# Properties

 Label 2.83.abk_sw 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: $( 1 - 18 x + 83 x^{2} )^{2}$ Frobenius angles: $\pm0.0496118990883$, $\pm0.0496118990883$ Angle rank: $1$ (numerical) Jacobians: 1

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 1 curves, and hence is principally polarizable:

• $y^2=x^6+71x^5+78x^4+78x^2+12x+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4356 45319824 325399511844 2251230714602496 15515337706155211716 106889563803136084469136 736365002320757424498321636 5072820162870579714748580511744 34946658985415969361029626940784516 240747534120087751567660403032270932624

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 48 6574 569088 47435950 3938862048 326939015518 27136041371760 2252292171719134 186940254978265104 15516041187013750414

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{83}$
 The isogeny class factors as 1.83.as 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$
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_agc $2$ (not in LMFDB) 2.83.bk_sw $2$ (not in LMFDB) 2.83.s_jh $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.83.a_agc $2$ (not in LMFDB) 2.83.bk_sw $2$ (not in LMFDB) 2.83.s_jh $3$ (not in LMFDB) 2.83.a_gc $4$ (not in LMFDB) 2.83.as_jh $6$ (not in LMFDB) 2.83.ae_i $8$ (not in LMFDB) 2.83.e_i $8$ (not in LMFDB)