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

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

Invariants

Base field:	$\F_{23}$
Dimension:	$2$
L-polynomial:	$( 1 + 8 x + 23 x^{2} )^{2}$
	$1 + 16 x + 110 x^{2} + 368 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.813988011405$, $\pm0.813988011405$
Angle rank:	$1$ (numerical)
Jacobians:	$3$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1024$	$262144$	$147088384$	$78722891776$	$41362803057664$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$494$	$12088$	$281310$	$6426440$	$148081358$	$3404689496$	$78311027134$	$1801155453544$	$41426487914414$

Jacobians and polarizations

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

• $y^2=7 x^6+4 x^5+5 x^4+13 x^3+5 x^2+4 x+7$
• $y^2=20 x^6+9 x^4+9 x^2+20$
• $y^2=x^6+12 x^4+12 x^2+1$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{23}$

The isogeny class factors as 1.23.i^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
2.23.aq_eg	$2$	(not in LMFDB)
2.23.a_as	$2$	(not in LMFDB)
2.23.ai_bp	$3$	(not in LMFDB)