Timeline for Rational points on the "quintic circle" $x^5 + y^5 = 7$

9 events

when toggle format	what		by	license	comment
Jun 15, 2020 at 7:27	history	edited	CommunityBot		Commonmark migration
Jun 30, 2019 at 3:52	comment	added	pre-kidney		It is worth pointing out that passage from $x^5+y^5=7$ to the hyperelliptic curve is achieved by rewriting as $x^{10}+(xy)^5=7x^5$ changing variables to $u=x^5$ and $v=xy$ and completing the square.
Nov 22, 2015 at 18:54	history	edited	Michael Stoll	CC BY-SA 3.0	found better way of getting "P<x>" correctly displayed
Nov 22, 2015 at 18:16	vote	accept	pre-kidney
Nov 22, 2015 at 12:42	comment	added	Michael Stoll		@Arul : This is explained in detail in my paper "Implementing 2-descent for Jacobians of hyperelliptic curves" (Acta Arith. 98, 245-277, 2001). If you are familiar with 2-descent on elliptic curves, then it can be seen as a generalization of that: one computes the 2-Selmer group of the Jacobian as a subgroup of ${\mathbb Q}(\theta)^\times$ mod squares, where $\theta = \sqrt[5]{49/4}$. There is an injection of $J({\mathbb Q})/2 J({\mathbb Q})$ into the 2-Selmer group, so we obtain an upper bound for the rank.
Nov 22, 2015 at 12:31	comment	added	user76479		Could you explain 'By a 2-descent, one can show that the Jacobian variety of C′'?
Nov 22, 2015 at 12:12	history	edited	Michael Stoll	CC BY-SA 3.0	The "<x>" after the "P" in the code was not shown. Added reference to online Magma calculator.
Nov 22, 2015 at 12:05	history	edited	Michael Stoll	CC BY-SA 3.0	added a comma, corrected a typo
Nov 22, 2015 at 11:59	history	answered	Michael Stoll	CC BY-SA 3.0