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

• Label: 2.113.abl_vu
• 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:	$( 1 - 20 x + 113 x^{2} )( 1 - 17 x + 113 x^{2} )$
	$1 - 37 x + 566 x^{2} - 4181 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.110150159186$, $\pm0.205038125192$
Angle rank:	$2$ (numerical)
Jacobians:	$4$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9118$	$160057372$	$2081419729144$	$26587186403114944$	$339462519183652738798$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$77$	$12533$	$1442528$	$163064193$	$18424665517$	$2081955313298$	$235260577317997$	$26584442093583361$	$3004041938384160944$	$339456738990825303893$

Jacobians and polarizations

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

• $y^2=47x^6+111x^5+32x^4+36x^3+81x^2+33x+55$
• $y^2=108x^6+20x^5+101x^4+55x^3+30x^2+57x+47$
• $y^2=16x^6+51x^5+77x^4+84x^3+100x^2+10x+66$
• $y^2=26x^6+69x^5+88x^4+36x^3+25x^2+49x+30$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.au $\times$ 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.au : \(\Q(\sqrt{-13}) \).
• 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.ad_aek	$2$	(not in LMFDB)
2.113.d_aek	$2$	(not in LMFDB)
2.113.bl_vu	$2$	(not in LMFDB)