# Properties

 Label 20.459...249.70.a.a Dimension $20$ Group $S_7$ Conductor $4.597\times 10^{52}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $20$ Group: $S_7$ Conductor: $$459\!\cdots\!249$$$$\medspace = 184607^{10}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 7.1.184607.1 Galois orbit size: $1$ Smallest permutation container: 70 Parity: even Determinant: 1.1.1t1.a.a Projective image: $S_7$ Projective stem field: Galois closure of 7.1.184607.1

## Defining polynomial

 $f(x)$ $=$ $$x^{7} - x^{6} - x^{5} + x^{4} - x^{2} + x + 1$$ x^7 - x^6 - x^5 + x^4 - x^2 + x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $$x^{2} + 102x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$39 + 12\cdot 103 + 56\cdot 103^{2} + 59\cdot 103^{3} + 33\cdot 103^{4} +O(103^{5})$$ 39 + 12*103 + 56*103^2 + 59*103^3 + 33*103^4+O(103^5) $r_{ 2 }$ $=$ $$86 + 6\cdot 103 + 91\cdot 103^{2} + 65\cdot 103^{3} + 35\cdot 103^{4} +O(103^{5})$$ 86 + 6*103 + 91*103^2 + 65*103^3 + 35*103^4+O(103^5) $r_{ 3 }$ $=$ $$80 a + 78 + \left(100 a + 73\right)\cdot 103 + \left(57 a + 89\right)\cdot 103^{2} + \left(11 a + 79\right)\cdot 103^{3} + \left(19 a + 31\right)\cdot 103^{4} +O(103^{5})$$ 80*a + 78 + (100*a + 73)*103 + (57*a + 89)*103^2 + (11*a + 79)*103^3 + (19*a + 31)*103^4+O(103^5) $r_{ 4 }$ $=$ $$4 a + 14 + \left(22 a + 94\right)\cdot 103 + \left(53 a + 72\right)\cdot 103^{2} + \left(57 a + 83\right)\cdot 103^{3} + \left(97 a + 9\right)\cdot 103^{4} +O(103^{5})$$ 4*a + 14 + (22*a + 94)*103 + (53*a + 72)*103^2 + (57*a + 83)*103^3 + (97*a + 9)*103^4+O(103^5) $r_{ 5 }$ $=$ $$99 a + 18 + \left(80 a + 9\right)\cdot 103 + \left(49 a + 1\right)\cdot 103^{2} + \left(45 a + 88\right)\cdot 103^{3} + \left(5 a + 49\right)\cdot 103^{4} +O(103^{5})$$ 99*a + 18 + (80*a + 9)*103 + (49*a + 1)*103^2 + (45*a + 88)*103^3 + (5*a + 49)*103^4+O(103^5) $r_{ 6 }$ $=$ $$23 a + 55 + \left(2 a + 94\right)\cdot 103 + \left(45 a + 46\right)\cdot 103^{2} + \left(91 a + 33\right)\cdot 103^{3} + \left(83 a + 39\right)\cdot 103^{4} +O(103^{5})$$ 23*a + 55 + (2*a + 94)*103 + (45*a + 46)*103^2 + (91*a + 33)*103^3 + (83*a + 39)*103^4+O(103^5) $r_{ 7 }$ $=$ $$20 + 18\cdot 103 + 54\cdot 103^{2} + 103^{3} + 6\cdot 103^{4} +O(103^{5})$$ 20 + 18*103 + 54*103^2 + 103^3 + 6*103^4+O(103^5)

## Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

 Cycle notation $(1,2,3,4,5,6,7)$ $(1,2)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character value $1$ $1$ $()$ $20$ $21$ $2$ $(1,2)$ $0$ $105$ $2$ $(1,2)(3,4)(5,6)$ $0$ $105$ $2$ $(1,2)(3,4)$ $-4$ $70$ $3$ $(1,2,3)$ $2$ $280$ $3$ $(1,2,3)(4,5,6)$ $2$ $210$ $4$ $(1,2,3,4)$ $0$ $630$ $4$ $(1,2,3,4)(5,6)$ $0$ $504$ $5$ $(1,2,3,4,5)$ $0$ $210$ $6$ $(1,2,3)(4,5)(6,7)$ $2$ $420$ $6$ $(1,2,3)(4,5)$ $0$ $840$ $6$ $(1,2,3,4,5,6)$ $0$ $720$ $7$ $(1,2,3,4,5,6,7)$ $-1$ $504$ $10$ $(1,2,3,4,5)(6,7)$ $0$ $420$ $12$ $(1,2,3,4)(5,6,7)$ $0$

The blue line marks the conjugacy class containing complex conjugation.