Abelian variety isogeny class 2.23.ae_ah over $\F_{23}$

• Label: 2.23.ae_ah
• Base field: $\F_{23}$
• 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:	$\F_{23}$
Dimension:	$2$
L-polynomial:	$1 - 4 x - 7 x^{2} - 92 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.0297380745307$, $\pm0.696404741197$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-19})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$427$	$264313$	$142945936$	$78266515369$	$41399206944907$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$500$	$11744$	$279684$	$6432100$	$147994670$	$3404840620$	$78310450564$	$1801149081632$	$41426516352500$

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 $\F_{23^{3}}$.

Endomorphism algebra over $\F_{23}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-19})\).

Endomorphism algebra over $\overline{\F}_{23}$

The base change of $A$ to $\F_{23^{3}}$ is 1.12167.aie^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$

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
2.23.e_ah	$2$	(not in LMFDB)
2.23.i_ck	$3$	(not in LMFDB)
2.23.ai_ck	$6$	(not in LMFDB)