2G / 4 = 0.5G Does this point lie on the curve? If it does not lie on the curve,…
user306666
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Isomorphism between elliptic curves over $\mathbb Q$ and $\mathbb F_5$

Given are the Elliptic curves $E_1 : y^2 = x^3+x$ and $E_2 = y^2 = x^3+3x$. Are these isomorphic over a) $\mathbb Q$? b) $\mathbb F_5$? I see they are isomorphic over $\mathbb C$, as they have the same $j$-invariant. I suppose they aren't…
Banjobill27
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Silverman - MOV attack: why is $E[N] \subset E(\mathbb{F}_{q^d})$ important?

In Silverman's "Arithmetic of Elliptic Curves", the author describes the embedding degree $d$ of the integer $N$ in $\mathbb{F}_q$ as the smallest integer $d$ such that $\mu_N \subset \mathbb{F}_{q^d}^*$. Therefore we can use the Weil pairing to…
NightRain23
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Why does $\frac{X - aZ}{Z}$ have a double pole at the point $(0 : 1 : 0)$ and not just one (Divisors)?

If I have an Elliptic Curve E and the function $\frac{X - aZ}{Z}$, I would have expected the divisor to be, defining a point $P = (a,b)$ and $-P = (a,-b)$, $div(f) = [P] + [-P] - [\infty]$. Instead the correct solution would be $div(f) = [P] + [-P]…
NightRain23
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What does it mean to solve the equation of an ellipse as a quadratic?

An illustration required me to find out the area of the curve $$5x^2 + 6xy + 2y^2 + 7x + +6y + 6 = 0 $$ They proceeded to solve the equation as a quadratic obtaining $y_1$ and $y_2$ as the two branches. Why are $y_1$ and $y_2$ as depicted in…
user619072
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Rational $3$-torsion points of an elliptic curve.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with equation $y^2=x^3+(ax+b)^2$, where $a,b \in \mathbb{Q}$. I have to prove that $0$ and $(0, \pm b)$ are the rational $3$-torsion points of $E$. I know that if $Q$ is a $3$-torsion point…
user289143
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