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

• Label: 2.23.ao_dq
• 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:	$( 1 - 8 x + 23 x^{2} )( 1 - 6 x + 23 x^{2} )$
	$1 - 14 x + 94 x^{2} - 322 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.186011988595$, $\pm0.284877382774$
Angle rank:	$2$ (numerical)
Jacobians:	$8$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$288$	$276480$	$150964128$	$78785740800$	$41466087769248$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$522$	$12406$	$281534$	$6442490$	$148043754$	$3404776742$	$78310648126$	$1801151800618$	$41426511011082$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

• $y^2=5 x^6+x^5+12 x^4+22 x^3+9 x^2+2 x+19$
• $y^2=11 x^6+16 x^5+12 x^4+3 x^3+12 x^2+16 x+11$
• $y^2=18 x^6+10 x^5+5 x^4+19 x^3+5 x^2+10 x+18$
• $y^2=20 x^6+21 x^5+2 x^4+21 x^3+2 x^2+21 x+20$
• $y^2=8 x^6+16 x^5+15 x^4+22 x^3+11 x^2+13 x+4$
• $y^2=9 x^6+x^5+x^4+20 x^3+18 x^2+2 x+2$
• $y^2=17 x^6+15 x^5+5 x^4+6 x^3+14 x^2+21 x+5$
• $y^2=10 x^6+13 x^5+16 x^4+13 x^3+16 x^2+13 x+10$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{23}$

The isogeny class factors as 1.23.ai $\times$ 1.23.ag 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.ai : \(\Q(\sqrt{-7}) \).
• 1.23.ag : \(\Q(\sqrt{-14}) \).

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.ac_ac	$2$	(not in LMFDB)
2.23.c_ac	$2$	(not in LMFDB)
2.23.o_dq	$2$	(not in LMFDB)