Abelian variety isogeny class 1.491.abh over $\F_{491}$

• Label: 1.491.abh
• Base field: $\F_{491}$
• 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:	$\F_{491}$
Dimension:	$1$
L-polynomial:	$1 - 33 x + 491 x^{2}$
Frobenius angles:	$\pm0.232623770170$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-35}) \)
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$459$	$240975$	$118383444$	$58120519275$	$28536953155029$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$459$	$240975$	$118383444$	$58120519275$	$28536953155029$	$14011639503296400$	$6879714956664370839$	$3377940044627667539475$	$1658568561961436983082124$	$814357163924246300417049375$

Jacobians and polarizations

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

• $y^2=x^3+231 x+462$
• $y^2=x^3+190 x+190$
• $y^2=x^3+333 x+175$
• $y^2=x^3+135 x+135$
• $y^2=x^3+308 x+125$
• $y^2=x^3+325 x+159$
• $y^2=x^3+159 x+318$
• $y^2=x^3+67 x+67$
• $y^2=x^3+228 x+228$
• $y^2=x^3+344 x+344$
• $y^2=x^3+244 x+244$
• $y^2=x^3+257 x+257$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{491}$.

Endomorphism algebra over $\F_{491}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-35}) \).

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
1.491.bh	$2$	(not in LMFDB)