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

• Label: 1.97.aq
• 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:	$1 - 16 x + 97 x^{2}$
Frobenius angles:	$\pm0.198227810371$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-33}) \)
Galois group:	$C_2$
Jacobians:	$4$
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$82$	$9348$	$913234$	$88544256$	$8587525522$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$82$	$9348$	$913234$	$88544256$	$8587525522$	$832973516676$	$80798290695442$	$7837433547214848$	$760231057296890578$	$73742412672344756868$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which 0 are hyperelliptic):

• $y^2=x^3+18 x+90$
• $y^2=x^3+15 x+15$
• $y^2=x^3+94 x+94$
• $y^2=x^3+16 x+80$

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 \(\Q(\sqrt{-33}) \).

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
1.97.q	$2$	(not in LMFDB)