Abelian variety isogeny class 1.137.ar over $\F_{137}$

• Label: 1.137.ar
• Base field: $\F_{137}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{137}$
Dimension:	$1$
L-polynomial:	$1 - 17 x + 137 x^{2}$
Frobenius angles:	$\pm0.241282780251$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-259}) \)
Galois group:	$C_2$
Jacobians:	$4$
Isomorphism classes:	4

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$1$
Slopes:	$[0, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$121$	$18755$	$2573428$	$352312675$	$48262074641$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$121$	$18755$	$2573428$	$352312675$	$48262074641$	$6611857091840$	$905824272659273$	$124097929279971075$	$17001416398494506836$	$2329194047537287260275$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which 0 are hyperelliptic):

• $y^2=x^3+114 x+114$
• $y^2=x^3+134 x+134$
• $y^2=x^3+98 x+98$
• $y^2=x^3+61 x+46$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{137}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-259}) \).

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
1.137.r	$2$	(not in LMFDB)