Properties

Label 2.5.ad_h
Base Field $\F_{5}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{5}$
Dimension:  $2$
L-polynomial:  $1 - 3 x + 7 x^{2} - 15 x^{3} + 25 x^{4}$
Frobenius angles:  $\pm0.177952114464$, $\pm0.556618995437$
Angle rank:  $2$ (numerical)
Number field:  4.0.48069.2
Galois group:  $D_{4}$
Jacobians:  4

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class contains the Jacobians of 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 15 765 14715 386325 10381200 249904845 6093381435 152632757925 3818442864735 95307887059200

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 31 117 619 3318 15991 77997 390739 1955043 9759526

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The endomorphism algebra of this simple isogeny class is 4.0.48069.2.
All geometric endomorphisms are defined over $\F_{5}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.5.d_h$2$2.25.f_j