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

• Label: 1.109.ao
• 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:	$1 - 14 x + 109 x^{2}$
Frobenius angles:	$\pm0.266088896335$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-15}) \)
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$96$	$11904$	$1296864$	$141181440$	$15386365536$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$96$	$11904$	$1296864$	$141181440$	$15386365536$	$1677099337344$	$182803887520224$	$19925626157352960$	$2171893278485821536$	$236736367474111730304$

Jacobians and polarizations

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

• $y^2=x^3+5 x+10$
• $y^2=x^3+12 x+12$
• $y^2=x^3+77 x+45$
• $y^2=x^3+20 x+40$
• $y^2=x^3+93 x+77$
• $y^2=x^3+83 x+57$
• $y^2=x^3+96 x+96$
• $y^2=x^3+98 x+98$

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 \(\Q(\sqrt{-15}) \).

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
1.109.o	$2$	(not in LMFDB)