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

• Label: 2.13.i_bq
• 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:	$( 1 + 4 x + 13 x^{2} )^{2}$
	$1 + 8 x + 42 x^{2} + 104 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.687167041811$, $\pm0.687167041811$
Angle rank:	$1$ (numerical)
Jacobians:	$5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$324$	$32400$	$4435236$	$829440000$	$138040485444$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$190$	$2014$	$29038$	$371782$	$4818670$	$62774734$	$815731678$	$10604154742$	$137859857950$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{13}$

The isogeny class factors as 1.13.e^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.ai_bq	$2$	2.169.u_qw
2.13.a_k	$2$	2.169.u_qw
2.13.ae_d	$3$	(not in LMFDB)