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

• Label: 1.289.abe
• Base field: $\F_{17^{2}}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{17^{2}}$
Dimension:	$1$
L-polynomial:	$1 - 30 x + 289 x^{2}$
Frobenius angles:	$\pm0.155958260755$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-1}) \)
Galois group:	$C_2$
Jacobians:	$8$

This isogeny class is simple and geometrically simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$260$	$83200$	$24136580$	$6975820800$	$2015996087300$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$260$	$83200$	$24136580$	$6975820800$	$2015996087300$	$582622284524800$	$168377827346252420$	$48661191885604147200$	$14063084452138443271940$	$4064231406646822197280000$

Jacobians and polarizations

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

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 \(\Q(\sqrt{-1}) \).

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{17^{2}}$.

Subfield	Primitive Model
$\F_{17}$	1.17.ai
$\F_{17}$	1.17.i

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
1.289.be	$2$	(not in LMFDB)
1.289.aq	$4$	(not in LMFDB)
1.289.q	$4$	(not in LMFDB)