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

• Label: 2.23.an_de
• 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:	$( 1 - 9 x + 23 x^{2} )( 1 - 4 x + 23 x^{2} )$
	$1 - 13 x + 82 x^{2} - 299 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.112386341891$, $\pm0.363071407864$
Angle rank:	$2$ (numerical)
Jacobians:	$8$
Isomorphism classes:	52

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$300$	$277200$	$149302800$	$78309000000$	$41405518132500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$525$	$12272$	$279833$	$6433081$	$148027950$	$3404932087$	$78312051793$	$1801157133776$	$41426517985125$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{23}$

The isogeny class factors as 1.23.aj $\times$ 1.23.ae 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.ae : \(\Q(\sqrt{-19}) \).

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_k	$2$	(not in LMFDB)
2.23.f_k	$2$	(not in LMFDB)
2.23.n_de	$2$	(not in LMFDB)