# Properties

 Label 35.19e15_11149e15.70.1c1 Dimension 35 Group $S_7$ Conductor $19^{15} \cdot 11149^{15}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $35$ Group: $S_7$ Conductor: $77596609597017151433303632978312635660217736946038571721852745996145332043946951= 19^{15} \cdot 11149^{15}$ 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: Odd Determinant: 1.19_11149.2t1.1c1

## 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:
 $r_{ 1 }$ $=$ $261 + 63\cdot 263 + 35\cdot 263^{2} + 254\cdot 263^{3} + 54\cdot 263^{4} +O\left(263^{ 5 }\right)$ $r_{ 2 }$ $=$ $228 a + 251 + \left(258 a + 231\right)\cdot 263 + \left(61 a + 97\right)\cdot 263^{2} + \left(184 a + 242\right)\cdot 263^{3} + \left(15 a + 215\right)\cdot 263^{4} +O\left(263^{ 5 }\right)$ $r_{ 3 }$ $=$ $85 a + 177 + \left(48 a + 161\right)\cdot 263 + \left(5 a + 34\right)\cdot 263^{2} + \left(226 a + 27\right)\cdot 263^{3} + \left(260 a + 223\right)\cdot 263^{4} +O\left(263^{ 5 }\right)$ $r_{ 4 }$ $=$ $184 + 198\cdot 263 + 69\cdot 263^{2} + 140\cdot 263^{3} +O\left(263^{ 5 }\right)$ $r_{ 5 }$ $=$ $35 a + 181 + \left(4 a + 258\right)\cdot 263 + \left(201 a + 225\right)\cdot 263^{2} + \left(78 a + 22\right)\cdot 263^{3} + \left(247 a + 63\right)\cdot 263^{4} +O\left(263^{ 5 }\right)$ $r_{ 6 }$ $=$ $178 a + 84 + \left(214 a + 173\right)\cdot 263 + \left(257 a + 259\right)\cdot 263^{2} + \left(36 a + 210\right)\cdot 263^{3} + \left(2 a + 255\right)\cdot 263^{4} +O\left(263^{ 5 }\right)$ $r_{ 7 }$ $=$ $178 + 226\cdot 263 + 65\cdot 263^{2} + 154\cdot 263^{3} + 238\cdot 263^{4} +O\left(263^{ 5 }\right)$

### 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 $1$ $1$ $()$ $35$ $21$ $2$ $(1,2)$ $5$ $105$ $2$ $(1,2)(3,4)(5,6)$ $1$ $105$ $2$ $(1,2)(3,4)$ $-1$ $70$ $3$ $(1,2,3)$ $-1$ $280$ $3$ $(1,2,3)(4,5,6)$ $-1$ $210$ $4$ $(1,2,3,4)$ $-1$ $630$ $4$ $(1,2,3,4)(5,6)$ $1$ $504$ $5$ $(1,2,3,4,5)$ $0$ $210$ $6$ $(1,2,3)(4,5)(6,7)$ $-1$ $420$ $6$ $(1,2,3)(4,5)$ $-1$ $840$ $6$ $(1,2,3,4,5,6)$ $1$ $720$ $7$ $(1,2,3,4,5,6,7)$ $0$ $504$ $10$ $(1,2,3,4,5)(6,7)$ $0$ $420$ $12$ $(1,2,3,4)(5,6,7)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.