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

• Label: 1.289.f
• 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:	$1 + 5 x + 289 x^{2}$
Frobenius angles:	$\pm0.546980662974$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-1131}) \)
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$295$	$84075$	$24133360$	$6975618675$	$2015995810975$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$295$	$84075$	$24133360$	$6975618675$	$2015995810975$	$582622267780800$	$168377825854504015$	$48661191870362103075$	$14063084452297964026480$	$4064231406647954404426875$

Jacobians and polarizations

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

• $y^2=x^3+a^{85} x+a^{85}$
• $y^2=x^3+a^5 x+a^5$
• $y^2=x^3+a^{124} x+a^{125}$
• $y^2=x^3+a^{60} x+a^{61}$
• $y^2=x^3+a^{62} x+a^{62}$
• $y^2=x^3+a^{170} x+a^{170}$
• $y^2=x^3+a^{190} x+a^{190}$
• $y^2=x^3+a^{10} x+a^{10}$

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 \(\Q(\sqrt{-1131}) \).

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.af	$2$	(not in LMFDB)