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

• Label: 2.7.ah_ba
• 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:	$( 1 - 4 x + 7 x^{2} )( 1 - 3 x + 7 x^{2} )$
	$1 - 7 x + 26 x^{2} - 49 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.227185525829$, $\pm0.308124534521$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$20$	$2640$	$138320$	$6177600$	$285913100$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$53$	$400$	$2569$	$17011$	$117326$	$821437$	$5760241$	$40350640$	$282485693$

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.ae $\times$ 1.7.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.7.ae : \(\Q(\sqrt{-3}) \).
• 1.7.ad : \(\Q(\sqrt{-19}) \).

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.ab_c	$2$	2.49.d_dk
2.7.b_c	$2$	2.49.d_dk
2.7.h_ba	$2$	2.49.d_dk
2.7.ae_r	$3$	2.343.ce_ccc
2.7.c_ab	$3$	2.343.ce_ccc