Properties

Label 1.25.k
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:  $1$, $1$
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
$A(\F_{q^r})$ 36 576 15876 389376 9771876 244109376 6103671876 152587109376 3814701171876 95367412109376

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 36 576 15876 389376 9771876 244109376 6103671876 152587109376 3814701171876 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 isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{5^{2}}$.

SubfieldPrimitive Model
$\F_{5}$1.5.a

Twists

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