Abelian variety isogeny class 1.89.o over $\F_{89}$

• Label: 1.89.o
• Base field: $\F_{89}$
• 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:	$\F_{89}$
Dimension:	$1$
L-polynomial:	$1 + 14 x + 89 x^{2}$
Frobenius angles:	$\pm0.766121877123$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-10}) \)
Galois group:	$C_2$
Jacobians:	$6$
Isomorphism classes:	6

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$104$	$7904$	$703976$	$62757760$	$5583930664$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$104$	$7904$	$703976$	$62757760$	$5583930664$	$496981712864$	$44231340450856$	$3936588690378240$	$350356404827594984$	$31181719924548468704$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which 0 are hyperelliptic):

• $y^2=x^3+20 x+60$
• $y^2=x^3+2 x+2$
• $y^2=x^3+60 x+2$
• $y^2=x^3+59 x+59$
• $y^2=x^3+42 x+42$
• $y^2=x^3+14 x+14$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{89}$.

Endomorphism algebra over $\F_{89}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-10}) \).

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
1.89.ao	$2$	(not in LMFDB)