Abelian variety isogeny class 1.163.al over $\F_{163}$

• Label: 1.163.al
• Base field: $\F_{163}$
• 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:	$\F_{163}$
Dimension:	$1$
L-polynomial:	$1 - 11 x + 163 x^{2}$
Frobenius angles:	$\pm0.358233821351$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-59}) \)
Galois group:	$C_2$
Jacobians:	$9$
Isomorphism classes:	9

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$153$	$26775$	$4334796$	$705922875$	$115063079463$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$153$	$26775$	$4334796$	$705922875$	$115063079463$	$18755361853200$	$3057125243868261$	$498311415606445875$	$81224760547592909748$	$13239635966959293966375$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which 0 are hyperelliptic):

• $y^2=x^3+63 x+63$
• $y^2=x^3+143 x+143$
• $y^2=x^3+65 x+130$
• $y^2=x^3+64 x+64$
• $y^2=x^3+110 x+110$
• $y^2=x^3+54 x+54$
• $y^2=x^3+59 x+118$
• $y^2=x^3+150 x+150$
• $y^2=x^3+126 x+126$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{163}$.

Endomorphism algebra over $\F_{163}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-59}) \).

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
1.163.l	$2$	(not in LMFDB)