Abelian variety isogeny class 1.397.az over $\F_{397}$

• Label: 1.397.az
• Base field: $\F_{397}$
• 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:	$\F_{397}$
Dimension:	$1$
L-polynomial:	$1 - 25 x + 397 x^{2}$
Frobenius angles:	$\pm0.284136688472$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-107}) \)
Galois group:	$C_2$
Jacobians:	$9$
Isomorphism classes:	9

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$373$	$157779$	$62584924$	$24840883539$	$9861718510633$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$373$	$157779$	$62584924$	$24840883539$	$9861718510633$	$3915101558736576$	$1554295346133631789$	$617055253371855886275$	$244970935601702749541068$	$97253461433823039420684939$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which 0 are hyperelliptic):

• $y^2=x^3+301 x+301$
• $y^2=x^3+265 x+265$
• $y^2=x^3+303 x+303$
• $y^2=x^3+104 x+208$
• $y^2=x^3+x+1$
• $y^2=x^3+151 x+151$
• $y^2=x^3+145 x+290$
• $y^2=x^3+265 x+133$
• $y^2=x^3+25 x+25$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{397}$.

Endomorphism algebra over $\F_{397}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-107}) \).

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
1.397.z	$2$	(not in LMFDB)