Properties

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

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Invariants

Base field:  $\F_{5^{4}}$
Dimension:  $2$
L-polynomial:  $1 - 93 x + 3402 x^{2} - 58125 x^{3} + 390625 x^{4}$
Frobenius angles:  $\pm0.0347955215357$, $\pm0.166702725194$
Angle rank:  $2$ (numerical)
Number field:  4.0.31273324.1
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
$A(\F_{q^r})$ 335810 151868729260 59597425048484360 23283016022977712992000 9094947040070632896633300050 3552713684839204778072524355034880 1387778780847102233599970753057210031890 542101086241742107010649500028607661036928000 211758236813513482337264998965236754469905326869640 82718061255301100963389398229275802781041924129419688300

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 533 388781 244111052 152587573809 95367431874893 59604644876703266 37252902986381595437 23283064365343578996769 14551915228362618519898028 9094947017729099142926119901

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.31273324.1.
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_faw$2$(not in LMFDB)