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

• Label: 2.7.b_o
• Base field: $\F_{7}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$( 1 + 7 x^{2} )( 1 + x + 7 x^{2} )$
	$1 + x + 14 x^{2} + 7 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.560518859162$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$72$	$4032$	$111456$	$5370624$	$286055352$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$77$	$324$	$2233$	$17019$	$118622$	$821781$	$5759761$	$40366188$	$282497957$

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 $\F_{7^{2}}$.

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.a $\times$ 1.7.b 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.a : \(\Q(\sqrt{-7}) \).
• 1.7.b : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{7^{2}}$ is 1.49.n $\times$ 1.49.o. The endomorphism algebra for each factor is:

• 1.49.n : \(\Q(\sqrt{-3}) \).
• 1.49.o : the quaternion algebra over \(\Q\) ramified at $7$ and $\infty$.

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_o	$2$	2.49.bb_ku
2.7.af_o	$3$	2.343.au_bak
2.7.e_o	$3$	2.343.au_bak