Properties

Label 2.625.adt_fio
Base Field $\F_{5^{4}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{5^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 49 x + 625 x^{2} )( 1 - 48 x + 625 x^{2} )$
Frobenius angles:  $\pm0.0637685608585$, $\pm0.0903344706017$
Angle rank:  $2$ (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 is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 333506 151728554700 59593325970144968 23282931123531240192000 9094945732251198388598540066 3552713672138666320458752106393600 1387778780888870936100525479612747929058 542101086247553748357344467651129130339328000 211758236813693045399811901242684636196276418904776 82718061255305150026138072495087243750276259928699463500

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 529 388421 244094260 152587017409 95367418161409 59604644663623586 37252902987502815505 23283064365593187091969 14551915228374957999597364 9094947017729544342083496101

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The isogeny class factors as 1.625.abx $\times$ 1.625.abw and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ab_abqk$2$(not in LMFDB)
2.625.b_abqk$2$(not in LMFDB)
2.625.dt_fio$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.625.ab_abqk$2$(not in LMFDB)
2.625.b_abqk$2$(not in LMFDB)
2.625.dt_fio$2$(not in LMFDB)
2.625.acl_cwm$4$(not in LMFDB)
2.625.abj_vs$4$(not in LMFDB)
2.625.bj_vs$4$(not in LMFDB)
2.625.cl_cwm$4$(not in LMFDB)