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

• Label: 1.89.g
• 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:	$1 + 6 x + 89 x^{2}$
Frobenius angles:	$\pm0.603010988689$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-5}) \)
Galois group:	$C_2$
Jacobians:	$14$
Isomorphism classes:	14

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$96$	$8064$	$703584$	$62737920$	$5584208736$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$96$	$8064$	$703584$	$62737920$	$5584208736$	$496980779904$	$44231324675424$	$3936588912506880$	$350356403976245856$	$31181719918847992704$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which 0 are hyperelliptic):

• $y^2=x^3+76 x+50$
• $y^2=x^3+57 x+57$
• $y^2=x^3+25 x+75$
• $y^2=x^3+67 x+67$
• $y^2=x^3+40 x+31$
• $y^2=x^3+80 x+80$
• $y^2=x^3+27 x+27$
• $y^2=x^3+77 x+77$
• $y^2=x^3+25 x+25$
• $y^2=x^3+74 x+44$
• $y^2=x^3+36 x+19$
• $y^2=x^3+35 x+35$
• $y^2=x^3+71 x+35$
• $y^2=x^3+81 x+81$

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 \(\Q(\sqrt{-5}) \).

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.ag	$2$	(not in LMFDB)