Abelian variety isogeny class 2.11.ac_ah over $\F_{11}$

• Label: 2.11.ac_ah
• Base field: $\F_{11}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: no
• Contains a Jacobian: no

Invariants

Base field:	$\F_{11}$
Dimension:	$2$
L-polynomial:	$1 - 2 x - 7 x^{2} - 22 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.0691755521456$, $\pm0.735842218812$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-10})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Cyclic group of points:	yes

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$91$	$12649$	$1623076$	$215551609$	$25808743051$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$104$	$1216$	$14724$	$160250$	$1770158$	$19494590$	$214336324$	$2358020656$	$25937745704$

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 $\F_{11^{3}}$.

Endomorphism algebra over $\F_{11}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-10})\).

Endomorphism algebra over $\overline{\F}_{11}$

The base change of $A$ to $\F_{11^{3}}$ is 1.1331.acg^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$

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
2.11.c_ah	$2$	2.121.as_hv
2.11.e_ba	$3$	(not in LMFDB)
2.11.ae_ba	$6$	(not in LMFDB)