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

• Label: 1.89.aj
• 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:	$1 - 9 x + 89 x^{2}$
Frobenius angles:	$\pm0.341724512740$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-11}) \)
Galois group:	$C_2$
Jacobians:	$5$
Isomorphism classes:	5

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$81$	$8019$	$706644$	$62748675$	$5583968361$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$81$	$8019$	$706644$	$62748675$	$5583968361$	$496979898624$	$44231330471409$	$3936588889803075$	$350356404858140916$	$31181719932837096579$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which 0 are hyperelliptic):

• $y^2=x^3+3 x+3$
• $y^2=x^3+11 x+33$
• $y^2=x^3+82 x+82$
• $y^2=x^3+78 x+56$
• $y^2=x^3+38 x+25$

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 \(\Q(\sqrt{-11}) \).

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.j	$2$	(not in LMFDB)