# Properties

 Label 6.5_53_1327.7t7.1c1 Dimension 6 Group $S_7$ Conductor $5 \cdot 53 \cdot 1327$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $6$ Group: $S_7$ Conductor: $351655= 5 \cdot 53 \cdot 1327$ Artin number field: Splitting field of $f= x^{7} - x^{6} - 2 x^{4} + x^{3} + x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_7$ Parity: Odd Determinant: 1.5_53_1327.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 659 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $198 + 23\cdot 659 + 152\cdot 659^{2} + 170\cdot 659^{3} + 92\cdot 659^{4} +O\left(659^{ 5 }\right)$ $r_{ 2 }$ $=$ $307 + 109\cdot 659 + 175\cdot 659^{2} + 260\cdot 659^{3} + 652\cdot 659^{4} +O\left(659^{ 5 }\right)$ $r_{ 3 }$ $=$ $312 + 7\cdot 659 + 100\cdot 659^{2} + 302\cdot 659^{3} + 90\cdot 659^{4} +O\left(659^{ 5 }\right)$ $r_{ 4 }$ $=$ $357 + 397\cdot 659 + 594\cdot 659^{2} + 483\cdot 659^{3} + 535\cdot 659^{4} +O\left(659^{ 5 }\right)$ $r_{ 5 }$ $=$ $457 + 9\cdot 659 + 2\cdot 659^{2} + 251\cdot 659^{3} + 433\cdot 659^{4} +O\left(659^{ 5 }\right)$ $r_{ 6 }$ $=$ $480 + 621\cdot 659 + 17\cdot 659^{2} + 489\cdot 659^{3} + 569\cdot 659^{4} +O\left(659^{ 5 }\right)$ $r_{ 7 }$ $=$ $526 + 148\cdot 659 + 276\cdot 659^{2} + 20\cdot 659^{3} + 262\cdot 659^{4} +O\left(659^{ 5 }\right)$

### 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 $1$ $1$ $()$ $6$ $21$ $2$ $(1,2)$ $4$ $105$ $2$ $(1,2)(3,4)(5,6)$ $0$ $105$ $2$ $(1,2)(3,4)$ $2$ $70$ $3$ $(1,2,3)$ $3$ $280$ $3$ $(1,2,3)(4,5,6)$ $0$ $210$ $4$ $(1,2,3,4)$ $2$ $630$ $4$ $(1,2,3,4)(5,6)$ $0$ $504$ $5$ $(1,2,3,4,5)$ $1$ $210$ $6$ $(1,2,3)(4,5)(6,7)$ $-1$ $420$ $6$ $(1,2,3)(4,5)$ $1$ $840$ $6$ $(1,2,3,4,5,6)$ $0$ $720$ $7$ $(1,2,3,4,5,6,7)$ $-1$ $504$ $10$ $(1,2,3,4,5)(6,7)$ $-1$ $420$ $12$ $(1,2,3,4)(5,6,7)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.