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

• Label: 2.23.d_an
• 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:	$1 + 3 x - 13 x^{2} + 69 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.270744379230$, $\pm0.924942204553$
Angle rank:	$2$ (numerical)
Number field:	4.0.6525.1
Galois group:	$D_{4}$
Jacobians:	$14$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$589$	$262105$	$152361931$	$78423126525$	$41453483855344$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$27$	$495$	$12519$	$280243$	$6440532$	$148022835$	$3404705709$	$78310673683$	$1801150724997$	$41426528140350$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

• $y^2=3 x^6+5 x^5+2 x^4+18 x^3+3 x^2+7 x+4$
• $y^2=x^6+14 x^5+20 x^4+18 x^3+9 x+21$
• $y^2=2 x^6+x^5+7 x^4+15 x^3+13 x^2+20 x+4$
• $y^2=3 x^6+12 x^5+5 x^4+14 x^3+7 x^2+7 x+5$
• $y^2=8 x^6+6 x^5+18 x^4+9 x^3+12 x^2+3 x+12$
• $y^2=18 x^6+5 x^5+11 x^4+18 x^3+8 x^2+9 x+4$
• $y^2=x^6+5 x^5+10 x^4+x^3+x^2+9 x+4$
• $y^2=13 x^6+21 x^5+15 x^4+8 x^3+8 x^2+7 x+4$
• $y^2=5 x^6+10 x^5+20 x^4+22 x^3+15 x^2+4 x+9$
• $y^2=6 x^5+20 x^4+11 x^3+7 x^2+16 x+13$
• $y^2=8 x^6+16 x^5+8 x^4+21 x^3+19 x^2+12 x+4$
• $y^2=13 x^6+5 x^5+19 x^4+10 x^3+4 x^2+2$
• $y^2=13 x^6+12 x^5+x^4+7 x^3+10 x^2+14 x+21$
• $y^2=17 x^6+9 x^5+21 x^4+6 x^3+6 x^2+8 x$

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 4.0.6525.1.

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.ad_an	$2$	(not in LMFDB)