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

• Label: 1.409.i
• 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:	$1 + 8 x + 409 x^{2}$
Frobenius angles:	$\pm0.563375524330$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-393}) \)
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$418$	$168036$	$68408626$	$27982699008$	$11445025258018$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$418$	$168036$	$68408626$	$27982699008$	$11445025258018$	$4681013058920484$	$1914534317813406322$	$783044537101051619328$	$320265215674930758963874$	$130988473210585689553429476$

Jacobians and polarizations

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

• $y^2=x^3+86 x+86$
• $y^2=x^3+40 x+40$
• $y^2=x^3+135 x+135$
• $y^2=x^3+301 x+62$
• $y^2=x^3+78 x+78$
• $y^2=x^3+388 x+388$
• $y^2=x^3+235 x+235$
• $y^2=x^3+233 x+404$
• $y^2=x^3+144 x+190$
• $y^2=x^3+49 x+343$
• $y^2=x^3+262 x+262$
• $y^2=x^3+108 x+347$

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 \(\Q(\sqrt{-393}) \).

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.ai	$2$	(not in LMFDB)