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

• Label: 3.7.ae_v_abz
• 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: yes

Invariants

Base field:	$\F_{7}$
Dimension:	$3$
L-polynomial:	$( 1 + x + 7 x^{2} )( 1 - 5 x + 19 x^{2} - 35 x^{3} + 49 x^{4} )$
	$1 - 4 x + 21 x^{2} - 51 x^{3} + 147 x^{4} - 196 x^{5} + 343 x^{6}$
Frobenius angles:	$\pm0.260350433790$, $\pm0.415892662795$, $\pm0.560518859162$
Angle rank:	$3$ (numerical)
Isomorphism classes:	18

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:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$261$	$199143$	$44715564$	$13712389551$	$4778766099216$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$76$	$379$	$2380$	$16919$	$117793$	$821986$	$5762404$	$40351393$	$282445571$

Jacobians and polarizations

This isogeny class contains a Jacobian, and hence is principally polarizable.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.b $\times$ 2.7.af_t 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.b : \(\Q(\sqrt{-3}) \).
• 2.7.af_t : 4.0.9725.1.

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.ag_bf_adl	$2$	(not in LMFDB)
3.7.e_v_bz	$2$	(not in LMFDB)
3.7.g_bf_dl	$2$	(not in LMFDB)
3.7.ak_bz_agj	$3$	(not in LMFDB)
3.7.ab_g_g	$3$	(not in LMFDB)