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

• Label: 2.23.a_abq
• Base field: $\F_{23}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{23}$
Dimension:	$2$
L-polynomial:	$1 - 42 x^{2} + 529 x^{4}$
Frobenius angles:	$\pm0.0668628183960$, $\pm0.933137181604$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{22})\)
Galois group:	$C_2^2$
Jacobians:	$3$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$488$	$238144$	$148028456$	$77916906496$	$41426517718568$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$446$	$12168$	$278430$	$6436344$	$148021022$	$3404825448$	$78311107774$	$1801152661464$	$41426524223486$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

• $y^2=5 x^6+8 x^4+11 x^3+22 x^2+12 x+13$
• $y^2=21 x^6+8 x^5+7 x^4+6 x^3+14 x^2+3 x+21$
• $y^2=9 x^6+10 x^5+6 x^4+4 x^2+16 x+18$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{23^{2}}$.

Endomorphism algebra over $\F_{23}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{22})\).

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

The base change of $A$ to $\F_{23^{2}}$ is 1.529.abq^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-22}) \)$)$

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.ae_by	$4$	(not in LMFDB)
2.23.a_bq	$4$	(not in LMFDB)
2.23.e_by	$4$	(not in LMFDB)