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

• Label: 2.83.ak_hj
• 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:	$( 1 - 5 x + 83 x^{2} )^{2}$
	$1 - 10 x + 191 x^{2} - 830 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.411517245350$, $\pm0.411517245350$
Angle rank:	$1$ (numerical)
Jacobians:	$42$
Cyclic group of points:	no
Non-cyclic primes:	$79$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6241$	$49434961$	$328223576464$	$2251713088023961$	$15515068464242974561$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$7172$	$574028$	$47446116$	$3938793694$	$326940151718$	$27136070378218$	$2252292347479108$	$186940254234987764$	$15516041172469864772$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 42 curves (of which all are hyperelliptic):

• $y^2=74 x^6+47 x^5+76 x^4+19 x^3+21 x^2+5 x+71$
• $y^2=72 x^6+61 x^5+59 x^4+38 x^3+59 x^2+61 x+72$
• $y^2=16 x^6+37 x^5+50 x^4+51 x^3+50 x^2+37 x+16$
• $y^2=15 x^6+53 x^5+38 x^4+7 x^3+52 x^2+63 x+73$
• $y^2=19 x^6+4 x^5+73 x^4+49 x^3+2 x^2+80 x+28$
• $y^2=4 x^6+62 x^5+49 x^4+73 x^3+49 x^2+62 x+4$
• $y^2=59 x^6+18 x^5+43 x^4+3 x^3+39 x+64$
• $y^2=43 x^6+73 x^5+10 x^4+69 x^3+36 x^2+30 x+73$
• $y^2=3 x^6+22 x^5+13 x^3+64 x^2+23 x+71$
• $y^2=22 x^6+9 x^5+11 x^4+21 x^3+3 x^2+44 x+28$
• $y^2=35 x^6+68 x^5+51 x^4+13 x^3+6 x^2+73 x+25$
• $y^2=17 x^6+63 x^5+17 x^4+10 x^3+17 x^2+63 x+17$
• $y^2=59 x^6+10 x^5+78 x^4+81 x^3+76 x^2+82 x+39$
• $y^2=69 x^6+51 x^5+55 x^4+5 x^3+28 x^2+20 x+54$
• $y^2=4 x^6+29 x^5+47 x^4+41 x^3+47 x^2+29 x+4$
• $y^2=58 x^6+42 x^5+44 x^4+3 x^3+15 x+69$
• $y^2=29 x^6+12 x^5+12 x^4+72 x^3+59 x^2+43 x+15$
• $y^2=82 x^6+81 x^5+52 x^4+52 x^2+81 x+82$
• $y^2=26 x^6+72 x^5+36 x^4+26 x^3+36 x^2+72 x+26$
• $y^2=36 x^6+25 x^5+17 x^4+66 x^3+60 x^2+5 x+34$
• and

• $y^2=21 x^6+65 x^5+4 x^4+71 x^3+28 x^2+19 x+50$
• $y^2=56 x^6+18 x^5+60 x^4+72 x^3+80 x^2+38 x+66$
• $y^2=36 x^6+46 x^5+19 x^4+5 x^3+63 x^2+80 x+35$
• $y^2=54 x^6+60 x^5+31 x^4+12 x^3+31 x^2+60 x+54$
• $y^2=2 x^6+15 x^5+80 x^4+74 x^3+21 x^2+77 x+23$
• $y^2=35 x^6+80 x^5+43 x^4+62 x^3+43 x^2+80 x+35$
• $y^2=75 x^6+9 x^5+32 x^4+77 x^3+56 x^2+2 x+34$
• $y^2=47 x^6+54 x^5+5 x^4+43 x^3+21 x^2+6 x+20$
• $y^2=70 x^6+67 x^5+81 x^4+37 x^3+16 x^2+20 x+5$
• $y^2=21 x^6+57 x^5+70 x^4+16 x^3+40 x^2+77 x+73$
• $y^2=78 x^6+29 x^5+58 x^4+13 x^3+58 x^2+29 x+78$
• $y^2=74 x^6+40 x^5+39 x^4+59 x^3+2 x^2+78 x+23$
• $y^2=38 x^6+65 x^5+81 x^4+30 x^3+52 x^2+76 x+44$
• $y^2=35 x^6+63 x^5+43 x^4+20 x^3+43 x^2+63 x+35$
• $y^2=50 x^6+31 x^5+6 x^4+41 x^3+28 x^2+62 x+39$
• $y^2=41 x^6+37 x^5+61 x^4+59 x^3+8 x^2+74 x+68$
• $y^2=42 x^6+15 x^5+18 x^4+7 x^3+44 x^2+26 x+58$
• $y^2=24 x^6+46 x^5+35 x^4+63 x^3+77 x^2+74 x+20$
• $y^2=18 x^6+29 x^5+24 x^4+25 x^3+5 x^2+29 x+20$
• $y^2=24 x^6+19 x^5+27 x^4+30 x^3+27 x^2+19 x+24$
• $y^2=26 x^6+20 x^5+19 x^4+71 x^3+56 x^2+23 x+20$
• $y^2=30 x^6+65 x^5+3 x^4+81 x^3+64 x^2+31 x+7$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{83}$

The isogeny class factors as 1.83.af^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-307}) \)$)$

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_fl	$2$	(not in LMFDB)
2.83.k_hj	$2$	(not in LMFDB)
2.83.f_acg	$3$	(not in LMFDB)