Abelian variety isogeny class 1.109.ar over $\F_{109}$

• Label: 1.109.ar
• Base field: $\F_{109}$
• 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:	$\F_{109}$
Dimension:	$1$
L-polynomial:	$1 - 17 x + 109 x^{2}$
Frobenius angles:	$\pm0.197202036359$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}) \)
Galois group:	$C_2$
Jacobians:	$3$
Isomorphism classes:	3

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$93$	$11811$	$1295676$	$141176883$	$15386487393$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$93$	$11811$	$1295676$	$141176883$	$15386487393$	$1677102283584$	$182803922003397$	$19925626348742403$	$2171893277202129324$	$236736367428558049851$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{109}$.

Endomorphism algebra over $\F_{109}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).

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
1.109.r	$2$	(not in LMFDB)
1.109.ac	$3$	(not in LMFDB)
1.109.t	$3$	(not in LMFDB)