# Properties

 Label 2.83.abe_oz 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: $1 - 30 x + 389 x^{2} - 2490 x^{3} + 6889 x^{4}$ Frobenius angles: $\pm0.142946520854$, $\pm0.232154927040$ Angle rank: $2$ (numerical) Number field: 4.0.590400.4 Galois group: $D_{4}$ Jacobians: 8

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 8 curves, and hence is principally polarizable:

• $y^2=77x^6+50x^5+71x^4+16x^3+15x^2+10x+65$
• $y^2=73x^6+61x^5+44x^4+63x^3+70x^2+56x+25$
• $y^2=x^6+74x^5+62x^4+15x^3+64x^2+69x+18$
• $y^2=6x^6+30x^5+57x^4+77x^3+39x^2+62x+60$
• $y^2=18x^6+55x^5+7x^4+56x^3+82x^2+73x+39$
• $y^2=73x^6+x^5+67x^4+54x^3+73x^2+62x+59$
• $y^2=76x^6+64x^5+44x^4+4x^3+54x^2+66x+57$
• $y^2=53x^6+60x^5+16x^4+29x^3+27x^2+12x+58$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4759 46633441 327249988996 2253076095172041 15516782728097419039 106890468271837432652176 736365437766878761288262311 5072820298716888275866325534025 34946658984642507326210066724785764 240747534084939352117680232914609509521

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 54 6768 572328 47474836 3939228894 326941781982 27136057418538 2252292232033828 186940254974127624 15516041184748456128

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{83}$
 The endomorphism algebra of this simple isogeny class is 4.0.590400.4.
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.be_oz $2$ (not in LMFDB)