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

• Label: 3.23.ay_jw_acim
• 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:	$( 1 - 9 x + 23 x^{2} )( 1 - 15 x + 98 x^{2} - 345 x^{3} + 529 x^{4} )$
	$1 - 24 x + 256 x^{2} - 1572 x^{3} + 5888 x^{4} - 12696 x^{5} + 12167 x^{6}$
Frobenius angles:	$\pm0.0252283536038$, $\pm0.112386341891$, $\pm0.308104979730$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4020$	$131068080$	$1785051927360$	$21884576249764800$	$266455388712937988100$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$466$	$12060$	$279458$	$6432000$	$148005292$	$3404714880$	$78311004482$	$1801155343860$	$41426526544786$

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 $\F_{23^{6}}$.

Endomorphism algebra over $\F_{23}$

The isogeny class factors as 1.23.aj $\times$ 2.23.ap_du 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 : \(\Q(\sqrt{-11}) \).
• 2.23.ap_du : \(\Q(\sqrt{-3}, \sqrt{17})\).

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

The base change of $A$ to $\F_{23^{6}}$ is 1.148035889.abgac^2 $\times$ 1.148035889.sti. The endomorphism algebra for each factor is:

• 1.148035889.abgac^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$
• 1.148035889.sti : \(\Q(\sqrt{-11}) \).

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{23^{2}}$
  The base change of $A$ to $\F_{23^{2}}$ is 1.529.abj $\times$ 2.529.abd_ma. The endomorphism algebra for each factor is:

  • 1.529.abj : \(\Q(\sqrt{-11}) \).
  • 2.529.abd_ma : \(\Q(\sqrt{-3}, \sqrt{17})\).
• Endomorphism algebra over $\F_{23^{3}}$
  The base change of $A$ to $\F_{23^{3}}$ is 1.12167.aee $\times$ 2.12167.a_abgac. The endomorphism algebra for each factor is:

  • 1.12167.aee : \(\Q(\sqrt{-11}) \).
  • 2.12167.a_abgac : \(\Q(\sqrt{-3}, \sqrt{17})\).

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.ag_ao_hk	$2$	(not in LMFDB)
3.23.g_ao_ahk	$2$	(not in LMFDB)
3.23.y_jw_cim	$2$	(not in LMFDB)
3.23.aj_ca_akb	$3$	(not in LMFDB)
3.23.g_ao_ahk	$3$	(not in LMFDB)