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

• Label: 3.7.ak_ca_ago
• 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:	$( 1 - 5 x + 7 x^{2} )( 1 - 3 x + 7 x^{2} )( 1 - 2 x + 7 x^{2} )$
	$1 - 10 x + 52 x^{2} - 170 x^{3} + 364 x^{4} - 490 x^{5} + 343 x^{6}$
Frobenius angles:	$\pm0.106147807505$, $\pm0.308124534521$, $\pm0.376624142786$
Angle rank:	$3$ (numerical)
Isomorphism classes:	16
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$90$	$128700$	$46539360$	$14131260000$	$4677800643450$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$54$	$394$	$2450$	$16558$	$116856$	$823954$	$5773346$	$40380118$	$282515814$

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 $\times$ 1.7.ad $\times$ 1.7.ac 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 : \(\Q(\sqrt{-3}) \).
• 1.7.ad : \(\Q(\sqrt{-19}) \).
• 1.7.ac : \(\Q(\sqrt{-6}) \).

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_u_acc	$2$	(not in LMFDB)
3.7.ae_k_aba	$2$	(not in LMFDB)
3.7.a_c_abe	$2$	(not in LMFDB)
3.7.a_c_be	$2$	(not in LMFDB)
3.7.e_k_ba	$2$	(not in LMFDB)
3.7.g_u_cc	$2$	(not in LMFDB)
3.7.k_ca_go	$2$	(not in LMFDB)
3.7.ae_w_aby	$3$	(not in LMFDB)
3.7.ab_h_k	$3$	(not in LMFDB)