Properties

Label 1.25.ak
Base Field $\F_{5^{2}}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

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

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 the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 16 576 15376 389376 9759376 244109376 6103359376 152587109376 3814693359376 95367412109376

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 16 576 15376 389376 9759376 244109376 6103359376 152587109376 3814693359376 95367412109376

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{2}}$
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 $\F_{5^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.25.k$2$1.625.aby
1.25.f$3$(not in LMFDB)
1.25.af$6$(not in LMFDB)