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

• Label: 2.23.a_abj
• 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:	$( 1 - 9 x + 23 x^{2} )( 1 + 9 x + 23 x^{2} )$
	$1 - 35 x^{2} + 529 x^{4}$
Frobenius angles:	$\pm0.112386341891$, $\pm0.887613658109$
Angle rank:	$1$ (numerical)
Jacobians:	$14$
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$495$	$245025$	$148048560$	$78218105625$	$41426523123975$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$460$	$12168$	$279508$	$6436344$	$148061230$	$3404825448$	$78312048868$	$1801152661464$	$41426535034300$

Jacobians and polarizations

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

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

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.aj $\times$ 1.23.j 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.aj : \(\Q(\sqrt{-11}) \).
• 1.23.j : \(\Q(\sqrt{-11}) \).

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

The base change of $A$ to $\F_{23^{2}}$ is 1.529.abj^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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.as_ex	$2$	(not in LMFDB)
2.23.s_ex	$2$	(not in LMFDB)
2.23.a_bj	$4$	(not in LMFDB)