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

• Label: 2.13.ak_by
• 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:	$( 1 - 6 x + 13 x^{2} )( 1 - 4 x + 13 x^{2} )$
	$1 - 10 x + 50 x^{2} - 130 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.187167041811$, $\pm0.312832958189$
Angle rank:	$1$ (numerical)
Jacobians:	$5$
Isomorphism classes:	20

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$80$	$28800$	$5074640$	$829440000$	$138211672400$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$170$	$2308$	$29038$	$372244$	$4826810$	$62744308$	$815731678$	$10604558884$	$137858491850$

Jacobians and polarizations

This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13^{4}}$.

Endomorphism algebra over $\F_{13}$

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

• 1.13.ag : \(\Q(\sqrt{-1}) \).
• 1.13.ae : \(\Q(\sqrt{-1}) \).

Endomorphism algebra over $\overline{\F}_{13}$

The base change of $A$ to $\F_{13^{4}}$ is 1.28561.je^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{13^{2}}$
  The base change of $A$ to $\F_{13^{2}}$ is 1.169.ak $\times$ 1.169.k. The endomorphism algebra for each factor is:

  • 1.169.ak : \(\Q(\sqrt{-1}) \).
  • 1.169.k : \(\Q(\sqrt{-1}) \).

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.ac_c	$2$	2.169.a_je
2.13.c_c	$2$	2.169.a_je
2.13.k_by	$2$	2.169.a_je