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

• Label: 3.7.ak_bt_aff
• 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:	$( 1 - 5 x + 7 x^{2} )( 1 - 5 x + 13 x^{2} - 35 x^{3} + 49 x^{4} )$
	$1 - 10 x + 45 x^{2} - 135 x^{3} + 315 x^{4} - 490 x^{5} + 343 x^{6}$
Frobenius angles:	$\pm0.0616448849068$, $\pm0.106147807505$, $\pm0.511587336964$
Angle rank:	$3$ (numerical)
Isomorphism classes:	1

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$69$	$92391$	$34480404$	$12787745919$	$4727062720944$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$40$	$289$	$2212$	$16733$	$118333$	$823954$	$5764372$	$40372453$	$282551675$

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_n 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_n : 4.0.24389.1.

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_af_af	$2$	(not in LMFDB)
3.7.a_af_f	$2$	(not in LMFDB)
3.7.k_bt_ff	$2$	(not in LMFDB)
3.7.ae_p_acf	$3$	(not in LMFDB)
3.7.ab_a_as	$3$	(not in LMFDB)