Properties

Label 2.625.adq_fcz
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:  $1 - 94 x + 3457 x^{2} - 58750 x^{3} + 390625 x^{4}$
Frobenius angles:  $\pm0.0803816721029$, $\pm0.134758112033$
Angle rank:  $2$ (numerical)
Number field:  4.0.4214336.1
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
$A(\F_{q^r})$ 335239 151838794553 59596842739424284 23283015554922457794041 9094947419959294321035686119 3552713701178447176015384894145936 1387778781317956686383862021583405766551 542101086252731904893524147516277936057745833 211758236813733889960472428744809317178403170798748 82718061255304982303597754676083188528868656040454968393

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 532 388704 244108666 152587570740 95367435858312 59604645150830262 37252902999020998408 23283064365815587221732 14551915228377764816433466 9094947017729525900795164144

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.4214336.1.
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.dq_fcz$2$(not in LMFDB)