Properties

Label 2.625.adu_fkl
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:  $( 1 - 49 x + 625 x^{2} )^{2}$
Frobenius angles:  $\pm0.0637685608585$, $\pm0.0637685608585$
Angle rank:  $1$ (numerical)

This isogeny class is not 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
$A(\F_{q^r})$ 332929 151690775625 59592060966981904 23282898487386511055625 9094945010197654878745852129 3552713657817541093814153379840000 1387778780628962625902609352984947641729 542101086243205216571805161996168304364175625 211758236813626185185908677694533192411001024850704 82718061255304221841869813267320554378529632778810015625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 528 388324 244089078 152586803524 95367410590128 59604644423354974 37252902980525954928 23283064365406419073924 14551915228370363400683478 9094947017729442287143929124

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The isogeny class factors as 1.625.abx 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
All geometric endomorphisms are defined over $\F_{5^{4}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.625.a_absh$2$(not in LMFDB)
2.625.du_fkl$2$(not in LMFDB)
2.625.bx_cqi$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.625.a_absh$2$(not in LMFDB)
2.625.du_fkl$2$(not in LMFDB)
2.625.bx_cqi$3$(not in LMFDB)
2.625.a_bsh$4$(not in LMFDB)
2.625.abx_cqi$6$(not in LMFDB)