1,1

These are the set of positive integers k which satisfy the following three conditions: (i) Let K = Q(zeta_5) be the cyclotomic field of order 5, and let J be the Jacobian of the genus 2 curve y^2 = x^5 - k over K. Then, for all places v of K of bad reduction for J, either v lies above 5, or -k is a nonsquare in the local field K_v (the completion of K at v), (ii) k is congruent to 2, 12, 13, or 23 mod 25, and (iii) 5 does not divide the class numbers of the imaginary quadratic fields Q(sqrt(-k)) and Q(sqrt(-5k)). Stoll proved that the Mordell-Weil group of y^2 = x^5 - k over Q is trivial for such values k. - Robin Visser, Jun 30 2024

Robin Visser, Table of n, a(n) for n = 1..5000

Michael Stoll, On the arithmetic of curves y^2=x^l+A and their Jacobians, J. Reine Angew. Math. 501 (1998), 171-189, see p. 179.

(Magma) // Uses the Genus2Conductor package by Dokchitser--Doris.

is_A034011 := function(k)

K := CyclotomicField(5); R<x> := PolynomialRing(K);

C := HyperellipticCurve(R!(x^5 - k));

if (not ((k mod 25) in [2, 12, 13, 23])) then return false; end if;

if 0 in [(ClassNumber(QuadraticField(-i)) mod 5) : i in [k, 5*k]] then

return false;

end if;

for v in Factorisation(Conductor_Genus2(C)) do

if (not (Norm(v[1]) eq 5)) and IsSquare(Completion(K, v[1])!(-k)) then

return false;

end if;

end for;

return true;

end function;

[k : k in [1..1000] | is_A034011(k)]; // Robin Visser, Jun 30 2024

Cf. A034012, A034013, A034014.

Sequence in context: A009350 A063092 A338222 * A085497 A320515 A265775

Adjacent sequences: A034008 A034009 A034010 * A034012 A034013 A034014

More terms from Sean A. Irvine, Jul 29 2020

Name edited by Robin Visser, Jul 02 2024

approved