Abelian variety isogeny class 2.137.abo_zx over $\F_{137}$

• Label: 2.137.abo_zx
• Base field: $\F_{137}$
• 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:	$\F_{137}$
Dimension:	$2$
L-polynomial:	$( 1 - 21 x + 137 x^{2} )( 1 - 19 x + 137 x^{2} )$
	$1 - 40 x + 673 x^{2} - 5480 x^{3} + 18769 x^{4}$
Frobenius angles:	$\pm0.145687313345$, $\pm0.198575263152$
Angle rank:	$2$ (numerical)
Jacobians:	$18$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$13923$	$347559849$	$6612683628096$	$124111887406693641$	$2329229191974585343443$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$98$	$18516$	$2571674$	$352314980$	$48262452658$	$6611865236622$	$905824386795874$	$124097930385537604$	$17001416403711033098$	$2329194047480002870836$

Jacobians and polarizations

This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=14x^6+80x^5+40x^4+12x^3+40x^2+80x+14$
• $y^2=86x^6+103x^5+97x^4+34x^3+97x^2+103x+86$
• $y^2=x^6+41x^5+71x^4+16x^3+71x^2+41x+1$
• $y^2=91x^6+29x^5+114x^4+84x^3+114x^2+29x+91$
• $y^2=67x^6+133x^5+33x^4+17x^3+33x^2+133x+67$
• $y^2=77x^6+x^5+107x^4+136x^3+107x^2+x+77$
• $y^2=29x^6+92x^5+134x^4+35x^3+134x^2+92x+29$
• $y^2=62x^6+68x^5+46x^4+22x^3+46x^2+68x+62$
• $y^2=79x^6+85x^5+132x^4+20x^3+132x^2+85x+79$
• $y^2=31x^6+62x^5+59x^4+122x^3+59x^2+62x+31$
• $y^2=79x^6+113x^5+38x^4+115x^3+38x^2+113x+79$
• $y^2=129x^6+98x^5+62x^4+43x^3+62x^2+98x+129$
• $y^2=118x^6+120x^5+74x^4+50x^3+74x^2+120x+118$
• $y^2=45x^6+77x^5+47x^4+94x^3+47x^2+77x+45$
• $y^2=134x^6+97x^5+94x^4+110x^3+94x^2+97x+134$
• $y^2=59x^6+54x^5+62x^4+62x^2+54x+59$
• $y^2=84x^6+40x^5+22x^4+18x^3+22x^2+40x+84$
• $y^2=99x^6+52x^5+67x^4+5x^3+67x^2+52x+99$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{137}$.

Endomorphism algebra over $\F_{137}$

The isogeny class factors as 1.137.av $\times$ 1.137.at and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.137.av : \(\Q(\sqrt{-107}) \).
• 1.137.at : \(\Q(\sqrt{-187}) \).

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
2.137.ac_aev	$2$	(not in LMFDB)
2.137.c_aev	$2$	(not in LMFDB)
2.137.bo_zx	$2$	(not in LMFDB)