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

• Label: 2.137.abp_bam
• 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:	$( 1 - 23 x + 137 x^{2} )( 1 - 18 x + 137 x^{2} )$
	$1 - 41 x + 688 x^{2} - 5617 x^{3} + 18769 x^{4}$
Frobenius angles:	$\pm0.0596181899068$, $\pm0.220793476886$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$13800$	$346600800$	$6608905228800$	$124100589740400000$	$2329201564749498945000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$97$	$18465$	$2570206$	$352282913$	$48261880217$	$6611856717870$	$905824282289681$	$124097929393585153$	$17001416398277562862$	$2329194047515465371825$

Jacobians and polarizations

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

• $y^2=131x^6+95x^5+45x^4+37x^3+44x^2+19x+37$
• $y^2=114x^6+14x^5+46x^4+20x^3+79x^2+104x+100$
• $y^2=42x^6+6x^5+70x^4+9x^3+63x^2+56x+21$
• $y^2=37x^6+18x^5+118x^4+106x^3+79x^2+3x+92$
• $y^2=93x^6+79x^5+72x^4+43x^3+30x^2+119x+86$
• $y^2=40x^6+48x^5+29x^4+8x^3+39x^2+116x+83$
• $y^2=69x^6+40x^5+60x^4+74x^3+88x^2+18x+67$
• $y^2=74x^6+33x^5+22x^4+x^3+8x^2+108x+51$
• $y^2=84x^6+129x^5+104x^4+82x^3+33x^2+38x+131$
• $y^2=104x^6+76x^5+75x^4+15x^3+33x^2+64x+98$
• $y^2=58x^6+112x^5+103x^4+82x^3+62x^2+76x+97$
• $y^2=68x^6+52x^5+83x^4+66x^3+14x^2+72x+108$
• $y^2=134x^6+41x^5+125x^4+123x^3+15x^2+34x+43$
• $y^2=122x^6+91x^5+81x^4+11x^3+124x^2+73x+77$
• $y^2=53x^6+8x^5+106x^4+67x^3+105x^2+7x+70$
• $y^2=119x^6+57x^5+110x^4+88x^3+22x^2+70x+12$
• $y^2=58x^6+119x^5+5x^4+46x^3+112x^2+121x+89$
• $y^2=133x^6+98x^5+132x^4+85x^3+88x^2+95x+123$
• $y^2=42x^6+46x^5+34x^4+55x^3+24x^2+47x+29$
• $y^2=58x^6+63x^5+x^4+49x^3+109x^2+47x+49$
• $y^2=75x^6+8x^5+14x^4+79x^3+37x^2+121x+46$
• $y^2=20x^6+7x^5+48x^4+135x^3+35x^2+18x+114$
• $y^2=61x^6+108x^5+129x^4+80x^2+34x+95$
• $y^2=102x^6+49x^5+80x^4+101x^3+44x^2+39x+99$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{137}$

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

• 1.137.ax : \(\Q(\sqrt{-19}) \).
• 1.137.as : \(\Q(\sqrt{-14}) \).

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.af_afk	$2$	(not in LMFDB)
2.137.f_afk	$2$	(not in LMFDB)
2.137.bp_bam	$2$	(not in LMFDB)