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

• Label: 2.13.ah_bl
• 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:	$1 - 7 x + 37 x^{2} - 91 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.278766070715$, $\pm0.392843588800$
Angle rank:	$2$ (numerical)
Number field:	4.0.35525.3
Galois group:	$D_{4}$
Jacobians:	$4$
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$109$	$33245$	$5191561$	$822647525$	$137557260544$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$7$	$195$	$2359$	$28803$	$370482$	$4822635$	$62744059$	$815739363$	$10604482147$	$137858351350$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.35525.3.

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.h_bl	$2$	2.169.z_qr