# Properties

 Label 2.83.abf_pn 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: $1 - 31 x + 403 x^{2} - 2573 x^{3} + 6889 x^{4}$ Frobenius angles: $\pm0.101476209234$, $\pm0.229218298306$ Angle rank: $2$ (numerical) Number field: 4.0.795821.3 Galois group: $D_{4}$ Jacobians: 12

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 12 curves, and hence is principally polarizable:

• $y^2=12x^6+68x^5+71x^4+45x^3+59x^2+13x+6$
• $y^2=67x^6+53x^5+46x^4+13x^3+55x^2+15x+63$
• $y^2=57x^6+44x^5+11x^4+18x^3+16x^2+69x+40$
• $y^2=78x^6+62x^5+76x^4+46x^3+2x^2+35x+47$
• $y^2=72x^6+33x^5+5x^4+2x^3+11x^2+14x+13$
• $y^2=48x^6+76x^5+72x^4+64x^3+60x^2+27x+5$
• $y^2=60x^6+53x^5+57x^4+9x^3+26x^2+23x+58$
• $y^2=24x^5+22x^4+65x^3+44x^2+53x+61$
• $y^2=53x^6+21x^5+60x^4+46x^3+6x^2+66x+80$
• $y^2=13x^6+7x^5+68x^4+20x^3+32x^2+12x+54$
• $y^2=20x^6+60x^5+25x^4+21x^3+12x^2+15x+29$
• $y^2=22x^6+33x^5+16x^4+68x^3+74x^2+19x+19$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4689 46407033 326923058475 2252733493647789 15516495934019559504 106890275864917346130225 736365345644157546859080111 5072820291169978109059863210069 34946659036684125576587004387745725 240747534170918087344880719193221494528

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 53 6735 571757 47467619 3939156088 326941193475 27136054023691 2252292228683059 186940255252513991 15516041190289736550

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{83}$
 The endomorphism algebra of this simple isogeny class is 4.0.795821.3.
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_pn $2$ (not in LMFDB)