Abelian variety isogeny class 2.139.abo_zy over $\F_{139}$

• Label: 2.139.abo_zy
• Base field: $\F_{139}$
• 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:	$\F_{139}$
Dimension:	$2$
L-polynomial:	$( 1 - 22 x + 139 x^{2} )( 1 - 18 x + 139 x^{2} )$
	$1 - 40 x + 674 x^{2} - 5560 x^{3} + 19321 x^{4}$
Frobenius angles:	$\pm0.117174211439$, $\pm0.223543330897$
Angle rank:	$2$ (numerical)
Jacobians:	$24$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$14396$	$368480016$	$7213089440924$	$139365886709412864$	$2692480546039269410876$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$100$	$19070$	$2685820$	$373333774$	$51889390900$	$7212555180686$	$1002544409117260$	$139353667356344734$	$19370159742301145860$	$2692452204209313701150$

Jacobians and polarizations

This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=138x^6+7x^5+120x^4+120x^2+7x+138$
• $y^2=32x^6+98x^5+51x^4+74x^3+119x^2+5x+34$
• $y^2=98x^6+26x^5+105x^4+111x^3+128x^2+2x+32$
• $y^2=85x^6+112x^5+83x^4+x^3+83x^2+112x+85$
• $y^2=28x^6+112x^5+73x^4+113x^3+24x^2+131x+45$
• $y^2=36x^6+110x^5+73x^4+134x^3+92x^2+27x+55$
• $y^2=68x^6+31x^5+104x^4+123x^3+90x^2+100x+47$
• $y^2=126x^6+93x^5+118x^4+6x^3+79x+129$
• $y^2=116x^6+51x^5+58x^4+32x^3+58x^2+51x+116$
• $y^2=96x^6+33x^5+120x^4+91x^3+52x^2+15x+120$
• $y^2=116x^6+27x^5+129x^4+111x^3+117x^2+114x+52$
• $y^2=106x^6+96x^5+15x^4+21x^3+78x^2+10x+15$
• $y^2=35x^6+19x^5+30x^4+129x^3+65x^2+15x+24$
• $y^2=43x^6+13x^5+28x^4+22x^3+118x^2+16x+110$
• $y^2=83x^6+17x^5+26x^4+110x^2+23x+4$
• $y^2=17x^6+54x^5+53x^4+77x^3+27x^2+66x+17$
• $y^2=64x^6+3x^5+126x^4+98x^3+49x^2+83x+73$
• $y^2=102x^6+101x^5+23x^4+130x^3+100x^2+68x+34$
• $y^2=93x^6+135x^5+36x^4+26x^3+36x^2+135x+93$
• $y^2=102x^6+100x^5+58x^4+29x^3+58x^2+100x+102$
• $y^2=37x^6+102x^5+108x^4+48x^3+79x^2+32x+91$
• $y^2=58x^6+31x^5+84x^4+55x^3+84x^2+31x+58$
• $y^2=3x^6+74x^5+131x^4+12x^3+49x^2+39x+26$
• $y^2=107x^6+89x^5+x^4+62x^3+110x^2+2x+125$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{139}$.

Endomorphism algebra over $\F_{139}$

The isogeny class factors as 1.139.aw $\times$ 1.139.as and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.139.aw : \(\Q(\sqrt{-2}) \).
• 1.139.as : \(\Q(\sqrt{-58}) \).

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
2.139.ae_aeo	$2$	(not in LMFDB)
2.139.e_aeo	$2$	(not in LMFDB)
2.139.bo_zy	$2$	(not in LMFDB)