# Properties

 Label 2.83.abf_pj 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: $1 - 31 x + 399 x^{2} - 2573 x^{3} + 6889 x^{4}$ Frobenius angles: $\pm0.0177383414019$, $\pm0.251888863097$ Angle rank: $2$ (numerical) Number field: 4.0.145493.1 Galois group: $D_{4}$ Jacobians: 3

This isogeny class is simple and geometrically 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 3 curves, and hence is principally polarizable:

• $y^2=58x^6+13x^5+63x^4+70x^3+38x^2+28x+66$
• $y^2=8x^6+12x^5+41x^4+10x^3+76x^2+51x+5$
• $y^2=43x^6+7x^5+24x^4+49x^3+54x^2+29x+49$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4685 46348705 326709463535 2252308135425725 15515904909644702800 106889641255220509663705 736364793795425368849505435 5072819892529475099757365830325 34946658789707347453493011297731905 240747534026575025470440128914064454400

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 53 6727 571385 47458659 3939006048 326939252419 27136033687319 2252292051689811 186940253931360455 15516041180986908582

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{83}$
 The endomorphism algebra of this simple isogeny class is 4.0.145493.1.
All geometric endomorphisms are defined over $\F_{83}$.

## 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.bf_pj $2$ (not in LMFDB)