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

• Label: 1.491.ay
• 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:	$1 - 24 x + 491 x^{2}$
Frobenius angles:	$\pm0.317836770105$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-347}) \)
Galois group:	$C_2$
Jacobians:	$20$
Isomorphism classes:	20

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$468$	$241488$	$118392300$	$58120365888$	$28536940889028$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$468$	$241488$	$118392300$	$58120365888$	$28536940889028$	$14011639200421200$	$6879714954732514908$	$3377940044748542477568$	$1658568561966234498539700$	$814357163924324277262104528$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which 0 are hyperelliptic):

• $y^2=x^3+279 x+279$
• $y^2=x^3+226 x+452$
• $y^2=x^3+79 x+158$
• $y^2=x^3+469 x+447$
• $y^2=x^3+242 x+484$
• $y^2=x^3+416 x+341$
• $y^2=x^3+162 x+162$
• $y^2=x^3+291 x+91$
• $y^2=x^3+49 x+49$
• $y^2=x^3+471 x+471$
• $y^2=x^3+355 x+355$
• $y^2=x^3+334 x+334$
• $y^2=x^3+379 x+379$
• $y^2=x^3+442 x+393$
• $y^2=x^3+200 x+400$
• $y^2=x^3+325 x+325$
• $y^2=x^3+321 x+321$
• $y^2=x^3+335 x+335$
• $y^2=x^3+417 x+417$
• $y^2=x^3+355 x+219$

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 \(\Q(\sqrt{-347}) \).

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.y	$2$	(not in LMFDB)