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

• Label: 2.7.ak_bn
• 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: no

Invariants

Base field:	$\F_{7}$
Dimension:	$2$
L-polynomial:	$( 1 - 5 x + 7 x^{2} )^{2}$
	$1 - 10 x + 39 x^{2} - 70 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.106147807505$, $\pm0.106147807505$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9$	$1521$	$104976$	$5659641$	$283349889$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$28$	$304$	$2356$	$16858$	$118222$	$826054$	$5773348$	$40378768$	$282541228$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.af^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_al	$2$	2.49.aw_il
2.7.k_bn	$2$	2.49.aw_il
2.7.ae_j	$3$	2.343.abo_bpu
2.7.ab_ag	$3$	2.343.abo_bpu
2.7.c_p	$3$	2.343.abo_bpu
2.7.f_s	$3$	2.343.abo_bpu
2.7.i_be	$3$	2.343.abo_bpu