# Properties

 Label 20.19e10_11149e10.70.1c1 Dimension 20 Group $S_7$ Conductor $19^{10} \cdot 11149^{10}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $20$ Group: $S_7$ Conductor: $181926164762284206907343555768608459562430993045958801= 19^{10} \cdot 11149^{10}$ Artin number field: Splitting field of $f=x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{2} - 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 70 Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 263 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 263 }$: $x^{2} + 261 x + 5$
Roots: \begin{aligned} r_{ 1 } &= 262977995577 +O\left(263^{ 5 }\right) \\ r_{ 2 } &= 75116772054 a - 225239726357 +O\left(263^{ 5 }\right) \\ r_{ 3 } &= -10241426107 a - 190880459105 +O\left(263^{ 5 }\right) \\ r_{ 4 } &= 2551627499 +O\left(263^{ 5 }\right) \\ r_{ 5 } &= -75116772054 a + 301829928237 +O\left(263^{ 5 }\right) \\ r_{ 6 } &= 10241426107 a - 34436640264 +O\left(263^{ 5 }\right) \\ r_{ 7 } &= -116802725586 +O\left(263^{ 5 }\right) \\ \end{aligned}

### 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.