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

• Label: 2.23.k_cp
• 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:	$( 1 + 3 x + 23 x^{2} )( 1 + 7 x + 23 x^{2} )$
	$1 + 10 x + 67 x^{2} + 230 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.601257449372$, $\pm0.760387042310$
Angle rank:	$2$ (numerical)
Jacobians:	$18$
Isomorphism classes:	30
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$837$	$298809$	$144191664$	$78517742121$	$41432617697157$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$34$	$564$	$11848$	$280580$	$6437294$	$148032558$	$3404794658$	$78310907524$	$1801155765784$	$41426494224564$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

• $y^2=14 x^6+5 x^5+21 x^4+16 x^3+17 x^2+19 x+19$
• $y^2=6 x^6+6 x^5+7 x^4+17 x^3+7 x^2+6 x+6$
• $y^2=16 x^6+20 x^5+13 x^4+2 x^3+3 x^2+2 x+17$
• $y^2=5 x^6+12 x^5+9 x^4+14 x^3+9 x^2+12 x+5$
• $y^2=9 x^6+11 x^5+11 x^4+8 x^3+3 x^2+2 x+17$
• $y^2=3 x^6+8 x^5+17 x^4+18 x^3+17 x^2+8 x+3$
• $y^2=x^6+17 x^5+9 x^4+14 x^3+9 x^2+14 x+21$
• $y^2=10 x^6+5 x^5+2 x^4+11 x^3+5 x^2+4 x+13$
• $y^2=9 x^6+6 x^5+4 x^4+5 x^3+x^2+2 x+12$
• $y^2=10 x^6+15 x^5+6 x^4+8 x^3+6 x^2+15 x+10$
• $y^2=21 x^6+3 x^5+9 x^4+12 x^3+9 x^2+3 x+21$
• $y^2=19 x^6+19 x^5+4 x^4+13 x^3+4 x^2+19 x+19$
• $y^2=20 x^6+20 x^5+3 x^4+10 x^3+5 x^2+15 x+8$
• $y^2=5 x^6+5 x^5+11 x^4+18 x^3+12 x^2+2 x+2$
• $y^2=12 x^6+20 x^5+20 x^4+18 x^3+20 x^2+20 x+12$
• $y^2=9 x^6+15 x^5+11 x^4+20 x^3+11 x^2+15 x+9$
• $y^2=13 x^6+21 x^5+16 x^4+13 x^3+16 x^2+21 x+13$
• $y^2=16 x^6+14 x^5+21 x^4+2 x^3+21 x^2+8 x+3$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{23}$

The isogeny class factors as 1.23.d $\times$ 1.23.h 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.d : \(\Q(\sqrt{-83}) \).
• 1.23.h : \(\Q(\sqrt{-43}) \).

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.ak_cp	$2$	(not in LMFDB)
2.23.ae_z	$2$	(not in LMFDB)
2.23.e_z	$2$	(not in LMFDB)