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

• Label: 2.83.bg_qg
• 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:	$( 1 + 16 x + 83 x^{2} )^{2}$
	$1 + 32 x + 422 x^{2} + 2656 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.841198311973$, $\pm0.841198311973$
Angle rank:	$1$ (numerical)
Jacobians:	$14$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10000$	$46240000$	$327069610000$	$2252831296000000$	$15515252262540250000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$116$	$6710$	$572012$	$47469678$	$3938840356$	$326942635430$	$27136031420572$	$2252292357492958$	$186940254886109396$	$15516041182904374550$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

• $y^2=78 x^6+63 x^5+62 x^4+6 x^3+62 x^2+63 x+78$
• $y^2=43 x^6+53 x^5+38 x^4+81 x^2+62 x$
• $y^2=47 x^6+22 x^5+61 x^4+77 x^3+61 x^2+22 x+47$
• $y^2=67 x^6+11 x^5+17 x^4+34 x^3+46 x^2+62 x+70$
• $y^2=75 x^6+40 x^5+79 x^4+11 x^3+79 x^2+40 x+75$
• $y^2=67 x^6+13 x^4+13 x^2+67$
• $y^2=40 x^6+34 x^5+64 x^4+64 x^2+34 x+40$
• $y^2=3 x^6+45 x^5+57 x^4+13 x^3+47 x^2+46 x+40$
• $y^2=2 x^6+32 x^5+29 x^4+56 x^3+48 x^2+51 x+43$
• $y^2=70 x^6+55 x^5+70 x^4+66 x^3+29 x^2+8 x+51$
• $y^2=19 x^6+79 x^4+79 x^2+19$
• $y^2=4 x^6+51 x^5+55 x^4+32 x^3+30 x^2+81 x+33$
• $y^2=3 x^6+x^4+x^2+3$
• $y^2=38 x^6+11 x^5+77 x^4+33 x^3+12 x^2+44 x+28$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{83}$

The isogeny class factors as 1.83.q^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$

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.abg_qg	$2$	(not in LMFDB)
2.83.a_adm	$2$	(not in LMFDB)
2.83.aq_gr	$3$	(not in LMFDB)