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

• Label: 2.23.ad_bq
• 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:	$( 1 - 4 x + 23 x^{2} )( 1 + x + 23 x^{2} )$
	$1 - 3 x + 42 x^{2} - 69 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.363071407864$, $\pm0.533246515430$
Angle rank:	$2$ (numerical)
Jacobians:	$16$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$500$	$322000$	$149798000$	$78085000000$	$41415487887500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$21$	$605$	$12312$	$279033$	$6434631$	$148034990$	$3404762697$	$78311144593$	$1801156619016$	$41426512541525$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

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

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 $\times$ 1.23.b 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.ae : \(\Q(\sqrt{-19}) \).
• 1.23.b : \(\Q(\sqrt{-91}) \).

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.af_by	$2$	(not in LMFDB)
2.23.d_bq	$2$	(not in LMFDB)
2.23.f_by	$2$	(not in LMFDB)