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

• Label: 2.23.an_dd
• 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:	$1 - 13 x + 81 x^{2} - 299 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.0921437596788$, $\pm0.370068381004$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-170 +26 \sqrt{29}})\)
Galois group:	$D_{4}$
Jacobians:	$4$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$299$	$275977$	$148821569$	$78211053869$	$41393392612624$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$523$	$12233$	$279483$	$6431196$	$148021183$	$3404909519$	$78311920131$	$1801156330649$	$41426516107518$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{-170 +26 \sqrt{29}})\).

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.n_dd	$2$	(not in LMFDB)