Abelian variety isogeny class 2.7.af_t over $\F_{7}$

• Label: 2.7.af_t
• Base field: $\F_{7}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{7}$
Dimension:	$2$
L-polynomial:	$1 - 5 x + 19 x^{2} - 35 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.260350433790$, $\pm0.415892662795$
Angle rank:	$2$ (numerical)
Number field:	4.0.9725.1
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$29$	$3161$	$138011$	$5882621$	$280790064$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$63$	$399$	$2451$	$16708$	$117507$	$823749$	$5762643$	$40338813$	$282456478$

Jacobians and polarizations

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

• $y^2=6x^6+3x^5+4x^4+x^3+4x^2+3$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7}$.

Endomorphism algebra over $\F_{7}$

The endomorphism algebra of this simple isogeny class is 4.0.9725.1.

Base change

This is a primitive isogeny class.

Twists

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

Twist	Extension degree	Common base change
2.7.f_t	$2$	2.49.n_ef