# Properties

 Label 10.3e14_5e6_7e6.30t88.2c1 Dimension 10 Group $A_6$ Conductor $3^{14} \cdot 5^{6} \cdot 7^{6}$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $10$ Group: $A_6$ Conductor: $8792367498140625= 3^{14} \cdot 5^{6} \cdot 7^{6}$ Artin number field: Splitting field of $f= x^{6} + 3 x^{4} - 12 x^{2} - 15 x + 16$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $A_6$ Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $x^{2} + 29 x + 3$
Roots:
 $r_{ 1 }$ $=$ $9 + 8\cdot 31 + 20\cdot 31^{2} + 27\cdot 31^{3} + 2\cdot 31^{4} + 26\cdot 31^{5} +O\left(31^{ 6 }\right)$ $r_{ 2 }$ $=$ $19 a + 7 + \left(6 a + 23\right)\cdot 31 + \left(15 a + 7\right)\cdot 31^{2} + \left(10 a + 26\right)\cdot 31^{3} + 23 a\cdot 31^{4} + \left(14 a + 25\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ $r_{ 3 }$ $=$ $20 + 7\cdot 31 + 29\cdot 31^{2} + 14\cdot 31^{3} + 20\cdot 31^{4} + 9\cdot 31^{5} +O\left(31^{ 6 }\right)$ $r_{ 4 }$ $=$ $27 a + 10 + \left(8 a + 7\right)\cdot 31 + \left(a + 5\right)\cdot 31^{2} + \left(17 a + 26\right)\cdot 31^{3} + \left(6 a + 17\right)\cdot 31^{4} + \left(18 a + 16\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ $r_{ 5 }$ $=$ $12 a + 14 + \left(24 a + 17\right)\cdot 31 + 15 a\cdot 31^{2} + \left(20 a + 1\right)\cdot 31^{3} + \left(7 a + 6\right)\cdot 31^{4} + 16 a\cdot 31^{5} +O\left(31^{ 6 }\right)$ $r_{ 6 }$ $=$ $4 a + 2 + \left(22 a + 29\right)\cdot 31 + \left(29 a + 29\right)\cdot 31^{2} + \left(13 a + 27\right)\cdot 31^{3} + \left(24 a + 13\right)\cdot 31^{4} + \left(12 a + 15\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $10$ $45$ $2$ $(1,2)(3,4)$ $-2$ $40$ $3$ $(1,2,3)(4,5,6)$ $1$ $40$ $3$ $(1,2,3)$ $1$ $90$ $4$ $(1,2,3,4)(5,6)$ $0$ $72$ $5$ $(1,2,3,4,5)$ $0$ $72$ $5$ $(1,3,4,5,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.