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

• Label: 2.83.ak_gr
• Base field: $\F_{83}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{83}$
Dimension:	$2$
L-polynomial:	$1 - 10 x + 173 x^{2} - 830 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.330659929291$, $\pm0.486765480163$
Angle rank:	$2$ (numerical)
Number field:	4.0.5230144.1
Galois group:	$D_{4}$
Jacobians:	$78$
Isomorphism classes:	78
Cyclic group of points:	yes

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6223$	$49180369$	$327913840996$	$2251993145710441$	$15515710120224123943$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$7136$	$573488$	$47452020$	$3938956594$	$326940339422$	$27136048248838$	$2252292184257124$	$186940255813721504$	$15516041198910248336$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 78 curves (of which all are hyperelliptic):

• $y^2=6 x^6+31 x^5+45 x^4+29 x^3+64 x^2+68 x+58$
• $y^2=52 x^6+67 x^5+55 x^4+8 x^3+40 x^2+31 x+67$
• $y^2=28 x^6+73 x^5+15 x^4+51 x^3+37 x^2+75 x+63$
• $y^2=24 x^6+47 x^5+72 x^4+59 x^3+71 x^2+59 x+75$
• $y^2=3 x^6+11 x^5+12 x^4+81 x^3+81 x^2+27 x+7$
• $y^2=41 x^6+25 x^5+22 x^4+16 x^3+15 x^2+27 x+25$
• $y^2=79 x^6+65 x^5+58 x^4+29 x^3+7 x^2+48 x+53$
• $y^2=7 x^6+31 x^5+42 x^4+75 x^3+5 x^2+10 x+76$
• $y^2=12 x^6+33 x^5+79 x^4+27 x^3+4 x^2+67 x+37$
• $y^2=6 x^6+47 x^5+80 x^4+64 x^3+39 x^2+49 x+63$
• $y^2=5 x^6+38 x^4+46 x^3+65 x^2+65 x+77$
• $y^2=54 x^6+18 x^5+67 x^4+20 x^3+54 x^2+59 x+65$
• $y^2=61 x^6+22 x^5+7 x^4+61 x^3+20 x^2+4 x+79$
• $y^2=45 x^6+40 x^5+45 x^4+26 x^3+69 x^2+60 x+31$
• $y^2=41 x^6+64 x^5+45 x^4+16 x^3+58 x^2+14 x+22$
• $y^2=64 x^6+12 x^5+61 x^4+34 x^3+44 x^2+6 x+24$
• $y^2=2 x^6+48 x^5+2 x^4+48 x^3+66 x^2+75 x+64$
• $y^2=6 x^6+73 x^5+72 x^4+48 x^3+65 x^2+47 x+52$
• $y^2=68 x^6+80 x^5+4 x^4+38 x^3+73 x^2+37 x+26$
• $y^2=7 x^6+55 x^5+61 x^4+39 x^3+20 x^2+36 x+50$
• and

• $y^2=55 x^6+55 x^5+81 x^4+32 x^3+41 x^2+77 x+42$
• $y^2=50 x^6+75 x^5+58 x^4+16 x^3+21 x^2+51 x+80$
• $y^2=47 x^6+19 x^5+15 x^4+2 x^3+x^2+59 x+81$
• $y^2=56 x^6+40 x^5+67 x^4+66 x^3+57 x^2+66 x+36$
• $y^2=16 x^6+50 x^5+39 x^4+42 x^3+53 x^2+50 x+28$
• $y^2=77 x^6+58 x^5+55 x^4+24 x^3+45 x^2+78 x+56$
• $y^2=28 x^6+23 x^5+31 x^4+3 x^3+48 x^2+74 x+20$
• $y^2=75 x^6+27 x^5+61 x^4+7 x^3+64 x^2+65 x+28$
• $y^2=39 x^6+67 x^5+43 x^4+68 x^3+61 x^2+18 x+34$
• $y^2=39 x^6+75 x^5+x^4+2 x^3+79 x^2+80 x+67$
• $y^2=10 x^6+11 x^5+82 x^4+77 x^3+70 x^2+57 x+56$
• $y^2=23 x^6+80 x^5+5 x^4+44 x^3+39 x^2+32 x+53$
• $y^2=77 x^6+17 x^5+12 x^4+16 x^3+3 x^2+20 x+38$
• $y^2=39 x^6+9 x^5+80 x^4+38 x^3+13 x^2+78 x+53$
• $y^2=18 x^6+28 x^5+68 x^4+19 x^3+41 x^2+8 x+50$
• $y^2=73 x^6+12 x^5+13 x^4+41 x^3+11 x^2+45 x+11$
• $y^2=50 x^6+63 x^5+54 x^4+78 x^3+47 x^2+78 x+17$
• $y^2=58 x^6+28 x^5+50 x^4+67 x^3+33 x^2+16 x+56$
• $y^2=32 x^6+6 x^5+34 x^4+55 x^3+74 x^2+80 x+71$
• $y^2=32 x^6+12 x^5+50 x^4+11 x^3+21 x^2+35 x+22$
• $y^2=77 x^6+29 x^5+21 x^4+11 x^3+28 x^2+8 x+22$
• $y^2=41 x^6+67 x^5+6 x^4+77 x^3+38 x^2+35 x+57$
• $y^2=35 x^6+61 x^5+70 x^4+67 x^3+75 x^2+38 x+16$
• $y^2=72 x^6+59 x^5+59 x^4+x^2+78 x+52$
• $y^2=22 x^6+62 x^4+5 x^3+42 x^2+39 x+15$
• $y^2=21 x^6+50 x^5+17 x^4+64 x^3+58 x^2+65 x+8$
• $y^2=50 x^6+35 x^5+40 x^4+17 x^3+41 x^2+21 x+42$
• $y^2=22 x^6+5 x^5+44 x^4+36 x^3+x^2+80 x+64$
• $y^2=42 x^6+60 x^5+31 x^4+52 x^3+8 x^2+57 x+15$
• $y^2=74 x^6+55 x^5+68 x^4+34 x^3+75 x^2+13 x+50$
• $y^2=54 x^6+38 x^5+65 x^4+31 x^3+36 x^2+29 x+35$
• $y^2=63 x^6+72 x^5+9 x^4+57 x^3+11 x^2+31 x+39$
• $y^2=67 x^6+21 x^5+29 x^4+64 x^3+60 x^2+25 x+55$
• $y^2=15 x^6+53 x^5+28 x^4+79 x^3+38 x^2+72 x+65$
• $y^2=62 x^6+61 x^5+64 x^4+15 x^3+62 x+70$
• $y^2=60 x^6+81 x^5+38 x^4+73 x^3+10 x^2+63 x+63$
• $y^2=6 x^6+65 x^5+14 x^4+30 x^3+80 x^2+81 x+64$
• $y^2=18 x^6+64 x^5+49 x^4+7 x^3+22 x^2+68 x+1$
• $y^2=74 x^6+52 x^5+66 x^4+3 x^3+50 x^2+51 x+33$
• $y^2=33 x^6+11 x^5+46 x^4+73 x^3+19 x^2+81 x+60$
• $y^2=58 x^6+7 x^5+48 x^4+11 x^3+45 x^2+60 x+74$
• $y^2=69 x^6+39 x^5+12 x^4+55 x^3+69 x^2+80 x+28$
• $y^2=22 x^6+13 x^5+30 x^4+70 x^3+54 x^2+43 x+79$
• $y^2=24 x^6+82 x^5+38 x^4+29 x^3+64 x^2+75 x+24$
• $y^2=50 x^6+6 x^5+79 x^4+67 x^3+63 x^2+67 x+17$
• $y^2=57 x^6+66 x^5+52 x^4+40 x^3+47 x^2+36 x+33$
• $y^2=14 x^6+67 x^5+54 x^3+57 x^2+44 x+59$
• $y^2=28 x^6+61 x^5+23 x^4+16 x^3+10 x^2+61 x+53$
• $y^2=19 x^6+40 x^5+52 x^4+75 x^3+36 x^2+15 x+72$
• $y^2=23 x^6+66 x^5+77 x^4+75 x^3+79 x^2+19 x+12$
• $y^2=x^6+35 x^5+58 x^4+41 x^3+43 x^2+x+67$
• $y^2=14 x^6+9 x^5+7 x^4+51 x^3+30 x^2+57 x+20$
• $y^2=54 x^6+56 x^5+64 x^4+62 x^3+69 x^2+74 x+18$
• $y^2=38 x^6+48 x^5+59 x^4+74 x^3+12 x^2+15 x+11$
• $y^2=10 x^6+28 x^5+47 x^4+71 x^3+4 x^2+10 x+57$
• $y^2=45 x^6+61 x^5+74 x^4+45 x^3+22 x^2+41 x+35$
• $y^2=32 x^6+19 x^5+9 x^4+58 x^3+10 x^2+36 x+32$
• $y^2=2 x^6+79 x^5+28 x^4+75 x^3+29 x^2+77 x+56$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{83}$

The endomorphism algebra of this simple isogeny class is 4.0.5230144.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.83.k_gr	$2$	(not in LMFDB)