Properties

Label 2.625.ado_ezf
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:  $1 - 92 x + 3359 x^{2} - 57500 x^{3} + 390625 x^{4}$
Frobenius angles:  $\pm0.0742530806974$, $\pm0.165990880443$
Angle rank:  $2$ (numerical)
Number field:  4.0.7219856.1
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
$A(\F_{q^r})$ 336393 151907341761 59598760502780676 23283052119724556073033 9094947887906422168688508393 3552713702944881815836324334845584 1387778781206832002074154197226089512297 542101086248477388847004605068707180036137353 211758236813633203058166124371156635777878191648324 82718061255303129729795447926066427326840902426866497601

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 534 388880 244116522 152587810372 95367440765094 59604645180466118 37252902996038017878 23283064365632857148932 14551915228370845665192810 9094947017729322208151469200

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.7219856.1.
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.do_ezf$2$(not in LMFDB)