Abelian variety isogeny class 2.13.ai_bh over $\F_{13}$

• Label: 2.13.ai_bh
• Base field: $\F_{13}$
• 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:	$\F_{13}$
Dimension:	$2$
L-polynomial:	$( 1 - 7 x + 13 x^{2} )( 1 - x + 13 x^{2} )$
	$1 - 8 x + 33 x^{2} - 104 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.0772104791556$, $\pm0.455715642762$
Angle rank:	$2$ (numerical)
Jacobians:	$6$
Isomorphism classes:	16

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$91$	$28665$	$4758208$	$802190025$	$137411044771$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$172$	$2166$	$28084$	$370086$	$4829254$	$62763630$	$815726116$	$10604422158$	$137859184732$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=5x^6+9x^5+x^4+8x^3+5x^2+7x+11$
• $y^2=6x^6+12x^5+8x^4+10x^3+8x^2+12x+6$
• $y^2=5x^6+3x^5+5x^4+x^3+11x^2+x+7$
• $y^2=11x^6+11x^5+2x^4+6x^3+5x^2+6$
• $y^2=6x^6+x^5+5x^4+2x^3+7x^2+6x+2$
• $y^2=3x^6+2x^5+12x^4+3x^3+12x^2+2x+3$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13}$.

Endomorphism algebra over $\F_{13}$

The isogeny class factors as 1.13.ah $\times$ 1.13.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.13.ah : \(\Q(\sqrt{-3}) \).
• 1.13.ab : \(\Q(\sqrt{-51}) \).

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
2.13.ag_t	$2$	2.169.c_ajd
2.13.g_t	$2$	2.169.c_ajd
2.13.i_bh	$2$	2.169.c_ajd
2.13.b_y	$3$	(not in LMFDB)
2.13.e_v	$3$	(not in LMFDB)