Abelian variety isogeny class 2.131.abs_bcs over $\F_{131}$

• Label: 2.131.abs_bcs
• Base field: $\F_{131}$
• 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:	$\F_{131}$
Dimension:	$2$
L-polynomial:	$( 1 - 22 x + 131 x^{2} )^{2}$
	$1 - 44 x + 746 x^{2} - 5764 x^{3} + 17161 x^{4}$
Frobenius angles:	$\pm0.0891048534084$, $\pm0.0891048534084$
Angle rank:	$1$ (numerical)
Jacobians:	$4$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12100$	$286963600$	$5044920288100$	$86721391666201600$	$1488371859723217562500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$88$	$16718$	$2244088$	$294469998$	$38579355848$	$5053914120638$	$662062660908968$	$86730204199283038$	$11361656665395838648$	$1488377021876982305198$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=44x^6+61x^5+34x^4+81x^3+34x^2+61x+44$
• $y^2=66x^6+127x^5+130x^4+34x^3+119x^2+60x+120$
• $y^2=67x^6+75x^5+15x^4+9x^3+80x^2+81x+82$
• $y^2=62x^6+99x^4+99x^2+62$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{131}$.

Endomorphism algebra over $\F_{131}$

The isogeny class factors as 1.131.aw^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$

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
2.131.a_aio	$2$	(not in LMFDB)
2.131.bs_bcs	$2$	(not in LMFDB)
2.131.w_np	$3$	(not in LMFDB)