# Properties

 Label 2.83.abe_pb 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: $( 1 - 15 x + 83 x^{2} )^{2}$ Frobenius angles: $\pm0.192168636682$, $\pm0.192168636682$ Angle rank: $1$ (numerical) Jacobians: 18

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 18 curves, and hence is principally polarizable:

• $y^2=78x^6+30x^5+31x^4+4x^3+31x^2+30x+78$
• $y^2=2x^6+69x^5+38x^4+38x^2+69x+2$
• $y^2=18x^6+52x^5+13x^4+37x^3+x+13$
• $y^2=60x^6+64x^5+68x^4+46x^3+11x^2+37x+42$
• $y^2=x^6+2x^5+48x^4+66x^3+39x^2+13x+47$
• $y^2=15x^6+13x^5+2x^4+49x^3+2x^2+13x+15$
• $y^2=43x^6+21x^5+35x^4+50x^3+35x^2+21x+43$
• $y^2=30x^6+79x^5+61x^4+7x^3+61x^2+79x+30$
• $y^2=57x^6+17x^5+8x^4+61x^3+8x^2+17x+57$
• $y^2=78x^6+56x^5+75x^4+59x^3+52x^2+8x+22$
• $y^2=54x^6+24x^5+70x^4+73x^3+23x^2+82x+79$
• $y^2=13x^6+63x^5+7x^4+23x^3+7x^2+63x+13$
• $y^2=55x^6+61x^5+78x^4+80x^3+7x^2+10x+35$
• $y^2=61x^6+67x^5+45x^4+21x^3+33x^2+33x+67$
• $y^2=35x^6+12x^5+63x^4+72x^3+13x^2+3x+79$
• $y^2=13x^6+75x^5+33x^4+37x^3+79x^2+54x+37$
• $y^2=16x^6+70x^5+72x^4+39x^3+72x^2+70x+16$
• $y^2=19x^6+41x^5+73x^4+4x^3+73x^2+41x+19$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4761 46662561 327353333904 2253269789767161 15517022622579317961 106890670758419443374336 736365527610609821192597169 5072820248900876298544704875625 34946658826054520474617686195605136 240747533885956964146479803383446900321

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 54 6772 572508 47478916 3939289794 326942401318 27136060729398 2252292209915908 186940254125792484 15516041171924154772

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{83}$
 The isogeny class factors as 1.83.ap 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-107})$$$)$
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_ach $2$ (not in LMFDB) 2.83.be_pb $2$ (not in LMFDB) 2.83.p_fm $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.83.a_ach $2$ (not in LMFDB) 2.83.be_pb $2$ (not in LMFDB) 2.83.p_fm $3$ (not in LMFDB) 2.83.a_ch $4$ (not in LMFDB) 2.83.ap_fm $6$ (not in LMFDB)