# Properties

 Label 3.3775.12t76.a Dimension $3$ Group $A_5\times C_2$ Conductor $3775$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5\times C_2$ Conductor: $$3775$$$$\medspace = 5^{2} \cdot 151$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 10.0.49064203594375.1 Galois orbit size: $2$ Smallest permutation container: 12T76 Parity: odd Projective image: $A_5$ Projective field: 5.1.570025.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $$x^{5} + 7 x + 28$$
Roots:
 $r_{ 1 }$ $=$ $$2 a^{4} + 29 a^{3} + 8 a^{2} + 16 a + 24 + \left(29 a^{4} + 29 a^{3} + 25 a + 7\right)\cdot 31 + \left(21 a^{4} + 11 a^{3} + 10 a^{2} + 3 a + 11\right)\cdot 31^{2} + \left(21 a^{4} + 8 a^{3} + 16 a^{2} + 27 a + 22\right)\cdot 31^{3} + \left(18 a^{4} + 10 a^{3} + 12 a^{2} + 28 a + 11\right)\cdot 31^{4} + \left(19 a^{4} + 25 a^{3} + 28 a^{2} + 11 a + 4\right)\cdot 31^{5} + \left(4 a^{4} + 29 a^{3} + 5 a^{2} + 12 a + 1\right)\cdot 31^{6} + \left(7 a^{4} + 12 a^{3} + 13 a^{2} + 23 a + 9\right)\cdot 31^{7} + \left(4 a^{4} + 20 a^{3} + 9 a^{2} + 18 a + 11\right)\cdot 31^{8} + \left(6 a^{4} + 16 a^{3} + 28 a^{2} + 29 a + 3\right)\cdot 31^{9} +O(31^{10})$$ $r_{ 2 }$ $=$ $$3 a^{4} + 21 a^{3} + 30 a^{2} + 19 a + 11 + \left(5 a^{4} + 25 a^{3} + 27 a^{2} + 29 a + 22\right)\cdot 31 + \left(18 a^{4} + 19 a^{3} + 15 a^{2} + 6 a + 8\right)\cdot 31^{2} + \left(28 a^{4} + 24 a^{3} + 11 a^{2} + a + 11\right)\cdot 31^{3} + \left(22 a^{4} + 19 a^{3} + 24 a^{2} + a + 4\right)\cdot 31^{4} + \left(18 a^{4} + 10 a^{3} + 15 a^{2} + a + 18\right)\cdot 31^{5} + \left(23 a^{4} + 19 a^{3} + 22 a^{2} + 7 a + 20\right)\cdot 31^{6} + \left(30 a^{4} + 26 a^{3} + 23 a^{2} + 17 a + 29\right)\cdot 31^{7} + \left(15 a^{4} + 22 a^{3} + 24 a^{2} + 20 a + 8\right)\cdot 31^{8} + \left(30 a^{4} + 29 a^{3} + 22 a^{2} + 28\right)\cdot 31^{9} +O(31^{10})$$ $r_{ 3 }$ $=$ $$8 a^{4} + 30 a^{3} + 22 a^{2} + 22 a + 8 + \left(29 a^{4} + 5 a^{3} + 27 a^{2} + 10 a + 21\right)\cdot 31 + \left(4 a^{4} + 3 a^{3} + 8 a^{2} + 17 a + 27\right)\cdot 31^{2} + \left(18 a^{4} + 8 a^{3} + a^{2} + 4 a + 14\right)\cdot 31^{3} + \left(22 a^{4} + 11 a^{3} + 4 a^{2} + 21 a + 2\right)\cdot 31^{4} + \left(25 a^{4} + 19 a^{3} + 28 a^{2} + 4 a + 20\right)\cdot 31^{5} + \left(19 a^{4} + 11 a^{3} + 11 a^{2} + 28 a + 5\right)\cdot 31^{6} + \left(7 a^{4} + 17 a^{3} + 9 a^{2} + 3 a + 24\right)\cdot 31^{7} + \left(20 a^{4} + 8 a^{3} + 29 a^{2} + 3 a + 1\right)\cdot 31^{8} + \left(22 a^{4} + 8 a^{3} + 9 a^{2} + 13 a + 9\right)\cdot 31^{9} +O(31^{10})$$ $r_{ 4 }$ $=$ $$17 a^{4} + 18 a^{3} + 22 a^{2} + 13 a + 15 + \left(6 a^{4} + 30 a^{3} + 10 a^{2} + 9 a + 30\right)\cdot 31 + \left(6 a^{4} + 8 a^{3} + 27 a^{2} + 7 a + 9\right)\cdot 31^{2} + \left(28 a^{4} + 5 a^{3} + 3 a^{2} + 3 a + 15\right)\cdot 31^{3} + \left(19 a^{4} + 18 a^{3} + 12 a^{2} + 24 a + 18\right)\cdot 31^{4} + \left(a^{4} + 29 a^{3} + 2 a^{2} + 2 a + 21\right)\cdot 31^{5} + \left(5 a^{4} + 13 a^{3} + 12 a^{2} + 20 a + 9\right)\cdot 31^{6} + \left(11 a^{4} + 29 a^{3} + 26 a^{2} + 28 a + 25\right)\cdot 31^{7} + \left(27 a^{4} + 6 a^{3} + 21 a^{2} + 4 a + 16\right)\cdot 31^{8} + \left(7 a^{4} + 20 a^{3} + 11 a^{2} + 12 a + 25\right)\cdot 31^{9} +O(31^{10})$$ $r_{ 5 }$ $=$ $$22 a^{4} + a^{3} + 14 a^{2} + 23 a + 12 + \left(8 a^{4} + 12 a^{3} + 7 a^{2} + 24 a + 5\right)\cdot 31 + \left(2 a^{4} + 25 a^{3} + 18 a^{2} + 25\right)\cdot 31^{2} + \left(2 a^{4} + 11 a^{2} + 5 a + 17\right)\cdot 31^{3} + \left(24 a^{4} + 2 a^{3} + 23 a^{2} + a + 4\right)\cdot 31^{4} + \left(26 a^{4} + 10 a^{3} + 15 a^{2} + 16 a + 1\right)\cdot 31^{5} + \left(26 a^{4} + 28 a^{3} + 2 a^{2} + 11 a + 14\right)\cdot 31^{6} + \left(25 a^{4} + 4 a^{3} + 27 a^{2} + 13 a + 8\right)\cdot 31^{7} + \left(8 a^{4} + 22 a^{3} + 23 a^{2} + 24 a + 12\right)\cdot 31^{8} + \left(13 a^{4} + 25 a^{3} + 23 a^{2} + 28 a + 12\right)\cdot 31^{9} +O(31^{10})$$ $r_{ 6 }$ $=$ $$23 a^{4} + 11 a^{3} + 19 a^{2} + 14 a + 30 + \left(29 a^{4} + 24 a^{3} + 5 a^{2} + 30 a + 23\right)\cdot 31 + \left(21 a^{4} + 29 a^{3} + 20 a^{2} + 5 a + 23\right)\cdot 31^{2} + \left(27 a^{4} + 29 a^{3} + 9 a^{2} + 10 a + 18\right)\cdot 31^{3} + \left(14 a^{4} + 11 a^{3} + 8 a^{2} + 28 a + 27\right)\cdot 31^{4} + \left(a^{4} + 7 a^{3} + 25 a^{2} + 16 a + 26\right)\cdot 31^{5} + \left(22 a^{4} + 12 a^{3} + 18 a^{2} + 14 a + 11\right)\cdot 31^{6} + \left(8 a^{4} + 28 a^{3} + 14 a^{2} + 13 a + 5\right)\cdot 31^{7} + \left(13 a^{4} + 19 a^{3} + 8 a^{2} + 26 a + 6\right)\cdot 31^{8} + \left(16 a^{4} + 12 a^{3} + 21 a^{2} + 24 a + 5\right)\cdot 31^{9} +O(31^{10})$$ $r_{ 7 }$ $=$ $$25 a^{4} + 10 a^{3} + 25 a^{2} + a + 4 + \left(9 a^{4} + 17 a^{3} + 6 a^{2} + 4 a + 30\right)\cdot 31 + \left(3 a^{3} + 6 a^{2} + 28 a + 7\right)\cdot 31^{2} + \left(10 a^{4} + 9 a^{3} + 8 a^{2}\right)\cdot 31^{3} + \left(11 a^{4} + 30 a^{3} + 9 a^{2} + 10 a + 20\right)\cdot 31^{4} + \left(9 a^{4} + 19 a^{3} + 10 a^{2} + 30 a + 27\right)\cdot 31^{5} + \left(27 a^{4} + 2 a^{3} + 24 a^{2} + 13 a + 28\right)\cdot 31^{6} + \left(11 a^{4} + 24 a^{3} + 16\right)\cdot 31^{7} + \left(a^{4} + 16 a^{3} + 30 a^{2} + 12 a + 1\right)\cdot 31^{8} + \left(13 a^{4} + 16 a^{3} + 26 a^{2} + 25 a + 11\right)\cdot 31^{9} +O(31^{10})$$ $r_{ 8 }$ $=$ $$26 a^{4} + 2 a^{3} + 14 a^{2} + 22 a + 22 + \left(30 a^{4} + 7 a^{3} + 27 a + 17\right)\cdot 31 + \left(25 a^{4} + 28 a^{3} + 7 a^{2} + 27 a + 21\right)\cdot 31^{2} + \left(27 a^{4} + 21 a^{3} + 23 a^{2} + 18 a + 25\right)\cdot 31^{3} + \left(13 a^{4} + 8 a^{3} + 12 a^{2} + 13 a + 15\right)\cdot 31^{4} + \left(27 a^{4} + 30 a^{3} + 26 a^{2} + 27 a + 23\right)\cdot 31^{5} + \left(22 a^{4} + 19 a^{3} + 2 a^{2} + 30 a + 22\right)\cdot 31^{6} + \left(7 a^{4} + 7 a^{3} + a^{2} + 3 a + 24\right)\cdot 31^{7} + \left(8 a^{4} + 21 a^{3} + 21 a^{2} + 24 a + 2\right)\cdot 31^{8} + \left(17 a^{4} + 25 a^{3} + 10 a^{2} + 24 a + 16\right)\cdot 31^{9} +O(31^{10})$$ $r_{ 9 }$ $=$ $$30 a^{4} + 4 a^{3} + 22 a^{2} + a + 1 + \left(22 a^{4} + 8 a^{3} + 20 a^{2} + 2 a + 17\right)\cdot 31 + \left(26 a^{4} + 29 a^{3} + 6 a^{2} + 12 a + 19\right)\cdot 31^{2} + \left(24 a^{4} + 10 a^{3} + 25 a^{2} + 28 a + 27\right)\cdot 31^{3} + \left(22 a^{3} + 24 a^{2} + 29 a + 16\right)\cdot 31^{4} + \left(28 a^{4} + 10 a^{3} + 16 a^{2} + 13 a + 20\right)\cdot 31^{5} + \left(24 a^{4} + 12 a^{3} + 9 a^{2} + 6 a + 21\right)\cdot 31^{6} + \left(7 a^{4} + 22 a^{3} + 22 a^{2} + 22 a + 12\right)\cdot 31^{7} + \left(26 a^{4} + 21 a^{3} + 21 a^{2} + 21 a + 4\right)\cdot 31^{8} + \left(23 a^{4} + 24 a^{3} + 28 a^{2} + 6 a + 28\right)\cdot 31^{9} +O(31^{10})$$ $r_{ 10 }$ $=$ $$30 a^{4} + 29 a^{3} + 10 a^{2} + 24 a + 1 + \left(13 a^{4} + 24 a^{3} + 16 a^{2} + 21 a + 10\right)\cdot 31 + \left(26 a^{4} + 25 a^{3} + 3 a^{2} + 13 a + 30\right)\cdot 31^{2} + \left(27 a^{4} + 4 a^{3} + 13 a^{2} + 24 a\right)\cdot 31^{3} + \left(5 a^{4} + 20 a^{3} + 23 a^{2} + 27 a + 2\right)\cdot 31^{4} + \left(27 a^{4} + 22 a^{3} + 16 a^{2} + 29 a + 22\right)\cdot 31^{5} + \left(8 a^{4} + 4 a^{3} + 13 a^{2} + 9 a + 18\right)\cdot 31^{6} + \left(5 a^{4} + 12 a^{3} + 16 a^{2} + 28 a + 29\right)\cdot 31^{7} + \left(29 a^{4} + 25 a^{3} + 26 a^{2} + 29 a + 26\right)\cdot 31^{8} + \left(3 a^{4} + 5 a^{3} + a^{2} + 19 a + 15\right)\cdot 31^{9} +O(31^{10})$$

### Generators of the action on the roots $r_1, \ldots, r_{ 10 }$

 Cycle notation $(1,7)(2,5)(3,9)(4,8)(6,10)$ $(2,3,6)(5,9,10)$ $(1,4,6,7,8,10)(2,5)(3,9)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 10 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $3$ $3$ $1$ $2$ $(1,7)(2,5)(3,9)(4,8)(6,10)$ $-3$ $-3$ $15$ $2$ $(1,4)(2,10)(3,9)(5,6)(7,8)$ $1$ $1$ $15$ $2$ $(1,8)(2,6)(4,7)(5,10)$ $-1$ $-1$ $20$ $3$ $(1,6,8)(4,7,10)$ $0$ $0$ $12$ $5$ $(1,3,8,2,6)(4,5,10,7,9)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $12$ $5$ $(1,2,3,6,8)(4,7,5,9,10)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $20$ $6$ $(1,4,6,7,8,10)(2,5)(3,9)$ $0$ $0$ $12$ $10$ $(1,5,3,10,8,7,2,9,6,4)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $12$ $10$ $(1,10,2,4,3,7,6,5,8,9)$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$
The blue line marks the conjugacy class containing complex conjugation.