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

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

Invariants

Base field:	$\F_{7}$
Dimension:	$2$
L-polynomial:	$1 - 7 x^{2} + 49 x^{4}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.833333333333$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-7})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Cyclic group of points:	yes

This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular.

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$43$	$1849$	$118336$	$6007401$	$282458443$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$36$	$344$	$2500$	$16808$	$119022$	$823544$	$5769604$	$40353608$	$282441636$

Jacobians and polarizations

This isogeny class is not principally polarizable, and therefore does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{6}}$.

Endomorphism algebra over $\F_{7}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-7})\).

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

The base change of $A$ to $\F_{7^{6}}$ is 1.117649.bak^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $7$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{7^{2}}$

  The base change of $A$ to $\F_{7^{2}}$ is the simple isogeny class 2.49.ao_fr and its endomorphism algebra is the quaternion algebra over \(\Q(\sqrt{-3}) \) with the following ramification data at primes above $7$, and unramified at all archimedean places:

  $v$	($ 7 $,\( \pi + 2 \))	($ 7 $,\( \pi + 4 \))
  $\operatorname{inv}_v$	$1/2$	$1/2$

  where $\pi$ is a root of $x^{2} - x + 1$.
• Endomorphism algebra over $\F_{7^{3}}$

  The base change of $A$ to $\F_{7^{3}}$ is 1.343.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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_o	$3$	2.343.a_bak
2.7.a_h	$4$	(not in LMFDB)
2.7.a_ao	$12$	(not in LMFDB)