Abelian variety isogeny class 1.97.o over $\F_{97}$

• Label: 1.97.o
• Base field: $\F_{97}$
• 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:	$\F_{97}$
Dimension:	$1$
L-polynomial:	$1 + 14 x + 97 x^{2}$
Frobenius angles:	$\pm0.751640801674$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}) \)
Galois group:	$C_2$
Jacobians:	$8$
Isomorphism classes:	8
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$112$	$9408$	$911344$	$88548096$	$8587205872$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$112$	$9408$	$911344$	$88548096$	$8587205872$	$832972061376$	$80798296723312$	$7837433417468928$	$760231059943493488$	$73742412688607909568$

Jacobians and polarizations

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

• $y^2=x^3+9 x+9$
• $y^2=x^3+7 x+35$
• $y^2=x^3+8 x+8$
• $y^2=x^3+77 x+77$
• $y^2=x^3+19$
• $y^2=x^3+38 x+38$
• $y^2=x^3+60 x+60$
• $y^2=x^3+89 x+89$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{97}$.

Endomorphism algebra over $\F_{97}$

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
1.97.ao	$2$	(not in LMFDB)
1.97.at	$3$	(not in LMFDB)
1.97.f	$3$	(not in LMFDB)