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

• Label: 2.23.ai_ck
• 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:	$( 1 - 4 x + 23 x^{2} )^{2}$
	$1 - 8 x + 62 x^{2} - 184 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.363071407864$, $\pm0.363071407864$
Angle rank:	$1$ (numerical)
Jacobians:	$14$
Cyclic group of points:	no
Non-cyclic primes:	$2, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$400$	$313600$	$153264400$	$78400000000$	$41371910410000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$590$	$12592$	$280158$	$6427856$	$147994670$	$3404855792$	$78312054718$	$1801156241296$	$41426500935950$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{23}$

The isogeny class factors as 1.23.ae^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.23.a_be	$2$	(not in LMFDB)
2.23.i_ck	$2$	(not in LMFDB)
2.23.e_ah	$3$	(not in LMFDB)