Properties

Label 2.3.b_d
Base field $\F_{3}$
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:  $\F_{3}$
Dimension:  $2$
L-polynomial:  $1 + x + 3 x^{2} + 3 x^{3} + 9 x^{4}$
Frobenius angles:  $\pm0.377272149103$, $\pm0.731463671465$
Angle rank:  $2$ (numerical)
Number field:  4.0.11661.1
Galois group:  $D_{4}$
Jacobians:  $2$
Isomorphism classes:  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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $17$ $153$ $731$ $8109$ $49232$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $15$ $29$ $99$ $200$ $675$ $2315$ $6579$ $19847$ $58950$

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 $\F_{3}$.

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is 4.0.11661.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.3.ab_d$2$2.9.f_v