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

• Label: 2.23.al_cm
• 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:	$( 1 - 9 x + 23 x^{2} )( 1 - 2 x + 23 x^{2} )$
	$1 - 11 x + 64 x^{2} - 253 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.112386341891$, $\pm0.433137181604$
Angle rank:	$2$ (numerical)
Jacobians:	$8$
Isomorphism classes:	50
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$330$	$283140$	$148313880$	$78067360800$	$41404501035150$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$13$	$537$	$12190$	$278969$	$6432923$	$148055994$	$3405033029$	$78311578321$	$1801152795730$	$41426516619057$

Jacobians and polarizations

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

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

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.ac 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.ac : \(\Q(\sqrt{-22}) \).

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.ah_bc	$2$	(not in LMFDB)
2.23.h_bc	$2$	(not in LMFDB)
2.23.l_cm	$2$	(not in LMFDB)