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

• Label: 1.97.g
• 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:	$1 + 6 x + 97 x^{2}$
Frobenius angles:	$\pm0.598524067447$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-22}) \)
Galois group:	$C_2$
Jacobians:	$6$
Isomorphism classes:	6
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$104$	$9568$	$911144$	$88523136$	$8587525544$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$104$	$9568$	$911144$	$88523136$	$8587525544$	$832971489376$	$80798269598696$	$7837433733662208$	$760231059262157288$	$73742412672336604768$

Jacobians and polarizations

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

• $y^2=x^3+85 x+85$
• $y^2=x^3+78 x+2$
• $y^2=x^3+90 x+90$
• $y^2=x^3+73 x+73$
• $y^2=x^3+19 x+19$
• $y^2=x^3+20 x+20$

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 \(\Q(\sqrt{-22}) \).

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.ag	$2$	(not in LMFDB)