Abelian variety isogeny class 1.83.aj over $\F_{83}$

• Label: 1.83.aj
• Base field: $\F_{83}$
• 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:	$\F_{83}$
Dimension:	$1$
L-polynomial:	$1 - 9 x + 83 x^{2}$
Frobenius angles:	$\pm0.335556542753$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-251}) \)
Galois group:	$C_2$
Jacobians:	$7$
Isomorphism classes:	7

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$75$	$6975$	$573300$	$47464875$	$3938974125$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$75$	$6975$	$573300$	$47464875$	$3938974125$	$326939230800$	$27136046227575$	$2252292284113875$	$186940256130564300$	$15516041190659157375$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which 0 are hyperelliptic):

• $y^2=x^3+47 x+11$
• $y^2=x^3+69 x+69$
• $y^2=x^3+28 x+56$
• $y^2=x^3+13 x+13$
• $y^2=x^3+44 x+5$
• $y^2=x^3+66 x+66$
• $y^2=x^3+25 x+50$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{83}$.

Endomorphism algebra over $\F_{83}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-251}) \).

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
1.83.j	$2$	(not in LMFDB)