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

• Label: 2.7.ag_v
• Base field: $\F_{7}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$1 - 6 x + 21 x^{2} - 42 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.185925252552$, $\pm0.403118263531$
Angle rank:	$2$ (numerical)
Number field:	4.0.13888.1
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$23$	$2737$	$130916$	$5848969$	$282342503$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$56$	$380$	$2436$	$16802$	$118118$	$826142$	$5768644$	$40342052$	$282411416$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2=3 x^6+3 x^5+2 x^3+x^2+6$
• $y^2=6 x^6+4 x^5+4 x^4+x^3+4 x^2+5 x+2$

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 4.0.13888.1.

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.g_v	$2$	2.49.g_bj