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

• Label: 2.23.a_bq
• 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:	$( 1 - 2 x + 23 x^{2} )( 1 + 2 x + 23 x^{2} )$
	$1 + 42 x^{2} + 529 x^{4}$
Frobenius angles:	$\pm0.433137181604$, $\pm0.566862818396$
Angle rank:	$1$ (numerical)
Jacobians:	$12$
Isomorphism classes:	40

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$572$	$327184$	$148043324$	$77916906496$	$41426504708732$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$614$	$12168$	$278430$	$6436344$	$148050758$	$3404825448$	$78311107774$	$1801152661464$	$41426498203814$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=5 x^6+5 x^5+10 x^4+16 x^3+3 x^2+20 x+6$
• $y^2=21 x^6+4 x^5+22 x^4+4 x^3+22 x^2+4 x+21$
• $y^2=13 x^6+20 x^5+18 x^4+20 x^3+18 x^2+20 x+13$
• $y^2=10 x^6+9 x^5+6 x^3+13 x+12$
• $y^2=18 x^6+7 x^5+9 x^4+x^3+12 x^2+15 x+12$
• $y^2=21 x^6+12 x^5+22 x^4+5 x^3+14 x^2+6 x+14$
• $y^2=7 x^6+3 x^4+15 x^2+1$
• $y^2=18 x^6+4 x^4+20 x^2+19$
• $y^2=8 x^6+16 x^5+18 x^4+17 x^3+14 x^2+15 x+7$
• $y^2=17 x^6+11 x^5+21 x^4+16 x^3+x^2+6 x+12$
• $y^2=3 x^6+4 x^5+14 x^4+7 x^2+6 x+19$
• $y^2=15 x^6+20 x^5+x^4+12 x^2+7 x+3$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{23}$

The isogeny class factors as 1.23.ac $\times$ 1.23.c 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.ac : \(\Q(\sqrt{-22}) \).
• 1.23.c : \(\Q(\sqrt{-22}) \).

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

The base change of $A$ to $\F_{23^{2}}$ is 1.529.bq^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	$2$	(not in LMFDB)
2.23.e_by	$2$	(not in LMFDB)
2.23.a_abq	$4$	(not in LMFDB)