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

• Label: 1.397.g
• 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:	$1 + 6 x + 397 x^{2}$
Frobenius angles:	$\pm0.548109501029$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-97}) \)
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$404$	$158368$	$62563844$	$24840337536$	$9861721269044$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$404$	$158368$	$62563844$	$24840337536$	$9861721269044$	$3915101710934176$	$1554295346452866596$	$617055253386767774208$	$244970935602493764293588$	$97253461433806885721764768$

Jacobians and polarizations

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

• $y^2=x^3+52 x+52$
• $y^2=x^3+102 x+204$
• $y^2=x^3+257 x+117$
• $y^2=x^3+138 x+138$
• $y^2=x^3+288 x+179$
• $y^2=x^3+211 x+25$
• $y^2=x^3+34 x+34$
• $y^2=x^3+286 x+286$
• $y^2=x^3+185 x+370$
• $y^2=x^3+112 x+224$
• $y^2=x^3+349 x+349$
• $y^2=x^3+372 x+372$

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 \(\Q(\sqrt{-97}) \).

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.ag	$2$	(not in LMFDB)