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

• Label: 2.83.abg_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:	$\pm0.158801688027$, $\pm0.158801688027$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4624$	$46240000$	$326813448976$	$2252831296000000$	$15516830147685260944$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$52$	$6710$	$571564$	$47469678$	$3939240932$	$326942635430$	$27136070558684$	$2252292357492958$	$186940255648971412$	$15516041182904374550$

Jacobians and polarizations

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

• $y^2=39 x^6+73 x^5+31 x^4+3 x^3+31 x^2+73 x+39$
• $y^2=3 x^6+23 x^5+76 x^4+79 x^2+41 x$
• $y^2=65 x^6+11 x^5+72 x^4+80 x^3+72 x^2+11 x+65$
• $y^2=51 x^6+22 x^5+34 x^4+68 x^3+9 x^2+41 x+57$
• $y^2=79 x^6+20 x^5+81 x^4+47 x^3+81 x^2+20 x+79$
• $y^2=75 x^6+48 x^4+48 x^2+75$
• $y^2=80 x^6+68 x^5+45 x^4+45 x^2+68 x+80$
• $y^2=6 x^6+7 x^5+31 x^4+26 x^3+11 x^2+9 x+80$
• $y^2=4 x^6+64 x^5+58 x^4+29 x^3+13 x^2+19 x+3$
• $y^2=57 x^6+27 x^5+57 x^4+49 x^3+58 x^2+16 x+19$
• $y^2=38 x^6+75 x^4+75 x^2+38$
• $y^2=2 x^6+67 x^5+69 x^4+16 x^3+15 x^2+82 x+58$
• $y^2=43 x^6+42 x^4+42 x^2+43$
• $y^2=76 x^6+22 x^5+71 x^4+66 x^3+24 x^2+5 x+56$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{83}$

The isogeny class factors as 1.83.aq^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.a_adm	$2$	(not in LMFDB)
2.83.bg_qg	$2$	(not in LMFDB)
2.83.q_gr	$3$	(not in LMFDB)