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

• Label: 2.13.ai_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:	$\pm0.312832958189$, $\pm0.312832958189$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$100$	$32400$	$5244100$	$829440000$	$137678102500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$190$	$2382$	$29038$	$370806$	$4818670$	$62722302$	$815731678$	$10604844006$	$137859857950$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{13}$

The isogeny class factors as 1.13.ae^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.a_k	$2$	2.169.u_qw
2.13.i_bq	$2$	2.169.u_qw
2.13.e_d	$3$	(not in LMFDB)