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2

Do you know the group law (of an elliptic curve)? Assuming we're on a field of characteristic $\;\neq 2,3\;$ ,we can define: $$t:=\frac{3\cdot 2^2}{2\cdot 1}=6$$ $$x_1:=t^2-2\cdot2=32\\y_1:=1+6(32-2)=181$$ and we get a new solution $\;(32\,,\,\,181)\;$


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One approach (see Example $15.5$ at Elliptic Curve Cryptography), is the following. Take $x = 0 \ldots 16$ and for each $x$ solve: $$y^2 = x^3 + 2x + 2 \pmod {17}$$ This yields the following sets of points: $x = 0, 7, 10, y = 6, 11$ $x = 3, 5, 9, y = 1, 16$ $x = 6, y = 3, 14$ $x = 13, y = 7, 10$ $x = 16, y = 4, 13$ You can also look into Point Counting ...


1

You can find quite comprehensive list of books that deal with elliptic curves on the official site of Andrej Dujella: http://web.math.pmf.unizg.hr/~duje/literatura.html#EK


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In addition to Dietrich's complete answer, using Sage you can compute $E(\mathbb F_{27})$: E = EllipticCurve(GF(27,'a'),[0,0,1,2,0]); E.abelian_group() Additive abelian group isomorphic to Z/14 + Z/2 embedded in Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 + 2*x over Finite Field in a of size 3^3 to find that $E(\mathbb F_{27}) ...


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The Hasse bound holds for all elliptic curves over finite fields. Since the discriminant is $\Delta=-539\equiv 1(3^k)$, we really have an elliptic curve over $\mathbb{F}_{3^k}$ for all $k\ge 1$. I count $\# E(\mathbb{F}_3)=7$, and this satisfies the Hasse bound, because $3\le 2\sqrt{3}$. The computation for $\mathbb{F}_9$ and $\mathbb{F}_{27}$ is probably ...


3

Yes, it can be proven that any elliptic curve over $\mathbb C$ is isomorphic to $\mathbb C / \Lambda $ for some lattice $\Lambda \subset \mathbb C$ (a lattice is a additive subgroup of $\mathbb C$ generated by two elements $\omega_1, \omega_2)$. But since any lattice is, after choosing a basis, isomorphic to $\mathbb Z^2$, and $\mathbb C = \mathbb R^2$, ...


2

Neil Koblitz, one of the major figures in the development of ECC, has a graduate level work on cryptography and some relevant sectors of number theory. Somewhat needless to say, it has an extended discussion of elliptic curves. The only thing to note is that it is from 1994, so it may be a bit dated in some places. ...


3

Lemma. If $a$ and $b$ are integers such that $3a^4+3a^2+1=b^2\!$, then $a=0$. Proof. Assume, contrary to the claim, that $a \ge 1$ and $b>1$ are integers satisfying the equation. Evidently $b$ is odd, say $b=2c+1$ for an integer $c \ge 1$. Hence \begin{align*} 3a^4 + 3a^2+1 &= (2c+1)^2 \\ 3a^2(a^2+1) &= 4c(c+1). \end{align*} Since $4 \nmid ...


0

What you want to do is more complicated for an ellipse than for a circle because it requires the use of special functions : First, the "complete elliptic integral of the second kind" to compute the perimeter. See formulas (63, 64, 65) in http://mathworld.wolfram.com/Ellipse.html If an high accuracy is not required, there are some approximate formulas (69, ...


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The map $x: E \to \mathbb P^1$ is of degree $2$ because for a generic choice of $x$, there are two values of $y$ for which $[x,y,1] \in E$, i.e. there are two values of $y$ with $F(x,y)=0$. Indeed, $F$ is a quadratic polynomial in $y$. The map will be ramified above $x_0 \in \mathbb A^1$ if there is only one point in the fibre above $x_0$, which is to say ...


2

In addition to guy-in-seoul's excellent answer: Modularity of an elliptic curve $E$ is absolutely essential to even talk about the $L$-function of $E$ outside of the initial region of convergence of the Euler product. Other than for CM curves, the only known way to analytically continue $L$-functions of elliptic curves uses modularity. Same goes for the ...


2

What's multiplication? It's simply repeated addition. Instead of thinking of the problem as $101 \cdot (2, \, 2)$, where $101$ is meaningless within the context of the elliptic curve, think of the operation as $(2, \, 2) + (2, \, 2) + \underbrace{\ldots}_{98} + (2, \, 2)$. Unfortunately, that seems like quite a lot of computations, and it is, especially in ...


2

The projective curve $E$ given by the equation $Y^2Z = X^3 + cZ^3$ is a smooth curve of genus $1$ over $R = \mathbb Q[c, c^{-1}]$. You are essentially asking for a description of $$\text{End}_{R}({E}).$$ The curve $E$ has the rational point $$\mathcal O := [0:1:0] \in E(R)$$ which we can use as a base-point, so that $E$ is an elliptic curve over $R$. I ...


1

Punctured curves can be used to construct extensions of Galois representations. This an example of Deligne's philosophy of mixed motives. A pure motive is a motive attached to a smooth projective variety; according to Deligne not necessarily projective smooth varieties should live in a suitable category of extensions of pure motives. I will explain how a ...


1

Given: $x^2+1=y^3$ Taking $\mod{3}$ on both sides, either $y=3b+1 \: \text{or} \: y=3k+2$ CASE $1$: $y=3k+2$ $x^2+1=y^3$ $\implies x^2=(y-1)(y^2+y+1)$ Also, by Remainder Factor Theorem, $y^2+y+1=(y-1)(y+2)+3$ $\implies \gcd[(y-1),(y^2+y+1)] = \gcd[(y-1),((y-1)(y+2)+3)] = \gcd[(y-1),3] = 1$ (since $y=3k+2$) $\implies y-1=a^2 \: ...


0

Me neither. For the first computation, I count at most 4 multiplications (and 1 addition) to get $\lambda$, then 2 multiplications (and 1 addition) to get $x_2$ and 2 multiplications (and 2 additions) to get $y_2$. So that's only 8 multiplications and some of them might be replaced by addition. On the other hand, I counted division as multiplication, which ...



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