Abelian variety isogeny class 1.289.i over $\F_{17^{2}}$

• Label: 1.289.i
• Base field: $\F_{17^{2}}$
• 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:	$\F_{17^{2}}$
Dimension:	$1$
L-polynomial:	$1 + 8 x + 289 x^{2}$
Frobenius angles:	$\pm0.575605336838$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-273}) \)
Galois group:	$C_2$
Jacobians:	$8$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$298$	$84036$	$24131146$	$6975660288$	$2015996534218$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$298$	$84036$	$24131146$	$6975660288$	$2015996534218$	$582622244237124$	$168377825742183082$	$48661191880179483648$	$14063084452267800027754$	$4064231406644667776324676$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which 0 are hyperelliptic):

• $y^2=x^3+a^{238} x+a^{238}$
• $y^2=x^3+a^{190} x+a^{191}$
• $y^2=x^3+a^{30} x+a^{31}$
• $y^2=x^3+a^{14} x+a^{14}$
• $y^2=x^3+a^{257} x+a^{257}$
• $y^2=x^3+a^{179} x+8$
• $y^2=x^3+a^{49} x+a^{49}$
• $y^2=x^3+a^{131} x+a^{132}$

where $a$ is a root of the Conway polynomial.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{17^{2}}$.

Endomorphism algebra over $\F_{17^{2}}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-273}) \).

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
1.289.ai	$2$	(not in LMFDB)