Properties

Label 2.5.ab_ad
Base field $\F_{5}$
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:  $\F_{5}$
Dimension:  $2$
L-polynomial:  $1 - x - 3 x^{2} - 5 x^{3} + 25 x^{4}$
Frobenius angles:  $\pm0.123439569429$, $\pm0.747770883447$
Angle rank:  $2$ (numerical)
Number field:  4.0.81461.1
Galois group:  $D_{4}$
Jacobians:  $1$
Isomorphism classes:  1

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $17$ $493$ $12869$ $422501$ $9656272$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $19$ $101$ $675$ $3090$ $15787$ $79049$ $390627$ $1957601$ $9770694$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$
The endomorphism algebra of this simple isogeny class is 4.0.81461.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.5.b_ad$2$2.25.ah_bx