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

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

Invariants

Base field:	$\F_{7}$
Dimension:	$2$
L-polynomial:	$( 1 - 4 x + 7 x^{2} )^{2}$
	$1 - 8 x + 30 x^{2} - 56 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.227185525829$, $\pm0.227185525829$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$16$	$2304$	$132496$	$6230016$	$290497936$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$46$	$384$	$2590$	$17280$	$118222$	$822528$	$5756734$	$40328448$	$282431086$

Jacobians and polarizations

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

• $y^2=3x^6+3$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.ae^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.7.a_ac	$2$	2.49.ae_dy
2.7.i_be	$2$	2.49.ae_dy
2.7.af_s	$3$	2.343.bo_bpu
2.7.ac_p	$3$	2.343.bo_bpu
2.7.b_ag	$3$	2.343.bo_bpu
2.7.e_j	$3$	2.343.bo_bpu
2.7.k_bn	$3$	2.343.bo_bpu