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

• Label: 2.7.a_o
• Base field: $\F_{7}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

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

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$64$	$4096$	$118336$	$5308416$	$282508864$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$78$	$344$	$2206$	$16808$	$119022$	$823544$	$5755198$	$40353608$	$282542478$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2=x^5+6 x$
• $y^2=x^5+x$

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^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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

The base change of $A$ to $\F_{7^{2}}$ is 1.49.o^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is 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.a_ah	$3$	2.343.a_bak
2.7.a_ao	$4$	(not in LMFDB)
2.7.a_a	$8$	(not in LMFDB)