Properties

Label 2.625.adq_fcw
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:  $1 - 94 x + 3454 x^{2} - 58750 x^{3} + 390625 x^{4}$
Frobenius angles:  $\pm0.0557136983427$, $\pm0.146976429221$
Angle rank:  $2$ (numerical)
Number field:  4.0.940400.1
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
$A(\F_{q^r})$ 335236 151836430064 59596636142208484 23283005702845137370880 9094947077065589467415204836 3552713691445935794461443973380464 1387778781080282739536857440222428501764 542101086247595083078705287136302963691089920 211758236813634246642815022051958454081011265338756 82718061255303241733195532825154572337319874560784353904

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 532 388698 244107820 152587506174 95367432262812 59604644987545818 37252902992640986980 23283064365594962404734 14551915228370917379783812 9094947017729334523055543578

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.940400.1.
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_fcw$2$(not in LMFDB)