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

• Label: 2.113.abm_wo
• 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:	$( 1 - 20 x + 113 x^{2} )( 1 - 18 x + 113 x^{2} )$
	$1 - 38 x + 586 x^{2} - 4294 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.110150159186$, $\pm0.178616545187$
Angle rank:	$2$ (numerical)
Jacobians:	$16$
Isomorphism classes:	56

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9024$	$159616512$	$2080583555904$	$26586267657928704$	$339462255327893461824$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$76$	$12498$	$1441948$	$163058558$	$18424651196$	$2081955962898$	$235260592792364$	$26584442304912766$	$3004041940300359724$	$339456738998371396818$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=78x^6+10x^5+75x^4+45x^3+86x^2+70x+43$
• $y^2=18x^6+106x^5+74x^4+26x^3+74x^2+106x+18$
• $y^2=108x^6+60x^5+110x^4+24x^3+43x^2+85x+67$
• $y^2=26x^6+59x^5+25x^4+24x^3+25x^2+59x+26$
• $y^2=x^6+8x^5+42x^4+94x^3+42x^2+8x+1$
• $y^2=93x^6+19x^5+13x^4+100x^3+13x^2+19x+93$
• $y^2=9x^6+75x^5+73x^4+103x^3+73x^2+75x+9$
• $y^2=108x^6+29x^5+102x^4+75x^3+51x^2+92x+70$
• $y^2=84x^6+17x^5+53x^4+33x^3+53x^2+17x+84$
• $y^2=31x^6+20x^5+55x^4+65x^3+55x^2+20x+31$
• $y^2=9x^6+19x^5+17x^4+31x^3+40x^2+70x+7$
• $y^2=26x^6+45x^5+93x^4+91x^3+108x^2+24x+11$
• $y^2=52x^6+34x^5+32x^4+6x^3+105x^2+101x+91$
• $y^2=51x^6+59x^5+93x^4+32x^3+34x^2+27x+49$
• $y^2=42x^6+37x^5+28x^4+36x^3+28x^2+37x+42$
• $y^2=108x^6+67x^5+30x^4+94x^3+30x^2+67x+108$

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.as 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.as : \(\Q(\sqrt{-2}) \).

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.ac_afe	$2$	(not in LMFDB)
2.113.c_afe	$2$	(not in LMFDB)
2.113.bm_wo	$2$	(not in LMFDB)