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

• Label: 2.113.abk_uv
• 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:	$( 1 - 21 x + 113 x^{2} )( 1 - 15 x + 113 x^{2} )$
	$1 - 36 x + 541 x^{2} - 4068 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.0498602789898$, $\pm0.250704227710$
Angle rank:	$2$ (numerical)
Jacobians:	$35$
Isomorphism classes:	100

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9207$	$160339905$	$2081327643648$	$26585232506580825$	$339456691899909109647$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$78$	$12556$	$1442466$	$163052212$	$18424349238$	$2081950011934$	$235260512009286$	$26584441501163428$	$3004041935026626498$	$339456738992875742236$

Jacobians and polarizations

This isogeny class contains the Jacobians of 35 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=19x^6+76x^5+8x^4+92x^3+3x^2+75x+60$
• $y^2=12x^6+109x^5+37x^4+71x^3+96x^2+69x+93$
• $y^2=17x^6+74x^5+22x^4+101x^3+90x^2+16x+26$
• $y^2=45x^6+9x^5+16x^4+64x^3+16x^2+9x+45$
• $y^2=21x^6+35x^5+40x^4+109x^3+40x^2+35x+21$
• $y^2=29x^6+63x^5+68x^4+80x^3+68x^2+63x+29$
• $y^2=95x^6+79x^5+15x^4+77x^3+18x^2+7x+90$
• $y^2=5x^6+31x^5+87x^4+106x^3+107x^2+22x+5$
• $y^2=76x^6+25x^5+92x^4+49x^3+76x^2+70x+54$
• $y^2=5x^6+12x^5+72x^4+20x^3+79x^2+83x+89$
• $y^2=5x^6+83x^5+110x^4+87x^3+110x^2+83x+5$
• $y^2=73x^6+101x^5+89x^4+70x^3+9x^2+18x+98$
• $y^2=27x^6+10x^5+20x^4+110x^3+111x^2+67x+66$
• $y^2=4x^6+105x^5+25x^4+99x^3+19x^2+59x+23$
• $y^2=43x^6+92x^5+13x^4+29x^3+13x^2+92x+43$
• $y^2=5x^6+97x^5+33x^4+110x^3+10x^2+79x+101$
• $y^2=28x^6+30x^5+101x^4+33x^3+101x^2+30x+28$
• $y^2=24x^6+94x^5+7x^4+85x^3+92x^2+111x+21$
• $y^2=3x^6+39x^5+5x^4+73x^3+5x^2+39x+3$
• $y^2=69x^6+42x^5+46x^4+14x^3+46x^2+42x+69$
• and

• $y^2=19x^6+109x^5+12x^4+22x^3+64x^2+54x+29$
• $y^2=42x^6+66x^5+82x^3+66x+42$
• $y^2=91x^6+38x^5+32x^4+24x^3+32x^2+38x+91$
• $y^2=2x^6+82x^5+103x^4+19x^3+97x^2+10x+50$
• $y^2=103x^6+104x^5+97x^4+87x^3+37x^2+84x+96$
• $y^2=102x^6+23x^5+36x^4+34x^3+36x^2+23x+102$
• $y^2=7x^6+25x^5+30x^4+111x^3+23x^2+94x+72$
• $y^2=2x^6+8x^5+68x^4+41x^3+94x^2+107x+87$
• $y^2=15x^6+47x^5+54x^4+45x^3+45x^2+38x+81$
• $y^2=89x^6+38x^5+41x^4+105x^3+41x^2+38x+89$
• $y^2=66x^6+69x^5+22x^4+77x^3+23x+23$
• $y^2=42x^6+60x^5+51x^4+62x^3+51x^2+60x+42$
• $y^2=32x^6+78x^5+95x^4+7x^3+95x^2+78x+32$
• $y^2=40x^6+77x^5+50x^4+11x^3+50x^2+77x+40$
• $y^2=43x^6+51x^5+45x^4+61x^3+55x^2+15x+74$

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.ap 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.ap : \(\Q(\sqrt{-227}) \).

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.ag_adl	$2$	(not in LMFDB)
2.113.g_adl	$2$	(not in LMFDB)
2.113.bk_uv	$2$	(not in LMFDB)