︠c4b0e29f-edc8-46e8-b20c-4758ef69968fi︠ %md # 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. ︡bb98471a-9863-41c6-866e-69c2cee4fbc4︡︡{"done":true,"md":"# Math 582: computational number theory\n\n## Homework 4 -- due by Friday Feb 5 at 9am\n\n\nWARNING: I haven't exactly tried the problems below. I don't know how computationally difficult they are for sure."} ︠6bb6a1a5-7551-40be-b113-3531a4cadc89︠ ︠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 $\rho_{E,2}(\text{Frob}_P)$. ︡6e8e8e60-15e5-4616-822d-097358d8fa30︡︡{"done":true,"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 $\\rho_{E,2}(\\text{Frob}_P)$."} ︠a0253cc5-3baf-4539-a3ab-ee179ffc8280︠ ︠fd00381e-10ad-454a-aefb-3bb40d7a131f︠ ︠d8b26975-ea3e-4b5d-8626-cb24fe336e60︠ ︠81f969b4-b3ab-4008-a2ad-004aa6f0aa23︠ ︠b7d0ad95-db24-4d1f-8eb5-af5f7ede042a︠ ︠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. ︡ecd806f7-8a59-486c-a06b-e1f7f84e43ce︡︡{"done":true,"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 $\\rho_{E,2}(\\text{Frob}_P)$.\n\nTry 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."} ︠538abfb8-27c3-4336-91dc-2476925320a6︠ ︠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 ︠c7349679-e04f-4c1d-adc4-f279e791dbca︠ ︠809ffae5-6a4e-4115-927e-9392867ca575︠ ︠537fa1a1-325f-4988-a744-34fef7f59ff3︠