Abelian variety isogeny class 1.431.aw over $\F_{431}$

• Label: 1.431.aw
• Base field: $\F_{431}$
• 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:	$\F_{431}$
Dimension:	$1$
L-polynomial:	$1 - 22 x + 431 x^{2}$
Frobenius angles:	$\pm0.322247710563$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-310}) \)
Galois group:	$C_2$
Jacobians:	$8$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$410$	$186140$	$80080790$	$34507377760$	$14872578630250$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$410$	$186140$	$80080790$	$34507377760$	$14872578630250$	$6410082371223260$	$2762745567202367590$	$1190743340475670035840$	$513210379739162260140410$	$221193673667014265784033500$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which 0 are hyperelliptic):

• $y^2=x^3+357 x+357$
• $y^2=x^3+15 x+15$
• $y^2=x^3+264 x+264$
• $y^2=x^3+296 x+348$
• $y^2=x^3+107 x+318$
• $y^2=x^3+211 x+211$
• $y^2=x^3+329 x+148$
• $y^2=x^3+159 x+159$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{431}$.

Endomorphism algebra over $\F_{431}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-310}) \).

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
1.431.w	$2$	(not in LMFDB)