Abelian variety isogeny class 2.113.abm_wl over $\F_{113}$

• Label: 2.113.abm_wl
• Base field: $\F_{113}$
• 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:	$\F_{113}$
Dimension:	$2$
L-polynomial:	$( 1 - 21 x + 113 x^{2} )( 1 - 17 x + 113 x^{2} )$
	$1 - 38 x + 583 x^{2} - 4294 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.0498602789898$, $\pm0.205038125192$
Angle rank:	$2$ (numerical)
Jacobians:	$6$
Isomorphism classes:	10

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9021$	$159536385$	$2080088593488$	$26584585616734425$	$339458180535141361221$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$76$	$12492$	$1441606$	$163048244$	$18424430036$	$2081952213534$	$235260540229700$	$26584441688080036$	$3004041934363214758$	$339456738955672464732$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=9x^6+66x^5+6x^4+33x^3+7x^2+13x+66$
• $y^2=68x^6+14x^5+52x^4+98x^3+30x^2+7x+37$
• $y^2=24x^6+111x^5+28x^4+107x^3+106x^2+26x+92$
• $y^2=53x^6+19x^5+45x^4+79x^3+6x^2+19x+29$
• $y^2=25x^6+77x^5+93x^4+77x^3+96x^2+61x+15$
• $y^2=79x^6+87x^5+76x^4+5x^3+37x^2+87x+34$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{113}$.

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.av $\times$ 1.113.ar and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.113.av : \(\Q(\sqrt{-11}) \).
• 1.113.ar : \(\Q(\sqrt{-163}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.113.ae_afb	$2$	(not in LMFDB)
2.113.e_afb	$2$	(not in LMFDB)
2.113.bm_wl	$2$	(not in LMFDB)