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

• Label: 2.83.au_kg
• 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:	$( 1 - 10 x + 83 x^{2} )^{2}$
	$1 - 20 x + 266 x^{2} - 1660 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.315076740302$, $\pm0.315076740302$
Angle rank:	$1$ (numerical)
Jacobians:	$20$
Cyclic group of points:	no
Non-cyclic primes:	$2, 37$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5476$	$48385936$	$328647665284$	$2253186720449536$	$15515809186457305636$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$64$	$7022$	$574768$	$47477166$	$3938981744$	$326938220318$	$27136034347808$	$2252292244424158$	$186940256771662624$	$15516041201227411022$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

• $y^2=66 x^6+7 x^5+42 x^4+43 x^3+28 x^2+48 x+60$
• $y^2=40 x^6+23 x^5+29 x^4+54 x^3+44 x^2+64 x+51$
• $y^2=62 x^6+2 x^5+40 x^4+8 x^3+77 x^2+52 x+55$
• $y^2=60 x^6+38 x^5+67 x^4+62 x^3+39 x^2+50 x+29$
• $y^2=16 x^6+68 x^5+75 x^4+72 x^3+52 x^2+33 x+43$
• $y^2=78 x^6+33 x^5+68 x^4+15 x^3+65 x^2+68 x+60$
• $y^2=5 x^6+3 x^5+46 x^4+74 x^3+54 x^2+81 x+18$
• $y^2=18 x^6+13 x^5+74 x^4+4 x^3+24 x^2+74 x+46$
• $y^2=40 x^6+15 x^5+25 x^4+25 x^3+36 x^2+48 x+58$
• $y^2=5 x^6+24 x^5+33 x^4+12 x^3+42 x^2+70 x+25$
• $y^2=7 x^6+50 x^5+2 x^4+65 x^3+6 x^2+65 x+43$
• $y^2=57 x^6+55 x^4+55 x^2+57$
• $y^2=67 x^6+48 x^5+55 x^4+4 x^3+47 x^2+38 x+80$
• $y^2=74 x^6+62 x^5+49 x^4+69 x^3+72 x^2+69 x+67$
• $y^2=12 x^6+70 x^4+63 x^3+67 x^2+35 x+79$
• $y^2=72 x^6+13 x^5+12 x^4+13 x^3+71 x^2+39 x+34$
• $y^2=58 x^6+52 x^5+50 x^4+77 x^3+50 x^2+52 x+58$
• $y^2=24 x^6+67 x^5+79 x^4+7 x^3+14 x^2+49 x+2$
• $y^2=62 x^6+15 x^5+46 x^4+78 x^3+8 x^2+11 x+6$
• $y^2=60 x^6+52 x^5+16 x^4+81 x^3+9 x^2+53 x+12$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{83}$

The isogeny class factors as 1.83.ak^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-58}) \)$)$

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_co	$2$	(not in LMFDB)
2.83.u_kg	$2$	(not in LMFDB)
2.83.k_r	$3$	(not in LMFDB)