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

• Label: 2.13.aj_bt
• Base field: $\F_{13}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{13}$
Dimension:	$2$
L-polynomial:	$1 - 9 x + 45 x^{2} - 117 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.215685987913$, $\pm0.344616475996$
Angle rank:	$2$ (numerical)
Number field:	4.0.20725.1
Galois group:	$D_{4}$
Jacobians:	$3$
Isomorphism classes:	3

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$89$	$30349$	$5131829$	$828193861$	$138005338064$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$179$	$2333$	$28995$	$371690$	$4825163$	$62743721$	$815734819$	$10604492489$	$137857962614$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{13}$

The endomorphism algebra of this simple isogeny class is 4.0.20725.1.

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
2.13.j_bt	$2$	2.169.j_jx