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

• Label: 1.397.aw
• 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:	$1 - 22 x + 397 x^{2}$
Frobenius angles:	$\pm0.313836518327$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-69}) \)
Galois group:	$C_2$
Jacobians:	$24$
Isomorphism classes:	24

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$376$	$157920$	$62586328$	$24840816000$	$9861715607416$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$376$	$157920$	$62586328$	$24840816000$	$9861715607416$	$3915101517032160$	$1554295346593954648$	$617055253406015424000$	$244970935602368984321656$	$97253461433823604192197600$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which 0 are hyperelliptic):

• $y^2=x^3+180 x+180$
• $y^2=x^3+123 x+246$
• $y^2=x^3+101 x+202$
• $y^2=x^3+221 x+45$
• $y^2=x^3+196 x+196$
• $y^2=x^3+379 x+379$
• $y^2=x^3+218 x+39$
• $y^2=x^3+289 x+181$
• $y^2=x^3+254 x+111$
• $y^2=x^3+193 x+386$
• $y^2=x^3+277 x+277$
• $y^2=x^3+31 x+62$
• $y^2=x^3+214 x+31$
• $y^2=x^3+88 x+88$
• $y^2=x^3+62 x+124$
• $y^2=x^3+67 x+134$
• $y^2=x^3+53 x+53$
• $y^2=x^3+176 x+352$
• $y^2=x^3+178 x+178$
• $y^2=x^3+63 x+126$
• $y^2=x^3+30 x+60$
• $y^2=x^3+147 x+147$
• $y^2=x^3+89 x+89$
• $y^2=x^3+83 x+83$

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 \(\Q(\sqrt{-69}) \).

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.w	$2$	(not in LMFDB)