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

• Label: 3.7.ak_bv_afp
• 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:	$( 1 - 5 x + 7 x^{2} )( 1 - 5 x + 15 x^{2} - 35 x^{3} + 49 x^{4} )$
	$1 - 10 x + 47 x^{2} - 145 x^{3} + 329 x^{4} - 490 x^{5} + 343 x^{6}$
Frobenius angles:	$\pm0.106147807505$, $\pm0.139519842760$, $\pm0.487441680688$
Angle rank:	$3$ (numerical)
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$75$	$102375$	$37770300$	$13270359375$	$4783635606000$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$44$	$319$	$2300$	$16933$	$119201$	$827104$	$5768996$	$40370593$	$282550139$

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.af_p 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.af_p : 4.0.62181.1.

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.a_ad_af	$2$	(not in LMFDB)
3.7.a_ad_f	$2$	(not in LMFDB)
3.7.k_bv_fp	$2$	(not in LMFDB)
3.7.ae_r_acd	$3$	(not in LMFDB)
3.7.ab_c_ak	$3$	(not in LMFDB)