Properties

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

Learn more about

Invariants

Base field:  $\F_{5}$
Dimension:  $2$
L-polynomial:  $1 - 3 x + 9 x^{2} - 15 x^{3} + 25 x^{4}$
Frobenius angles:  $\pm0.235523574971$, $\pm0.521566933142$
Angle rank:  $2$ (numerical)
Number field:  4.0.30589.1
Galois group:  $D_{4}$
Jacobians:  1

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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 17 901 16949 390133 10188032 248828269 6064126253 151715311173 3812745837521 95388179992576

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 35 135 627 3258 15923 77619 388387 1952127 9767750

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The endomorphism algebra of this simple isogeny class is 4.0.30589.1.
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_j$2$2.25.j_bp