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

• Label: 1.409.aq
• 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:	$1 - 16 x + 409 x^{2}$
Frobenius angles:	$\pm0.370545491461$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-345}) \)
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$394$	$167844$	$68433466$	$27982951680$	$11445013526314$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$394$	$167844$	$68433466$	$27982951680$	$11445013526314$	$4681012904117604$	$1914534321341341786$	$783044537155435729920$	$320265215674387532531914$	$130988473210581257691072804$

Jacobians and polarizations

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

• $y^2=x^3+170 x+170$
• $y^2=x^3+163 x+323$
• $y^2=x^3+269 x+269$
• $y^2=x^3+183 x+183$
• $y^2=x^3+209 x+236$
• $y^2=x^3+101 x+101$
• $y^2=x^3+35 x+245$
• $y^2=x^3+189 x+189$

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 \(\Q(\sqrt{-345}) \).

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.q	$2$	(not in LMFDB)