# Properties

 Label 6.2e6_3e8_11e2.7t6.1c1 Dimension 6 Group $A_7$ Conductor $2^{6} \cdot 3^{8} \cdot 11^{2}$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $6$ Group: $A_7$ Conductor: $50808384= 2^{6} \cdot 3^{8} \cdot 11^{2}$ Artin number field: Splitting field of $f= x^{7} - 2 x^{6} + 2 x + 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $A_7$ Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $x^{2} + 102 x + 5$
Roots:
 $r_{ 1 }$ $=$ $21 a + 28 + \left(22 a + 5\right)\cdot 103 + \left(42 a + 60\right)\cdot 103^{2} + \left(13 a + 62\right)\cdot 103^{3} + \left(17 a + 70\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 2 }$ $=$ $67 + 7\cdot 103 + 16\cdot 103^{2} + 44\cdot 103^{3} + 46\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 3 }$ $=$ $84 + 21\cdot 103 + 32\cdot 103^{2} + 43\cdot 103^{3} + 59\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 4 }$ $=$ $82 a + 49 + \left(80 a + 6\right)\cdot 103 + \left(60 a + 80\right)\cdot 103^{2} + \left(89 a + 33\right)\cdot 103^{3} + \left(85 a + 74\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 5 }$ $=$ $66 + 36\cdot 103 + 56\cdot 103^{2} + 11\cdot 103^{3} + 73\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 6 }$ $=$ $77 a + 73 + \left(68 a + 16\right)\cdot 103 + \left(42 a + 45\right)\cdot 103^{2} + \left(20 a + 16\right)\cdot 103^{3} + \left(58 a + 25\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 7 }$ $=$ $26 a + 47 + \left(34 a + 8\right)\cdot 103 + \left(60 a + 19\right)\cdot 103^{2} + \left(82 a + 97\right)\cdot 103^{3} + \left(44 a + 62\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$

### 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 $1$ $1$ $()$ $6$ $105$ $2$ $(1,2)(3,4)$ $2$ $70$ $3$ $(1,2,3)$ $3$ $280$ $3$ $(1,2,3)(4,5,6)$ $0$ $630$ $4$ $(1,2,3,4)(5,6)$ $0$ $504$ $5$ $(1,2,3,4,5)$ $1$ $210$ $6$ $(1,2,3)(4,5)(6,7)$ $-1$ $360$ $7$ $(1,2,3,4,5,6,7)$ $-1$ $360$ $7$ $(1,3,4,5,6,7,2)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.