Abelian variety isogeny class 2.7.aj_bi over $\F_{7}$

• Label: 2.7.aj_bi
• Base field: $\F_{7}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{7}$
Dimension:	$2$
L-polynomial:	$( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 7 x^{2} )$
	$1 - 9 x + 34 x^{2} - 63 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.106147807505$, $\pm0.227185525829$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Isomorphism classes:	2

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12$	$1872$	$117936$	$5937984$	$286901652$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$37$	$344$	$2473$	$17069$	$118222$	$824291$	$5765041$	$40353608$	$282486157$

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 $\F_{7^{6}}$.

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.af $\times$ 1.7.ae 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.ae : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{7}$

The base change of $A$ to $\F_{7^{6}}$ is 1.117649.la^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{7^{2}}$
  The base change of $A$ to $\F_{7^{2}}$ is 1.49.al $\times$ 1.49.ac. The endomorphism algebra for each factor is:

  • 1.49.al : \(\Q(\sqrt{-3}) \).
  • 1.49.ac : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{7^{3}}$
  The base change of $A$ to $\F_{7^{3}}$ is 1.343.au $\times$ 1.343.u. The endomorphism algebra for each factor is:

  • 1.343.au : \(\Q(\sqrt{-3}) \).
  • 1.343.u : \(\Q(\sqrt{-3}) \).

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
2.7.ab_ag	$2$	2.49.an_eq
2.7.b_ag	$2$	2.49.an_eq
2.7.j_bi	$2$	2.49.an_eq
2.7.ag_t	$3$	2.343.a_la
2.7.ad_k	$3$	2.343.a_la
2.7.a_al	$3$	2.343.a_la
2.7.a_ac	$3$	2.343.a_la
2.7.a_n	$3$	2.343.a_la
2.7.d_k	$3$	2.343.a_la
2.7.g_t	$3$	2.343.a_la
2.7.j_bi	$3$	2.343.a_la