Abelian variety isogeny class 1.409.ar over $\F_{409}$

• Label: 1.409.ar
• Base field: $\F_{409}$
• 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:	$\F_{409}$
Dimension:	$1$
L-polynomial:	$1 - 17 x + 409 x^{2}$
Frobenius angles:	$\pm0.361925556984$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-1347}) \)
Galois group:	$C_2$
Jacobians:	$6$
Isomorphism classes:	6

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$393$	$167811$	$68433876$	$27982987683$	$11445013989393$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$393$	$167811$	$68433876$	$27982987683$	$11445013989393$	$4681012891209984$	$1914534320827981497$	$783044537152791705603$	$320265215674608163979124$	$130988473210586650891096851$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which 0 are hyperelliptic):

• $y^2=x^3+231 x+390$
• $y^2=x^3+305 x+90$
• $y^2=x^3+27 x+27$
• $y^2=x^3+179 x+179$
• $y^2=x^3+296 x+27$
• $y^2=x^3+51 x+51$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{409}$.

Endomorphism algebra over $\F_{409}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1347}) \).

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
1.409.r	$2$	(not in LMFDB)