Abelian variety isogeny class 3.7.al_ci_ahq over $\F_{7}$

• Label: 3.7.al_ci_ahq
• Base field: $\F_{7}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{7}$
Dimension:	$3$
L-polynomial:	$( 1 - 4 x + 7 x^{2} )( 1 - 7 x + 25 x^{2} - 49 x^{3} + 49 x^{4} )$
	$1 - 11 x + 60 x^{2} - 198 x^{3} + 420 x^{4} - 539 x^{5} + 343 x^{6}$
Frobenius angles:	$\pm0.162349854003$, $\pm0.227185525829$, $\pm0.351370772325$
Angle rank:	$3$ (numerical)
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:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$76$	$119472$	$47436844$	$14841767616$	$4822415452096$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$49$	$399$	$2569$	$17072$	$117973$	$824457$	$5767585$	$40353663$	$282439004$

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 $\times$ 2.7.ah_z 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 : \(\Q(\sqrt{-3}) \).
• 2.7.ah_z : 4.0.3725.1.

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
3.7.ad_e_c	$2$	(not in LMFDB)
3.7.d_e_ac	$2$	(not in LMFDB)
3.7.l_ci_hq	$2$	(not in LMFDB)
3.7.ai_bn_aet	$3$	(not in LMFDB)
3.7.ac_ad_bb	$3$	(not in LMFDB)