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

• Label: 2.23.ak_cs
• 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:	$( 1 - 6 x + 23 x^{2} )( 1 - 4 x + 23 x^{2} )$
	$1 - 10 x + 70 x^{2} - 230 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.284877382774$, $\pm0.363071407864$
Angle rank:	$2$ (numerical)
Jacobians:	$24$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$360$	$302400$	$153091080$	$78624000000$	$41406888169800$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$570$	$12578$	$280958$	$6433294$	$148000410$	$3404723938$	$78311161918$	$1801154986574$	$41426517521850$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):

• $y^2=15 x^6+13 x^5+20 x^4+4 x^3+20 x^2+13 x+15$
• $y^2=17 x^6+14 x^5+9 x^4+12 x^3+6 x^2+19 x+11$
• $y^2=21 x^6+21 x^5+18 x^4+5 x^3+18 x^2+21 x+21$
• $y^2=21 x^6+10 x^5+2 x^4+16 x^3+2 x^2+10 x+21$
• $y^2=x^6+19 x^5+19 x^4+13 x^3+19 x^2+19 x+1$
• $y^2=19 x^6+20 x^5+8 x^4+22 x^3+6 x^2+17 x+17$
• $y^2=14 x^6+7 x^5+5 x^4+22 x^3+5 x^2+7 x+14$
• $y^2=15 x^6+10 x^5+14 x^4+12 x^3+14 x^2+10 x+15$
• $y^2=15 x^6+12 x^5+6 x^4+11 x^3+6 x^2+12 x+15$
• $y^2=21 x^6+11 x^5+19 x^4+10 x^3+19 x^2+11 x+21$
• $y^2=9 x^6+7 x^5+12 x^4+22 x^3+12 x^2+7 x+9$
• $y^2=15 x^6+19 x^5+15 x^4+12 x^3+15 x^2+19 x+15$
• $y^2=3 x^5+5 x^4+17 x^3+19 x^2+x$
• $y^2=x^5+17 x^4+15 x^3+17 x^2+x$
• $y^2=16 x^5+9 x^4+21 x^3+16 x^2+4 x$
• $y^2=17 x^5+x^4+5 x^3+16 x^2+5 x$
• $y^2=19 x^6+15 x^5+21 x^4+13 x^3+21 x^2+15 x+19$
• $y^2=17 x^6+21 x^5+17 x^4+5 x^3+5 x^2+5 x+22$
• $y^2=20 x^6+16 x^5+4 x^4+13 x^3+4 x^2+16 x+20$
• $y^2=9 x^6+11 x^5+4 x^4+16 x^3+4 x^2+11 x+9$
• $y^2=7 x^6+17 x^5+2 x^4+5 x^3+2 x^2+17 x+7$
• $y^2=15 x^6+13 x^5+7 x^4+20 x^3+7 x^2+13 x+15$
• $y^2=22 x^6+17 x^5+21 x^4+22 x^3+14 x^2+5 x+21$
• $y^2=16 x^6+9 x^5+14 x^4+12 x^3+14 x^2+9 x+16$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{23}$

The isogeny class factors as 1.23.ag $\times$ 1.23.ae 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.ag : \(\Q(\sqrt{-14}) \).
• 1.23.ae : \(\Q(\sqrt{-19}) \).

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.ac_w	$2$	(not in LMFDB)
2.23.c_w	$2$	(not in LMFDB)
2.23.k_cs	$2$	(not in LMFDB)