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

• Label: 2.83.abc_ny
• 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:	$( 1 - 14 x + 83 x^{2} )^{2}$
	$1 - 28 x + 362 x^{2} - 2324 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.221078141621$, $\pm0.221078141621$
Angle rank:	$1$ (numerical)
Jacobians:	$26$
Cyclic group of points:	no
Non-cyclic primes:	$2, 5, 7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4900$	$47059600$	$327790600900$	$2253514829440000$	$15516976384692422500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$56$	$6830$	$573272$	$47484078$	$3939278056$	$326941559390$	$27136047888712$	$2252292090286558$	$186940253538981656$	$15516041174779787150$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):

• $y^2=36 x^6+81 x^5+17 x^4+12 x^3+38 x^2+59 x+25$
• $y^2=45 x^6+69 x^5+30 x^4+10 x^3+30 x^2+69 x+45$
• $y^2=17 x^6+60 x^5+15 x^4+38 x^3+32 x^2+13 x+37$
• $y^2=30 x^6+32 x^4+32 x^2+30$
• $y^2=34 x^6+24 x^5+43 x^4+59 x^3+43 x^2+24 x+34$
• $y^2=62 x^6+79 x^5+x^4+82 x^3+41 x^2+82 x+13$
• $y^2=82 x^6+73 x^5+24 x^4+15 x^3+62 x^2+52 x+66$
• $y^2=46 x^6+36 x^5+55 x^4+55 x^3+55 x^2+36 x+46$
• $y^2=21 x^6+6 x^5+65 x^4+50 x^3+69 x^2+57 x+13$
• $y^2=58 x^6+30 x^5+76 x^4+6 x^3+43 x^2+46 x+62$
• $y^2=6 x^6+43 x^5+69 x^4+54 x^3+9 x^2+58 x+5$
• $y^2=73 x^6+27 x^5+47 x^4+5 x^3+24 x^2+50 x+68$
• $y^2=60 x^6+42 x^5+3 x^4+79 x^3+3 x^2+42 x+60$
• $y^2=55 x^6+65 x^4+65 x^2+55$
• $y^2=19 x^6+56 x^5+42 x^4+46 x^3+35 x^2+10 x+60$
• $y^2=50 x^6+37 x^5+46 x^4+22 x^3+34 x^2+29 x+79$
• $y^2=78 x^6+69 x^5+56 x^4+82 x^3+2 x^2+76 x+14$
• $y^2=53 x^6+4 x^5+57 x^4+3 x^3+3 x^2+18 x+24$
• $y^2=8 x^6+62 x^5+14 x^4+53 x^3+67 x^2+20 x+20$
• $y^2=82 x^6+68 x^5+46 x^4+30 x^3+41 x^2+27 x+71$
• $y^2=44 x^6+66 x^5+53 x^4+75 x^3+11 x^2+67 x$
• $y^2=71 x^6+78 x^5+71 x^4+47 x^3+27 x^2+24 x+26$
• $y^2=51 x^6+17 x^5+69 x^4+54 x^3+69 x^2+17 x+51$
• $y^2=43 x^6+37 x^5+21 x^4+77 x^3+21 x^2+37 x+43$
• $y^2=6 x^6+24 x^5+32 x^4+12 x^3+36 x^2+68 x+15$
• $y^2=30 x^6+29 x^5+22 x^4+12 x^3+78 x^2+x+8$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{83}$

The isogeny class factors as 1.83.ao^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-34}) \)$)$

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_abe	$2$	(not in LMFDB)
2.83.bc_ny	$2$	(not in LMFDB)
2.83.o_ej	$3$	(not in LMFDB)