An Explicit Sextic Twist of a Supersingular Elliptic Curve in Characteristic Two
- 1Facultad de Ciencias Económicas y Empresariales, Universidad Panamericana, Álvaro del Portillo 49, Zapopan 45010, Jalisco, Mexico
- 2Laboratory for Reputation Management in Education, HSE University Saint-Petersburg, Griboedova Embankment 123, Office 502, Saint Petersburg 190068, Russia
Abstract
Let E:y2+y=x3+x over F2, a supersingular curve whose automorphism group is isomorphic to SL(2,3), of order twenty-four. Using an automorphism of order six, we construct the sextic twist of E along the fiber product of an Artin-Schreier and a Kummer covering, obtaining the family Z2+Z=B(t)X3+A(t)+c, in which the constant c is not removable by a constant translation of Z over F16 when A is fixed. We analyze two members. For A(t)=t3+t and B(t)=t2(t+1), the associated elliptic surface is K3, the L-polynomial has degree eight, and the Mordell-Weil rank of the generic fiber is zero over F16(t) and four over F256(t). For A(t)=t7+t and B(t)=t17, the surface is again K3, the L-polynomial has degree fourteen, and the ranks are zero and two. In both cases, the rank follows from an exact factorization of the zeta numerator of the auxiliary curve, isolating in the non-invariant part of its cohomology the factors geometrically isogenous to E, the isogeny being defined over F256 but not over F16.
Related
- Iwasawa theory of CM elliptic curves at potentially supersingular primes
- An Explicit Belyi Map for the Wiman Sextic and Cusp Forms for a Noncongruence Subgroup
- Supersingular Elliptic Curves Without Inseparable Small Degree Endomorphisms
- On Sun's Conjectures for Truncated Jacobi-Symbol Determinants via Supersingular Elliptic Curves
- Classification of torsion of elliptic curves over sextic fields
BibTeX
@article{sosagomez2026explicit,
title = {{An Explicit Sextic Twist of a Supersingular Elliptic Curve in Characteristic Two}},
author = {Guillermo Sosa-Gómez},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203725},
url = {https://www.mdpi.com/2227-7390/14/20/3725},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - An Explicit Sextic Twist of a Supersingular Elliptic Curve in Characteristic Two
AU - Guillermo Sosa-Gómez
PY - 2026
DA - 2026/10/14
JO - Mathematics
DO - 10.3390/math14203725
UR - https://www.mdpi.com/2227-7390/14/20/3725
AB - Let E:y2+y=x3+x over F2, a supersingular curve whose automorphism group is isomorphic to SL(2,3), of order twenty-four. Using an automorphism of order six, we construct the sextic twist of E along the fiber product of an Artin-Schreier and a Kummer covering, obtaining the family Z2+Z=B(t)X3+A(t)+c, in which the constant c is not removable by a constant translation of Z over F16 when A is fixed. We analyze two members. For A(t)=t3+t and B(t)=t2(t+1), the associated elliptic surface is K3, the L-polynomial has degree eight, and the Mordell-Weil rank of the generic fiber is zero over F16(t) and four over F256(t). For A(t)=t7+t and B(t)=t17, the surface is again K3, the L-polynomial has degree fourteen, and the ranks are zero and two. In both cases, the rank follows from an exact factorization of the zeta numerator of the auxiliary curve, isolating in the non-invariant part of its cohomology the factors geometrically isogenous to E, the isogeny being defined over F256 but not over F16.
SN - 2227-7390
ER -