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

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

Invariants

Base field:	$\F_{83}$
Dimension:	$2$
L-polynomial:	$( 1 - 17 x + 83 x^{2} )^{2}$
	$1 - 34 x + 455 x^{2} - 2822 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.117184483028$, $\pm0.117184483028$
Angle rank:	$1$ (numerical)
Jacobians:	$2$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4489$	$45792289$	$326164347664$	$2252164096494841$	$15516304899219329689$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$50$	$6644$	$570428$	$47455620$	$3939107590$	$326941735718$	$27136068593026$	$2252292418321924$	$186940256971567364$	$15516041200721132564$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=61x^6+34x^5+61x^4+50x^3+61x^2+34x+61$
• $y^2=4x^6+78x^5+35x^4+x^3+63x^2+50x+23$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{83}$

The isogeny class factors as 1.83.ar^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.83.a_aet	$2$	(not in LMFDB)
2.83.bi_rn	$2$	(not in LMFDB)
2.83.r_hy	$3$	(not in LMFDB)