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

• Label: 2.23.ap_dx
• Base field: $\F_{23}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{23}$
Dimension:	$2$
L-polynomial:	$1 - 15 x + 101 x^{2} - 345 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.144663500024$, $\pm0.268275520367$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-33 +6 \sqrt{5}})\)
Galois group:	$D_{4}$
Jacobians:	$2$
Cyclic group of points:	yes

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$271$	$268561$	$149694709$	$78672527901$	$41463219189136$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$507$	$12303$	$281131$	$6442044$	$148049943$	$3404833833$	$78310976323$	$1801153513179$	$41426520353022$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2=21 x^5+7 x^4+22 x^3+6 x^2+11 x+17$
• $y^2=19 x^6+7 x^4+8 x^3+16 x^2+x+5$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{23}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-33 +6 \sqrt{5}})\).

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.p_dx	$2$	(not in LMFDB)