Properties

Label 1.625.aby
Base Field $\F_{5^{4}}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

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

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

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
$A(\F_{q^r})$ 576 389376 244109376 152587109376 95367412109376 59604644287109376 37252902972412109376 23283064365081787109376 14551915228359222412109376 9094947017729091644287109376

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 576 389376 244109376 152587109376 95367412109376 59604644287109376 37252902972412109376 23283064365081787109376 14551915228359222412109376 9094947017729091644287109376

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class 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}$1.5.a

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.625.by$2$(not in LMFDB)
1.625.z$3$(not in LMFDB)
1.625.az$6$(not in LMFDB)