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

• Label: 2.23.a_bt
• 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:	$( 1 - x + 23 x^{2} )( 1 + x + 23 x^{2} )$
	$1 + 45 x^{2} + 529 x^{4}$
Frobenius angles:	$\pm0.466753484570$, $\pm0.533246515430$
Angle rank:	$1$ (numerical)
Jacobians:	$10$
Isomorphism classes:	16
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$575$	$330625$	$148055600$	$77771265625$	$41426517680375$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$620$	$12168$	$277908$	$6436344$	$148075310$	$3404825448$	$78310234468$	$1801152661464$	$41426524147100$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

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

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

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

The base change of $A$ to $\F_{23^{2}}$ is 1.529.bt^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-91}) \)$)$

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.ac_bv	$2$	(not in LMFDB)
2.23.c_bv	$2$	(not in LMFDB)
2.23.a_abt	$4$	(not in LMFDB)