Properties

Label 2.625.adv_fmi
Base Field $\F_{5^{4}}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{5^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 25 x )^{2}( 1 - 49 x + 625 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.0637685608585$
Angle rank:  $1$ (numerical)

This isogeny class is not simple.

Newton polygon

$p$-rank:  $1$
Slopes:  $[0, 1/2, 1/2, 1]$

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
$A(\F_{q^r})$ 332352 151652217600 59590724194052352 23282862217526698675200 9094944151318564946239474752 3552713639205190684910894599372800 1387778780250456799889751804783109657152 542101086235873289430185046325370798302310400 211758236813489612674032135393014923177567564599552 82718061255301759941029364420712619518542369687883488000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 527 388225 244083602 152586565825 95367401584127 59604644111091550 37252902970365516527 23283064365091515201025 14551915228360978209130802 9094947017729171598283878625

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The isogeny class factors as 1.625.aby $\times$ 1.625.abx 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_abue$2$(not in LMFDB)
2.625.b_abue$2$(not in LMFDB)
2.625.dv_fmi$2$(not in LMFDB)
2.625.ay_z$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.625.ab_abue$2$(not in LMFDB)
2.625.b_abue$2$(not in LMFDB)
2.625.dv_fmi$2$(not in LMFDB)
2.625.ay_z$3$(not in LMFDB)
2.625.acw_drf$6$(not in LMFDB)
2.625.y_z$6$(not in LMFDB)
2.625.cw_drf$6$(not in LMFDB)