CoCalc Public Fileshomework / 2016-01-29-homework-4 / 2016-01-29-homework-4.sagews
Author: William A. Stein

# Math 582: computational number theory

## Homework 4 -- due by Friday Feb 5 at 9am

WARNING: I haven't exactly tried the problems below. I don't know how computationally difficult they are for sure.

︠fcaf67de-0ff6-4544-b9f2-d6d494060f6bi︠
%md
**Problem 1:** Let $\rho_{E,2}$ denote the mod 2 representation attached to $E$ for each of the following elliptic curves: 17a1, 32a1, 32a2, 37a1.   For each, compute explicitly the matrix of $\rho_{E,2}(\text{Frob}_P)$, where $P$ is a choice of prime ideal over each of 3,5,7,11,13.     Be sure to check that $X^2 - a_p(E)X + p$ is the charpoly of $\rho_{E,2}(\text{Frob}_P)$.



Problem 1: Let $\rho_{E,2}$ denote the mod 2 representation attached to $E$ for each of the following elliptic curves: 17a1, 32a1, 32a2, 37a1. For each, compute explicitly the matrix of $\rho_{E,2}(\text{Frob}_P)$, where $P$ is a choice of prime ideal over each of 3,5,7,11,13. Be sure to check that $X^2 - a_p(E)X + p$ is the charpoly of $\rho_{E,2}(\text{Frob}_P)$.



︠d8b26975-ea3e-4b5d-8626-cb24fe336e60︠

︠679c8d56-85fd-4034-bd07-3af1eefd70e3︠

︠6a1f1acc-7cb0-44a3-959b-7865ed4fea2a︠

︠56f2221a-e6a2-4dca-bd56-bae196a60aa2i︠
%md
**Problem 2:** Let $\rho_{E,4}$ denote the mod 4 representation attached to $E$ for the curve 32a1.   This is the homomorphism $G_\QQ\to\text{GL}_2(\ZZ/4\ZZ)$ defined by the action of $G_\QQ$ on $E[4]$.   Be sure to check that $X^2 - a_p(E)X + p$ is the charpoly of $\rho_{E,2}(\text{Frob}_P)$.

Try to compute explicitly the matrix of $\rho_{E,4}(\text{Frob}_P)$, where $P$ is a choice of prime ideal over each of 3,5,7,11,13.



Problem 2: Let $\rho_{E,4}$ denote the mod 4 representation attached to $E$ for the curve 32a1. This is the homomorphism $G_\QQ\to\text{GL}_2(\ZZ/4\ZZ)$ defined by the action of $G_\QQ$ on $E[4]$. Be sure to check that $X^2 - a_p(E)X + p$ is the charpoly of $\rho_{E,2}(\text{Frob}_P)$.

Try to compute explicitly the matrix of $\rho_{E,4}(\text{Frob}_P)$, where $P$ is a choice of prime ideal over each of 3,5,7,11,13.


︠ee1e4474-8413-42d7-9b50-1d5482cb6db2︠

︠706970bc-3997-404d-a300-82d455cf4b81︠

︠6293159f-22be-4675-b607-5ff1160e2668︠

︠6a517ed1-19c5-43f6-8b06-a6359f979c5f︠

︠c49d596e-916c-4cbe-a7e8-6ff12f6cf434︠

︠809ffae5-6a4e-4115-927e-9392867ca575︠

︠537fa1a1-325f-4988-a744-34fef7f59ff3︠