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

• Label: 2.23.ak_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:	$( 1 - 8 x + 23 x^{2} )( 1 - 2 x + 23 x^{2} )$
	$1 - 10 x + 62 x^{2} - 230 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.186011988595$, $\pm0.433137181604$
Angle rank:	$2$ (numerical)
Jacobians:	$20$
Isomorphism classes:	80
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$352$	$292864$	$150133984$	$78318862336$	$41430042276832$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$554$	$12338$	$279870$	$6436894$	$148066058$	$3405009538$	$78311067454$	$1801148717294$	$41426493059114$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{23}$

The isogeny class factors as 1.23.ai $\times$ 1.23.ac 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.ai : \(\Q(\sqrt{-7}) \).
• 1.23.ac : \(\Q(\sqrt{-22}) \).

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.ag_be	$2$	(not in LMFDB)
2.23.g_be	$2$	(not in LMFDB)
2.23.k_ck	$2$	(not in LMFDB)