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

• Label: 2.137.abs_bde
• 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:	$( 1 - 22 x + 137 x^{2} )^{2}$
	$1 - 44 x + 758 x^{2} - 6028 x^{3} + 18769 x^{4}$
Frobenius angles:	$\pm0.111017258455$, $\pm0.111017258455$
Angle rank:	$1$ (numerical)
Jacobians:	$10$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$13456$	$344473600$	$6603604783504$	$124093307453440000$	$2329201350436737613456$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$94$	$18350$	$2568142$	$352262238$	$48261875774$	$6611861377550$	$905824398395822$	$124097931290662078$	$17001416422065255454$	$2329194047744990576750$

Jacobians and polarizations

This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=24x^6+6x^5+42x^4+13x^3+110x^2+57x+10$
• $y^2=45x^6+102x^5+93x^4+109x^3+93x^2+102x+45$
• $y^2=78x^6+29x^4+29x^2+78$
• $y^2=86x^6+24x^5+39x^4+31x^3+68x^2+20x+91$
• $y^2=50x^6+32x^4+32x^2+50$
• $y^2=95x^6+79x^5+104x^4+40x^3+102x^2+12x+111$
• $y^2=117x^6+93x^5+101x^4+100x^3+101x^2+93x+117$
• $y^2=79x^6+24x^4+24x^2+79$
• $y^2=49x^6+126x^5+29x^4+72x^3+29x^2+126x+49$
• $y^2=78x^6+68x^5+58x^4+14x^3+128x^2+112x+55$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{137}$

The isogeny class factors as 1.137.aw^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.137.a_aic	$2$	(not in LMFDB)
2.137.bs_bde	$2$	(not in LMFDB)
2.137.w_nj	$3$	(not in LMFDB)