# Properties

 Label 10.149e6_211e6.70.1c1 Dimension 10 Group $A_7$ Conductor $149^{6} \cdot 211^{6}$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $10$ Group: $A_7$ Conductor: $965633540591074128803235361= 149^{6} \cdot 211^{6}$ Artin number field: Splitting field of $f= x^{7} - 2 x^{6} - 7 x^{5} + 11 x^{4} + 16 x^{3} - 14 x^{2} - 11 x + 2$ over $\Q$ Size of Galois orbit: 2 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 $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $x^{2} + 49 x + 2$
Roots:
 $r_{ 1 }$ $=$ $44 + 46\cdot 53 + 14\cdot 53^{2} + 25\cdot 53^{3} + 50\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 2 }$ $=$ $5 a + 28 + \left(18 a + 26\right)\cdot 53 + \left(41 a + 19\right)\cdot 53^{2} + \left(29 a + 22\right)\cdot 53^{3} + \left(14 a + 17\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 3 }$ $=$ $49 + 42\cdot 53 + 16\cdot 53^{2} + 6\cdot 53^{3} + 22\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 4 }$ $=$ $48 a + 48 + \left(34 a + 40\right)\cdot 53 + \left(11 a + 7\right)\cdot 53^{2} + \left(23 a + 47\right)\cdot 53^{3} + \left(38 a + 45\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 5 }$ $=$ $16 a + 30 + \left(13 a + 11\right)\cdot 53 + \left(26 a + 41\right)\cdot 53^{2} + \left(33 a + 29\right)\cdot 53^{3} + \left(36 a + 3\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 6 }$ $=$ $37 a + 41 + \left(39 a + 48\right)\cdot 53 + \left(26 a + 26\right)\cdot 53^{2} + \left(19 a + 31\right)\cdot 53^{3} + \left(16 a + 10\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 7 }$ $=$ $27 + 47\cdot 53 + 31\cdot 53^{2} + 49\cdot 53^{3} + 8\cdot 53^{4} +O\left(53^{ 5 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character value $1$ $1$ $()$ $10$ $105$ $2$ $(1,2)(3,4)$ $-2$ $70$ $3$ $(1,2,3)$ $1$ $280$ $3$ $(1,2,3)(4,5,6)$ $1$ $630$ $4$ $(1,2,3,4)(5,6)$ $0$ $504$ $5$ $(1,2,3,4,5)$ $0$ $210$ $6$ $(1,2,3)(4,5)(6,7)$ $1$ $360$ $7$ $(1,2,3,4,5,6,7)$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ $360$ $7$ $(1,3,4,5,6,7,2)$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
The blue line marks the conjugacy class containing complex conjugation.