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

• Label: 2.23.a_as
• 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:	$( 1 - 8 x + 23 x^{2} )( 1 + 8 x + 23 x^{2} )$
	$1 - 18 x^{2} + 529 x^{4}$
Frobenius angles:	$\pm0.186011988595$, $\pm0.813988011405$
Angle rank:	$1$ (numerical)
Jacobians:	$25$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$512$	$262144$	$148058624$	$78722891776$	$41426499564032$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$494$	$12168$	$281310$	$6436344$	$148081358$	$3404825448$	$78311027134$	$1801152661464$	$41426487914414$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 25 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{23^{2}}$.

Endomorphism algebra over $\F_{23}$

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

Endomorphism algebra over $\overline{\F}_{23}$

The base change of $A$ to $\F_{23^{2}}$ is 1.529.as^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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.aq_eg	$2$	(not in LMFDB)
2.23.q_eg	$2$	(not in LMFDB)
2.23.a_s	$4$	(not in LMFDB)