# Tagged Questions

For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.

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### Properties of the elliptic curve $y^2 \equiv x^3 – 2 \pmod 7$

Can someone help me: 1) to list the points on the elliptic curve $E: y^2\equiv x^3 – 2\pmod 7$. 2) to find the sum $(3, 2) + (5, 5)$ on $E$.
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### Cube root of discriminant of elliptic curve

Let $E/K$ be an elliptic curve over a field $K$, with discriminant $\Delta$. Then the polynomial $x^3-\Delta$ has a root (and hence all roots since Galois) in $K(E[3])$; this can be shown laboriously ...
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### Birational Equivalence of Diophantine Equations and Elliptic Curves

A while ago I saw this question Quartic diophantine equation: $16r^4+112r^3+200r^2-112r+16=s^2$ which was very relevant to a undergraduate research paper I am currently working on. The answer given ...
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### Understanding the group structure of quotient group derived from elliptic curve group

I am working through some content in L.C. Washington's Elliptic Curves, Number Theory, and Cryptography and I am unsure about what the group structure of a certain group looks like. Some background: ...
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### Is the coordinate ring of an elliptic curve principal?

Let $K$ a field, $E$ an elliptic curve. I would like to know if the coordinate ring of $K[E]=K[X,Y]/(E)$ is principal. I think the answer is no. I tried to prove that the ideal $J=\langle y,x^2\rangle$...
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### find the structure of an elliptic curve over a finite field

For the elliptic curves E1,E2,E3, and E4 defined below, determine the structure of the groups Ek(F13) by using the information given below together with a minimal amount of extra (hand) ...
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### Is there something similar to $\mathbb{R}^2$ for elliptic curve point representation?

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_p$ and denote with $E(\mathbb{F}_p)$ its set of points over $\mathbb{F}_p$. Consider a coordinate system in $\mathbb{R}^2$. Every point is ...
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### Direct sum of two points on an elliptic curve

Given $E:y^{2} = x^{3}+9x$ over $\mathbb{Z}_{71}$, and $A = (0,0), \: B = (1,9)$, I'm asked to find $C=A\oplus B$. I just don't know how the direct sum of two points on an elliptic curve is defined, ...
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### The term “elliptic”

There are many things which are called “elliptic” in various branches of mathematics: Elliptic curves Elliptic functions Elliptic geometry Elliptic hyperboloid Elliptic integral Elliptic modulus ...
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### Proving some facts about the EC $y^2 = x^3 + ax + b$ [closed]

A solution to this question would be much appreciated! If $E/F$ is the EC defined by $y^2 = x^3 + ax + b$ then prove the following: If $P = (x, y)$ is element of $E(F)$ with order 3 then $x$ is a ...
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### Weil pairing of curve of genus 2

We know there is Weil pairing for elliptic curve satisfying several nice properties. So do we have Weil pairing for other curves also satisfying the nice property? Especially genus 2 curve?
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### What is the intuitive explanation of (non-singular part of) a singular elliptic curve being isomorphic to either $K^{*}$ or $K$?

Is there a rough explanation without using explicit computation?
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### Quotient of invariant differentials is constant

In the proof of Proposition 2.1.1 in Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, he makes a comment about quotients of invariant differentials being constant, because their ...
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### Showing $N_E/F_p^r = p^r + 1$ for a given elliptic curve [closed]

Have been studying elliptic curves, and am syuck on this problem. A detalied explanation would be much appreciated! a. Let $E/F_p^r$ be the elliptic curve $y^2 = x^3 − x$. Prove that if $p ≡ 3 (mod 4)$...
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### Proving an eliptic curve is cyclic, and determining it's order

I need a solution with an explanation for the following. Thanks! Let $E/F_q$ be an elliptic curve and let $P ∈ E(F_q)$ be a point a. if $n=ord(P)>1/2(q^{0.5}+1)^2$ prove that $E(F_q)$ is cyclic ...
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### Finding zeta function of an elliptic curve

Let p=3 (mod 4) be a prime, and $E/F_{p^r}$ be the elliptic curve given by $y^2 = x^3 − x$ Find the zeta-function of $E/F_p$ and use it to determine $|E(F_{p^r} )|$ for all r>0.
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### 2-torsion points in a curve with genus 2

Let X be a genus 2 curve with affine equation y^2 = f(x), where f is a polynomial of degree 6. Write $P_1, ..., P_6$ for the points on X(C) with y=0. Then why every $P_i-P_j$ is a 2-torsion points in ...
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### Finding order of a point on eliptic curve

Just started studying eliptic curves and am having trouble with this question. An explanation/solution would be much appreciated. Find the order of the point X on the elliptic curve $E/Q$ for the ...
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### Hensel lemma and elliptic curves [closed]

What is the cardinality of this homomorphism $E(\mathbb{Z}/p^r\mathbb{Z} ) \rightarrow E( \mathbb{Z}/p\mathbb{Z} )$ where $E(\mathbb{Z}/p^r\mathbb{Z} )$, $E( \mathbb{Z}/p\mathbb{Z} )$...
### Is there another methods for counting points on the curve $x^3 + y^3 =1$ over finite fields?
For the circle $(C): x^2 + y^2=1$ over finite field, we can use simple method to count the number of points. The case $p\equiv 1\mod 4$ is not difficult to find, because $-1$ is a square on $F_p$. ...