Abelian variety isogeny class 1.347.au over $\F_{347}$

• Label: 1.347.au
• Base field: $\F_{347}$
• 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:	$\F_{347}$
Dimension:	$1$
L-polynomial:	$1 - 20 x + 347 x^{2}$
Frobenius angles:	$\pm0.319622357604$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-247}) \)
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$328$	$120704$	$41794744$	$14498481664$	$5030918205608$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$328$	$120704$	$41794744$	$14498481664$	$5030918205608$	$1745729008789376$	$605767992940002584$	$210201493952151705600$	$72939918400105196986888$	$25310151684671190586504064$

Jacobians and polarizations

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

• $y^2=x^3+81 x+81$
• $y^2=x^3+126 x+252$
• $y^2=x^3+99 x+198$
• $y^2=x^3+148 x+296$
• $y^2=x^3+202 x+57$
• $y^2=x^3+112 x+112$
• $y^2=x^3+175 x+175$
• $y^2=x^3+291 x+291$
• $y^2=x^3+310 x+310$
• $y^2=x^3+62 x+62$
• $y^2=x^3+344 x+344$
• $y^2=x^3+156 x+312$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{347}$.

Endomorphism algebra over $\F_{347}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-247}) \).

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
1.347.u	$2$	(not in LMFDB)