Properties

Label 2.625.adw_fog
Base field $\F_{5^{4}}$
Dimension $2$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple no
Geometrically simple no
Primitive no
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{5^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 25 x )^{4}$
  $1 - 100 x + 3750 x^{2} - 62500 x^{3} + 390625 x^{4}$
Frobenius angles:  $0$, $0$, $0$, $0$
Angle rank:  $0$ (numerical)

This isogeny class is not simple, not primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $331776$ $151613669376$ $59589387451109376$ $23282825947723387109376$ $9094943292439556121787109376$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class contains a Jacobian, and hence is principally polarizable.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{4}}$.

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 $\infty$.

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)