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

• Label: 3.7.al_cg_ahh
• Base field: $\F_{7}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• 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:	$( 1 - 5 x + 7 x^{2} )( 1 - 6 x + 21 x^{2} - 42 x^{3} + 49 x^{4} )$
	$1 - 11 x + 58 x^{2} - 189 x^{3} + 406 x^{4} - 539 x^{5} + 343 x^{6}$
Frobenius angles:	$\pm0.106147807505$, $\pm0.185925252552$, $\pm0.403118263531$
Angle rank:	$3$ (numerical)
Isomorphism classes:	4

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$69$	$106743$	$42416784$	$13914697251$	$4752671352999$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$45$	$360$	$2413$	$16827$	$118404$	$827397$	$5772917$	$40354632$	$282444405$

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 $\times$ 2.7.ag_v 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.af : \(\Q(\sqrt{-3}) \).
• 2.7.ag_v : 4.0.13888.1.

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.ab_ac_v	$2$	(not in LMFDB)
3.7.b_ac_av	$2$	(not in LMFDB)
3.7.l_cg_hh	$2$	(not in LMFDB)
3.7.af_w_acl	$3$	(not in LMFDB)
3.7.ac_e_a	$3$	(not in LMFDB)