# Properties

 Label 10.3e14_29e6.30t88.2c1 Dimension 10 Group $A_6$ Conductor $3^{14} \cdot 29^{6}$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $10$ Group: $A_6$ Conductor: $2845021504820049= 3^{14} \cdot 29^{6}$ Artin number field: Splitting field of $f= x^{6} - 3 x^{5} - 3 x^{4} + 11 x^{3} + 6 x^{2} + 75 x + 50$ 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 $\Q_{ 19 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $x^{2} + 18 x + 2$
Roots:
 $r_{ 1 }$ $=$ $4 + 19 + 15\cdot 19^{2} + 17\cdot 19^{4} + 19^{5} + 19^{6} + 5\cdot 19^{7} +O\left(19^{ 8 }\right)$ $r_{ 2 }$ $=$ $3 a + 4 + \left(6 a + 2\right)\cdot 19 + \left(16 a + 10\right)\cdot 19^{2} + \left(2 a + 6\right)\cdot 19^{3} + \left(a + 4\right)\cdot 19^{4} + \left(5 a + 18\right)\cdot 19^{5} + \left(15 a + 18\right)\cdot 19^{6} + \left(14 a + 1\right)\cdot 19^{7} +O\left(19^{ 8 }\right)$ $r_{ 3 }$ $=$ $13 + 18\cdot 19 + 19^{2} + 2\cdot 19^{3} + 19^{4} + 5\cdot 19^{5} + 4\cdot 19^{6} + 18\cdot 19^{7} +O\left(19^{ 8 }\right)$ $r_{ 4 }$ $=$ $9 a + 2 + \left(5 a + 7\right)\cdot 19 + \left(11 a + 11\right)\cdot 19^{2} + \left(18 a + 4\right)\cdot 19^{3} + \left(15 a + 17\right)\cdot 19^{4} + \left(18 a + 12\right)\cdot 19^{5} + \left(7 a + 16\right)\cdot 19^{6} + \left(a + 8\right)\cdot 19^{7} +O\left(19^{ 8 }\right)$ $r_{ 5 }$ $=$ $16 a + 7 + \left(12 a + 5\right)\cdot 19 + \left(2 a + 1\right)\cdot 19^{2} + \left(16 a + 12\right)\cdot 19^{3} + \left(17 a + 2\right)\cdot 19^{4} + \left(13 a + 3\right)\cdot 19^{5} + \left(3 a + 10\right)\cdot 19^{6} + \left(4 a + 1\right)\cdot 19^{7} +O\left(19^{ 8 }\right)$ $r_{ 6 }$ $=$ $10 a + 11 + \left(13 a + 3\right)\cdot 19 + \left(7 a + 17\right)\cdot 19^{2} + 11\cdot 19^{3} + \left(3 a + 14\right)\cdot 19^{4} + 15\cdot 19^{5} + \left(11 a + 5\right)\cdot 19^{6} + \left(17 a + 2\right)\cdot 19^{7} +O\left(19^{ 8 }\right)$

### 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.