Abelian variety isogeny class 2.83.abf_pj over $\F_{83}$

• Label: 2.83.abf_pj
• Base field: $\F_{83}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{83}$
Dimension:	$2$
L-polynomial:	$1 - 31 x + 399 x^{2} - 2573 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.0177383414019$, $\pm0.251888863097$
Angle rank:	$2$ (numerical)
Number field:	4.0.145493.1
Galois group:	$D_{4}$
Jacobians:	$3$
Isomorphism classes:	3

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:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4685$	$46348705$	$326709463535$	$2252308135425725$	$15515904909644702800$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$53$	$6727$	$571385$	$47458659$	$3939006048$	$326939252419$	$27136033687319$	$2252292051689811$	$186940253931360455$	$15516041180986908582$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=58x^6+13x^5+63x^4+70x^3+38x^2+28x+66$
• $y^2=8x^6+12x^5+41x^4+10x^3+76x^2+51x+5$
• $y^2=43x^6+7x^5+24x^4+49x^3+54x^2+29x+49$

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 4.0.145493.1.

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
2.83.bf_pj	$2$	(not in LMFDB)