Properties

Label 2.625.adp_fax
Base Field $\F_{5^{4}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{5^{4}}$
Dimension:  $2$
L-polynomial:  $1 - 93 x + 3403 x^{2} - 58125 x^{3} + 390625 x^{4}$
Frobenius angles:  $\pm0.0431458011821$, $\pm0.164652188420$
Angle rank:  $2$ (numerical)
Number field:  4.0.38207421.1
Galois group:  $D_{4}$

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

This isogeny class contains a Jacobian, and hence is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 335811 151869517317 59597493181061283 23283019225214002927701 9094947149560580455789122096 3552713687885537937492289054833309 1387778780919998540205212600740905673779 542101086243290233999187692782045805794918213 211758236813543244151056888538462529505175047739179 82718061255301626215128062170876309882223692158380133632

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 533 388783 244111331 152587594795 95367433022978 59604644927812255 37252902988338390767 23283064365410070544627 14551915228364663736168521 9094947017729156894965536958

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.38207421.1.
All geometric endomorphisms are defined over $\F_{5^{4}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.625.dp_fax$2$(not in LMFDB)