Properties

Label 1.5.a
Base Field $\F_{5}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{5}$
Dimension:  $1$
L-polynomial:  $1 + 5 x^{2}$
Frobenius angles:  $\pm0.5$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-5}) \)
Galois group:  $C_2$
Jacobians:  2

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 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 6 36 126 576 3126 15876 78126 389376 1953126 9771876

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 36 126 576 3126 15876 78126 389376 1953126 9771876

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-5}) \).
Endomorphism algebra over $\overline{\F}_{5}$
The base change of $A$ to $\F_{5^{2}}$ is the simple isogeny class 1.25.k and its endomorphism algebra 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

This isogeny class has no twists.