# Properties

 Label 21.255...176.42t418.a.a Dimension $21$ Group $S_7$ Conductor $2.555\times 10^{39}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $21$ Group: $S_7$ Conductor: $$255\!\cdots\!176$$$$\medspace = 2^{58} \cdot 3^{46}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 7.1.241864704.2 Galois orbit size: $1$ Smallest permutation container: 42T418 Parity: odd Determinant: 1.4.2t1.a.a Projective image: $S_7$ Projective stem field: Galois closure of 7.1.241864704.2

## Defining polynomial

 $f(x)$ $=$ $$x^{7} - 3x^{6} + 3x^{5} + x^{4} - 3x^{3} - 3x^{2} - 5x - 3$$ x^7 - 3*x^6 + 3*x^5 + x^4 - 3*x^3 - 3*x^2 - 5*x - 3 .

The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: $$x^{2} + 149x + 6$$

Roots:
 $r_{ 1 }$ $=$ $$102 + 81\cdot 151 + 14\cdot 151^{2} + 114\cdot 151^{3} + 106\cdot 151^{4} +O(151^{5})$$ 102 + 81*151 + 14*151^2 + 114*151^3 + 106*151^4+O(151^5) $r_{ 2 }$ $=$ $$66 a + 48 + \left(89 a + 135\right)\cdot 151 + \left(10 a + 61\right)\cdot 151^{2} + \left(123 a + 12\right)\cdot 151^{3} + \left(61 a + 86\right)\cdot 151^{4} +O(151^{5})$$ 66*a + 48 + (89*a + 135)*151 + (10*a + 61)*151^2 + (123*a + 12)*151^3 + (61*a + 86)*151^4+O(151^5) $r_{ 3 }$ $=$ $$72 + 54\cdot 151 + 103\cdot 151^{2} + 147\cdot 151^{3} + 37\cdot 151^{4} +O(151^{5})$$ 72 + 54*151 + 103*151^2 + 147*151^3 + 37*151^4+O(151^5) $r_{ 4 }$ $=$ $$26 + 52\cdot 151 + 8\cdot 151^{2} + 43\cdot 151^{3} + 46\cdot 151^{4} +O(151^{5})$$ 26 + 52*151 + 8*151^2 + 43*151^3 + 46*151^4+O(151^5) $r_{ 5 }$ $=$ $$85 a + 29 + \left(61 a + 97\right)\cdot 151 + \left(140 a + 144\right)\cdot 151^{2} + \left(27 a + 96\right)\cdot 151^{3} + \left(89 a + 86\right)\cdot 151^{4} +O(151^{5})$$ 85*a + 29 + (61*a + 97)*151 + (140*a + 144)*151^2 + (27*a + 96)*151^3 + (89*a + 86)*151^4+O(151^5) $r_{ 6 }$ $=$ $$83 + 58\cdot 151 + 116\cdot 151^{2} + 130\cdot 151^{3} + 14\cdot 151^{4} +O(151^{5})$$ 83 + 58*151 + 116*151^2 + 130*151^3 + 14*151^4+O(151^5) $r_{ 7 }$ $=$ $$96 + 124\cdot 151 + 3\cdot 151^{2} + 59\cdot 151^{3} + 74\cdot 151^{4} +O(151^{5})$$ 96 + 124*151 + 3*151^2 + 59*151^3 + 74*151^4+O(151^5)

## 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 $1$ $1$ $()$ $21$ $21$ $2$ $(1,2)$ $-1$ $105$ $2$ $(1,2)(3,4)(5,6)$ $3$ $105$ $2$ $(1,2)(3,4)$ $1$ $70$ $3$ $(1,2,3)$ $-3$ $280$ $3$ $(1,2,3)(4,5,6)$ $0$ $210$ $4$ $(1,2,3,4)$ $1$ $630$ $4$ $(1,2,3,4)(5,6)$ $-1$ $504$ $5$ $(1,2,3,4,5)$ $1$ $210$ $6$ $(1,2,3)(4,5)(6,7)$ $1$ $420$ $6$ $(1,2,3)(4,5)$ $-1$ $840$ $6$ $(1,2,3,4,5,6)$ $0$ $720$ $7$ $(1,2,3,4,5,6,7)$ $0$ $504$ $10$ $(1,2,3,4,5)(6,7)$ $-1$ $420$ $12$ $(1,2,3,4)(5,6,7)$ $1$

The blue line marks the conjugacy class containing complex conjugation.