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

• Label: 2.23.ao_dn
• 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:	$( 1 - 9 x + 23 x^{2} )( 1 - 5 x + 23 x^{2} )$
	$1 - 14 x + 91 x^{2} - 322 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.112386341891$, $\pm0.325452467839$
Angle rank:	$2$ (numerical)
Jacobians:	$6$
Isomorphism classes:	10
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$285$	$272745$	$149399280$	$78437370825$	$41420124422925$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$516$	$12280$	$280292$	$6435350$	$148024494$	$3404842010$	$78311696068$	$1801157961640$	$41426532096036$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):

• $y^2=5 x^6+11 x^5+13 x^4+5 x^3+13 x^2+11 x+5$
• $y^2=6 x^6+4 x^5+x^4+17 x^3+2 x+21$
• $y^2=10 x^6+18 x^5+12 x^4+22 x^3+7 x^2+15 x+5$
• $y^2=13 x^6+4 x^5+3 x^4+20 x^3+3 x^2+4 x+13$
• $y^2=19 x^6+16 x^5+22 x^4+18 x^3+22 x^2+16 x+19$
• $y^2=15 x^6+13 x^4+5 x^3+2 x^2+8$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{23}$

The isogeny class factors as 1.23.aj $\times$ 1.23.af and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.23.aj : \(\Q(\sqrt{-11}) \).
• 1.23.af : \(\Q(\sqrt{-67}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.23.ae_b	$2$	(not in LMFDB)
2.23.e_b	$2$	(not in LMFDB)
2.23.o_dn	$2$	(not in LMFDB)