Properties

Label 2.25.an_dh
Base field $\F_{5^{2}}$
Dimension $2$
$p$-rank $1$
Ordinary no
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:  $1 - 13 x + 85 x^{2} - 325 x^{3} + 625 x^{4}$
Frobenius angles:  $\pm0.128789547124$, $\pm0.375668685983$
Angle rank:  $2$ (numerical)
Number field:  4.0.44573.1
Galois group:  $D_{4}$
Jacobians:  $2$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $373$ $391277$ $246386269$ $152607811925$ $95337474393808$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $13$ $627$ $15769$ $390675$ $9762558$ $244142427$ $6103726369$ $152589449475$ $3814702120333$ $95367429615502$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 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 4.0.44573.1.

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_dh$2$2.625.b_z