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

• Label: 2.13.ab_ae
• 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:	$( 1 - 6 x + 13 x^{2} )( 1 + 5 x + 13 x^{2} )$
	$1 - x - 4 x^{2} - 13 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.187167041811$, $\pm0.743877145822$
Angle rank:	$2$ (numerical)
Jacobians:	$12$
Isomorphism classes:	124
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$152$	$27360$	$4715648$	$832291200$	$138013881272$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$13$	$161$	$2146$	$29137$	$371713$	$4830374$	$62770021$	$815674753$	$10604504938$	$137857950761$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{13}$

The isogeny class factors as 1.13.ag $\times$ 1.13.f 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.f : \(\Q(\sqrt{-3}) \).

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.al_ce	$2$	2.169.aj_mq
2.13.b_ae	$2$	2.169.aj_mq
2.13.l_ce	$2$	2.169.aj_mq
2.13.an_cq	$3$	(not in LMFDB)
2.13.ae_o	$3$	(not in LMFDB)