Properties

Label 2.625.adw_fog
Base Field $\F_{5^{4}}$
Dimension $2$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

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

This isogeny class is not simple.

Newton polygon

This isogeny class is supersingular.

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

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
$A(\F_{q^r})$ 331776 151613669376 59589387451109376 23282825947723387109376 9094943292439556121787109376 3552713620592840373516081787109376 1387778779871950973980128765081787109376 542101086228541362288664095103740081787109376 211758236813353040162155681173317134365081787109376 82718061255299298040188915647377143613993740081787109376

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 526 388126 244078126 152586328126 95367392578126 59604643798828126 37252902960205078126 23283064364776611328126 14551915228351593017578126 9094947017728900909423828126

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The isogeny class factors as 1.625.aby 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.
All geometric endomorphisms are defined over $\F_{5^{4}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{5^{4}}$.

SubfieldPrimitive Model
$\F_{5}$2.5.a_ak
$\F_{5}$2.5.a_k

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.625.a_abwc$2$(not in LMFDB)
2.625.dw_fog$2$(not in LMFDB)
2.625.az_a$3$(not in LMFDB)
2.625.by_cud$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.625.a_abwc$2$(not in LMFDB)
2.625.dw_fog$2$(not in LMFDB)
2.625.az_a$3$(not in LMFDB)
2.625.by_cud$3$(not in LMFDB)
2.625.a_bwc$4$(not in LMFDB)
2.625.z_yb$5$(not in LMFDB)
2.625.acx_dse$6$(not in LMFDB)
2.625.aby_cud$6$(not in LMFDB)
2.625.a_yb$6$(not in LMFDB)
2.625.z_a$6$(not in LMFDB)
2.625.cx_dse$6$(not in LMFDB)
2.625.a_a$8$(not in LMFDB)
2.625.az_yb$10$(not in LMFDB)
2.625.a_ayb$12$(not in LMFDB)