Abelian variety isogeny class 3.23.ax_jk_acfr over $\F_{23}$

• Label: 3.23.ax_jk_acfr
• Base field: $\F_{23}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{23}$
Dimension:	$3$
L-polynomial:	$( 1 - 9 x + 23 x^{2} )( 1 - 7 x + 23 x^{2} )^{2}$
	$1 - 23 x + 244 x^{2} - 1499 x^{3} + 5612 x^{4} - 12167 x^{5} + 12167 x^{6}$
Frobenius angles:	$\pm0.112386341891$, $\pm0.239612957690$, $\pm0.239612957690$
Angle rank:	$2$ (numerical)

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4335$	$137475855$	$1826931579840$	$22066288666668675$	$267017763776291771925$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$489$	$12340$	$281773$	$6445571$	$148058028$	$3404793533$	$78310435637$	$1801150611580$	$41426514871089$

Jacobians and polarizations

This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.

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.ah^2 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.ah^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$

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
3.23.aj_u_bb	$2$	(not in LMFDB)
3.23.af_ai_id	$2$	(not in LMFDB)
3.23.f_ai_aid	$2$	(not in LMFDB)
3.23.j_u_abb	$2$	(not in LMFDB)
3.23.x_jk_cfr	$2$	(not in LMFDB)
3.23.ac_ao_dk	$3$	(not in LMFDB)