Abelian variety isogeny class 2.13.c_c over F13\F_{13}

• Label: 2.13.c_c
• Base field: $\F_{13}$
• Dimension: $2$
• pp-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:	$( 1 - 4 x + 13 x^{2} )( 1 + 6 x + 13 x^{2} )$
	$1 + 2 x + 2 x^{2} + 26 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.312832958189$, $\pm0.812832958189$
Angle rank:	$1$ (numerical)
Jacobians:	$23$
Isomorphism classes:	206

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$200$	$28800$	$4992200$	$829440000$	$137325605000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$170$	$2272$	$29038$	$369856$	$4826810$	$62726512$	$815731678$	$10604784496$	$137858491850$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 23 curves (of which all are hyperelliptic):

• $y^2=8 x^6+2 x^5+11 x^4+10 x^3+5 x^2+7 x+8$
• $y^2=8 x^6+2 x^4+5 x^3+8$
• $y^2=10 x^6+x^5+5 x^4+5 x^2+12 x+10$
• $y^2=8 x^6+2 x^5+8 x^4+8 x^3+8 x^2+10 x+5$
• $y^2=7 x^6+7 x^5+12 x^4+3 x^3+12 x^2+11 x+10$
• $y^2=9 x^6+2 x^5+9 x^4+8 x^3+9 x^2+2 x+9$
• $y^2=2 x^6+9 x^5+5 x^4+9 x^3+12 x^2+x+1$
• $y^2=6 x^6+11 x^5+12 x^4+7 x^3+12 x^2+6 x$
• $y^2=11 x^5+6 x^4+x^3+6 x^2+11 x$
• $y^2=10 x^6+5 x^5+3 x^4+7 x^3+12 x^2+12 x+12$
• $y^2=6 x^6+9 x^5+5 x^4+12 x^3+3 x^2+5 x$
• $y^2=x^6+4 x^5+11 x^4+12 x^3+8 x^2+5 x+9$
• $y^2=9 x^6+x^5+6 x^3+9 x^2+2 x+4$
• $y^2=3 x^6+2 x^5+2 x^4+5 x^3+10 x^2+8 x+12$
• $y^2=8 x^6+12 x^5+2 x^4+5 x^3+6 x^2+4 x+8$
• $y^2=12 x^6+9 x^5+8 x^4+11 x^3+8 x^2+9 x+12$
• $y^2=8 x^6+10 x^4+12 x^3+10 x^2+12 x+12$
• $y^2=5 x^6+11 x^5+x^4+8 x^3+x^2+5 x+7$
• $y^2=10 x^6+2 x^4+x^3+11 x^2+4 x+6$
• $y^2=3 x^6+9 x^5+12 x^4+7 x^3+2 x^2+7 x+4$
• $y^2=5 x^5+10 x^4+10 x^3+9 x^2+5 x+12$
• $y^2=x^6+10 x^5+12 x^3+10 x+1$
• $y^2=11 x^5+8 x^4+12 x^3+x^2+12 x+10$

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.ae $\times$ 1.13.g 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.ae : $\Q(\sqrt{-1})$.
• 1.13.g : $\Q(\sqrt{-1})$.

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