# Properties

 Label 2.83.abg_qg Base Field $\F_{83}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{83}$ Dimension: $2$ L-polynomial: $( 1 - 16 x + 83 x^{2} )^{2}$ Frobenius angles: $\pm0.158801688027$, $\pm0.158801688027$ Angle rank: $1$ (numerical) Jacobians: 14

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 14 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4624 46240000 326813448976 2252831296000000 15516830147685260944 106890747299335219360000 736365794338687930037512336 5072820581287523460473856000000 34946659110797977803551557216904464 240747534056326506412743217552996000000

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

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{83}$
 The isogeny class factors as 1.83.aq 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-19})$$$)$
All geometric endomorphisms are defined over $\F_{83}$.

## 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)
Below is a list of all 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) 2.83.a_dm $4$ (not in LMFDB) 2.83.aq_gr $6$ (not in LMFDB)