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

• Label: 2.83.abk_sw
• 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:	$( 1 - 18 x + 83 x^{2} )^{2}$
	$1 - 36 x + 490 x^{2} - 2988 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.0496118990883$, $\pm0.0496118990883$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4356$	$45319824$	$325399511844$	$2251230714602496$	$15515337706155211716$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$48$	$6574$	$569088$	$47435950$	$3938862048$	$326939015518$	$27136041371760$	$2252292171719134$	$186940254978265104$	$15516041187013750414$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=x^6+71x^5+78x^4+78x^2+12x+1$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{83}$

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

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_agc	$2$	(not in LMFDB)
2.83.bk_sw	$2$	(not in LMFDB)
2.83.s_jh	$3$	(not in LMFDB)