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

• Label: 1.397.ak
• 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:	$1 - 10 x + 397 x^{2}$
Frobenius angles:	$\pm0.419259390957$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-93}) \)
Galois group:	$C_2$
Jacobians:	$12$
Isomorphism classes:	12

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$388$	$158304$	$62581684$	$24840430464$	$9861710966308$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$388$	$158304$	$62581684$	$24840430464$	$9861710966308$	$3915101639930976$	$1554295351066887124$	$617055253426333171200$	$244970935600776385650628$	$97253461433789493013748064$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which 0 are hyperelliptic):

• $y^2=x^3+245 x+93$
• $y^2=x^3+339 x+281$
• $y^2=x^3+161 x+161$
• $y^2=x^3+51 x+51$
• $y^2=x^3+96 x+192$
• $y^2=x^3+395 x+393$
• $y^2=x^3+355 x+313$
• $y^2=x^3+319 x+241$
• $y^2=x^3+223 x+223$
• $y^2=x^3+296 x+296$
• $y^2=x^3+360 x+360$
• $y^2=x^3+66 x+132$

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 \(\Q(\sqrt{-93}) \).

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.k	$2$	(not in LMFDB)