Properties

Label 2.25.an_dj
Base field $\F_{5^{2}}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{5^{2}}$
Dimension:  $2$
L-polynomial:  $1 - 13 x + 87 x^{2} - 325 x^{3} + 625 x^{4}$
Frobenius angles:  $\pm0.158125293436$, $\pm0.361724426820$
Angle rank:  $2$ (numerical)
Number field:  4.0.824229.1
Galois group:  $D_{4}$
Jacobians:  $12$
Isomorphism classes:  12

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $375$ $394125$ $247620375$ $152867293125$ $95365394430000$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $13$ $631$ $15847$ $391339$ $9765418$ $244144591$ $6103678867$ $152589154579$ $3814700913673$ $95367419676526$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{2}}$.

Endomorphism algebra over $\F_{5^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.824229.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.25.n_dj$2$2.625.f_of