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

• Label: 1.397.ac
• 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:	$1 - 2 x + 397 x^{2}$
Frobenius angles:	$\pm0.484017770379$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-11}) \)
Galois group:	$C_2$
Jacobians:	$30$
Isomorphism classes:	30

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$396$	$158400$	$62573148$	$24840288000$	$9861715401516$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$396$	$158400$	$62573148$	$24840288000$	$9861715401516$	$3915101753323200$	$1554295349483986428$	$617055253358620032000$	$244970935601093481004236$	$97253461433823004079352000$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which 0 are hyperelliptic):

• $y^2=x^3+48 x+96$
• $y^2=x^3+230 x+230$
• $y^2=x^3+289 x+289$
• $y^2=x^3+76 x+152$
• $y^2=x^3+37 x+37$
• $y^2=x^3+188 x+376$
• $y^2=x^3+349 x+301$
• $y^2=x^3+160 x+320$
• $y^2=x^3+156 x+312$
• $y^2=x^3+69 x+69$
• $y^2=x^3+149 x+149$
• $y^2=x^3+352 x+352$
• $y^2=x^3+375 x+375$
• $y^2=x^3+4 x+4$
• $y^2=x^3+157 x+157$
• $y^2=x^3+6 x+12$
• $y^2=x^3+39 x+78$
• $y^2=x^3+266 x+266$
• $y^2=x^3+386 x+386$
• $y^2=x^3+389 x+389$
• $y^2=x^3+354 x+354$
• $y^2=x^3+62 x+62$
• $y^2=x^3+305 x+213$
• $y^2=x^3+180 x+360$
• $y^2=x^3+373 x+349$
• $y^2=x^3+58 x+116$
• $y^2=x^3+131 x+262$
• $y^2=x^3+388 x+388$
• $y^2=x^3+245 x+245$
• $y^2=x^3+373 x+373$

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 \(\Q(\sqrt{-11}) \).

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.c	$2$	(not in LMFDB)