# Properties

 Label 7.2e6_3e8_7e7.8t43.1c1 Dimension 7 Group $\PGL(2,7)$ Conductor $2^{6} \cdot 3^{8} \cdot 7^{7}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $7$ Group: $\PGL(2,7)$ Conductor: $345808999872= 2^{6} \cdot 3^{8} \cdot 7^{7}$ Artin number field: Splitting field of $f= x^{8} - 3 x^{7} + 14 x^{5} - 84 x^{3} + 112 x^{2} - 24 x - 12$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $\PGL(2,7)$ Parity: Odd Determinant: 1.7.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 36.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $x^{3} + 2 x + 27$
Roots:
 $r_{ 1 }$ $=$ $4 + 3\cdot 29 + 21\cdot 29^{2} + 29^{3} + 28\cdot 29^{4} + 11\cdot 29^{5} + 11\cdot 29^{6} + 28\cdot 29^{7} + 6\cdot 29^{8} + 20\cdot 29^{9} + 21\cdot 29^{10} + 10\cdot 29^{11} + 11\cdot 29^{12} + 21\cdot 29^{13} + 2\cdot 29^{14} + 13\cdot 29^{15} + 11\cdot 29^{16} + 21\cdot 29^{17} + 28\cdot 29^{18} + 25\cdot 29^{19} + 13\cdot 29^{20} + 10\cdot 29^{21} + 13\cdot 29^{22} + 6\cdot 29^{23} + 21\cdot 29^{24} + 25\cdot 29^{25} + 25\cdot 29^{26} + 13\cdot 29^{27} + 8\cdot 29^{28} + 21\cdot 29^{29} + 19\cdot 29^{30} + 15\cdot 29^{31} + 2\cdot 29^{32} + 16\cdot 29^{33} + 10\cdot 29^{34} + 16\cdot 29^{35} +O\left(29^{ 36 }\right)$ $r_{ 2 }$ $=$ $21 a^{2} + 25 a + 20 + \left(8 a^{2} + 19 a + 16\right)\cdot 29 + \left(26 a^{2} + 23 a + 16\right)\cdot 29^{2} + \left(28 a^{2} + 16 a + 17\right)\cdot 29^{3} + \left(5 a^{2} + 22 a + 24\right)\cdot 29^{4} + \left(8 a^{2} + 24 a + 6\right)\cdot 29^{5} + \left(10 a^{2} + 7 a + 21\right)\cdot 29^{6} + \left(a^{2} + 16 a + 27\right)\cdot 29^{7} + \left(4 a^{2} + 27 a + 3\right)\cdot 29^{8} + \left(14 a^{2} + 4 a + 8\right)\cdot 29^{9} + \left(a^{2} + 9 a + 18\right)\cdot 29^{10} + \left(8 a^{2} + 25 a + 1\right)\cdot 29^{11} + \left(6 a^{2} + 19 a + 3\right)\cdot 29^{12} + \left(8 a^{2} + 8 a + 15\right)\cdot 29^{13} + \left(21 a^{2} + a + 14\right)\cdot 29^{14} + \left(2 a^{2} + 27 a + 4\right)\cdot 29^{15} + \left(6 a^{2} + 9 a + 20\right)\cdot 29^{16} + \left(5 a^{2} + 4 a + 4\right)\cdot 29^{17} + \left(4 a^{2} + 25 a + 2\right)\cdot 29^{18} + \left(7 a^{2} + a + 20\right)\cdot 29^{19} + \left(16 a^{2} + a + 10\right)\cdot 29^{20} + \left(2 a^{2} + 26 a + 3\right)\cdot 29^{21} + \left(24 a^{2} + 24 a + 9\right)\cdot 29^{22} + \left(5 a^{2} + 12 a + 20\right)\cdot 29^{23} + \left(28 a^{2} + 28 a + 20\right)\cdot 29^{24} + \left(8 a^{2} + 24 a + 21\right)\cdot 29^{25} + \left(18 a^{2} + 12 a + 20\right)\cdot 29^{26} + \left(25 a^{2} + 18 a + 24\right)\cdot 29^{27} + \left(20 a^{2} + 18\right)\cdot 29^{28} + \left(16 a^{2} + 4 a + 22\right)\cdot 29^{29} + \left(21 a + 5\right)\cdot 29^{30} + \left(5 a^{2} + 23 a + 9\right)\cdot 29^{31} + \left(17 a^{2} + 15 a + 20\right)\cdot 29^{32} + \left(2 a^{2} + a + 8\right)\cdot 29^{33} + \left(17 a^{2} + 21 a + 13\right)\cdot 29^{34} + \left(8 a^{2} + 12 a + 4\right)\cdot 29^{35} +O\left(29^{ 36 }\right)$ $r_{ 3 }$ $=$ $26 a^{2} + 24 a + 17 + \left(13 a^{2} + 4 a + 23\right)\cdot 29 + \left(8 a^{2} + 3 a + 21\right)\cdot 29^{2} + \left(9 a^{2} + 12 a + 10\right)\cdot 29^{3} + \left(18 a^{2} + 11 a + 2\right)\cdot 29^{4} + \left(17 a^{2} + 23 a\right)\cdot 29^{5} + \left(20 a^{2} + 26 a + 6\right)\cdot 29^{6} + \left(7 a^{2} + 3 a + 7\right)\cdot 29^{7} + \left(25 a^{2} + 7 a + 3\right)\cdot 29^{8} + \left(24 a^{2} + a + 3\right)\cdot 29^{9} + \left(28 a^{2} + 18 a + 16\right)\cdot 29^{10} + \left(27 a^{2} + 9 a + 18\right)\cdot 29^{11} + \left(9 a^{2} + 11 a + 17\right)\cdot 29^{12} + \left(a^{2} + 7 a + 15\right)\cdot 29^{13} + \left(9 a^{2} + 27 a + 17\right)\cdot 29^{14} + \left(19 a^{2} + 8 a + 26\right)\cdot 29^{15} + \left(28 a^{2} + 3 a + 1\right)\cdot 29^{16} + \left(26 a^{2} + 14 a + 24\right)\cdot 29^{17} + \left(5 a^{2} + 18 a + 23\right)\cdot 29^{18} + \left(20 a^{2} + 24 a + 27\right)\cdot 29^{19} + \left(21 a^{2} + 18 a + 17\right)\cdot 29^{20} + \left(25 a^{2} + 6 a + 24\right)\cdot 29^{21} + \left(7 a + 16\right)\cdot 29^{22} + \left(24 a^{2} + 27 a + 15\right)\cdot 29^{23} + \left(6 a^{2} + 20 a + 11\right)\cdot 29^{24} + \left(28 a^{2} + 23 a + 18\right)\cdot 29^{25} + \left(6 a^{2} + 19 a + 5\right)\cdot 29^{26} + \left(27 a^{2} + 12 a + 17\right)\cdot 29^{27} + \left(22 a^{2} + 2 a + 21\right)\cdot 29^{28} + \left(25 a^{2} + 14 a + 5\right)\cdot 29^{29} + \left(25 a^{2} + 18 a + 20\right)\cdot 29^{30} + \left(24 a^{2} + 21 a + 6\right)\cdot 29^{31} + \left(8 a^{2} + 25 a + 9\right)\cdot 29^{32} + \left(22 a^{2} + 10 a + 25\right)\cdot 29^{33} + \left(22 a^{2} + 12 a + 20\right)\cdot 29^{34} + \left(19 a^{2} + 19 a + 9\right)\cdot 29^{35} +O\left(29^{ 36 }\right)$ $r_{ 4 }$ $=$ $14 a^{2} + 25 a + 9 + \left(25 a^{2} + a + 8\right)\cdot 29 + \left(25 a^{2} + 26 a + 2\right)\cdot 29^{2} + \left(4 a^{2} + 10 a + 5\right)\cdot 29^{3} + \left(22 a^{2} + 3 a + 10\right)\cdot 29^{4} + \left(22 a^{2} + 14 a + 1\right)\cdot 29^{5} + \left(12 a^{2} + 23 a + 28\right)\cdot 29^{6} + \left(6 a^{2} + 7 a + 22\right)\cdot 29^{7} + \left(18 a^{2} + 13 a + 6\right)\cdot 29^{8} + \left(17 a^{2} + 9 a + 23\right)\cdot 29^{9} + \left(3 a^{2} + 19 a + 9\right)\cdot 29^{10} + \left(22 a^{2} + 18 a + 25\right)\cdot 29^{11} + \left(17 a^{2} + 19 a + 27\right)\cdot 29^{12} + \left(17 a^{2} + 25 a + 20\right)\cdot 29^{13} + \left(21 a^{2} + 8 a + 2\right)\cdot 29^{14} + \left(11 a^{2} + 27 a + 7\right)\cdot 29^{15} + \left(19 a^{2} + 23 a + 16\right)\cdot 29^{16} + \left(27 a^{2} + 28 a + 9\right)\cdot 29^{17} + \left(8 a^{2} + 17 a + 27\right)\cdot 29^{18} + \left(4 a^{2} + 24 a + 28\right)\cdot 29^{19} + \left(22 a^{2} + 26\right)\cdot 29^{20} + \left(20 a^{2} + 13 a + 14\right)\cdot 29^{21} + \left(28 a^{2} + 21 a + 28\right)\cdot 29^{22} + \left(3 a^{2} + 2 a + 6\right)\cdot 29^{23} + \left(8 a^{2} + 16 a + 15\right)\cdot 29^{24} + \left(22 a^{2} + 17 a + 25\right)\cdot 29^{25} + \left(27 a^{2} + 8 a + 20\right)\cdot 29^{26} + \left(16 a^{2} + 28 a + 8\right)\cdot 29^{27} + \left(10 a^{2} + 28 a + 7\right)\cdot 29^{28} + \left(2 a^{2} + 28 a + 18\right)\cdot 29^{29} + \left(17 a^{2} + 15 a + 1\right)\cdot 29^{30} + \left(21 a^{2} + 25 a + 1\right)\cdot 29^{31} + \left(5 a^{2} + a + 27\right)\cdot 29^{32} + \left(21 a^{2} + 21 a + 3\right)\cdot 29^{33} + \left(26 a^{2} + 14 a\right)\cdot 29^{34} + \left(9 a^{2} + 7 a\right)\cdot 29^{35} +O\left(29^{ 36 }\right)$ $r_{ 5 }$ $=$ $11 a^{2} + 9 a + 26 + \left(6 a^{2} + 4 a + 3\right)\cdot 29 + \left(23 a^{2} + 2 a + 22\right)\cdot 29^{2} + \left(19 a^{2} + 24\right)\cdot 29^{3} + \left(4 a^{2} + 24 a + 22\right)\cdot 29^{4} + \left(3 a^{2} + 9 a + 9\right)\cdot 29^{5} + \left(27 a^{2} + 23 a + 24\right)\cdot 29^{6} + \left(19 a^{2} + 8 a + 13\right)\cdot 29^{7} + \left(28 a^{2} + 23 a + 17\right)\cdot 29^{8} + \left(18 a^{2} + 22 a + 14\right)\cdot 29^{9} + \left(27 a^{2} + a + 14\right)\cdot 29^{10} + \left(21 a^{2} + 23 a + 10\right)\cdot 29^{11} + \left(12 a^{2} + 26 a + 21\right)\cdot 29^{12} + \left(19 a^{2} + 12 a + 10\right)\cdot 29^{13} + \left(27 a^{2} + 13\right)\cdot 29^{14} + \left(6 a^{2} + 22 a\right)\cdot 29^{15} + \left(23 a^{2} + 15 a + 14\right)\cdot 29^{16} + \left(25 a^{2} + 10 a + 22\right)\cdot 29^{17} + \left(18 a^{2} + 14 a + 21\right)\cdot 29^{18} + \left(a^{2} + 2 a + 12\right)\cdot 29^{19} + \left(20 a^{2} + 9 a + 25\right)\cdot 29^{20} + 25 a\cdot 29^{21} + \left(4 a^{2} + 25 a + 21\right)\cdot 29^{22} + \left(28 a^{2} + 17 a + 1\right)\cdot 29^{23} + \left(22 a^{2} + 8 a + 4\right)\cdot 29^{24} + \left(20 a^{2} + 9 a + 18\right)\cdot 29^{25} + \left(3 a^{2} + 25 a + 20\right)\cdot 29^{26} + \left(5 a^{2} + 26 a + 16\right)\cdot 29^{27} + \left(14 a^{2} + 25 a + 19\right)\cdot 29^{28} + \left(15 a^{2} + 10 a + 1\right)\cdot 29^{29} + \left(2 a^{2} + 18 a + 18\right)\cdot 29^{30} + \left(28 a^{2} + 12 a + 20\right)\cdot 29^{31} + \left(2 a^{2} + 16 a + 20\right)\cdot 29^{32} + \left(4 a^{2} + 16 a + 10\right)\cdot 29^{33} + \left(18 a^{2} + 24 a + 24\right)\cdot 29^{34} + \left(25 a + 22\right)\cdot 29^{35} +O\left(29^{ 36 }\right)$ $r_{ 6 }$ $=$ $11 a^{2} + 17 a + 5 + \left(a^{2} + 14 a + 5\right)\cdot 29 + \left(2 a^{2} + 20 a + 9\right)\cdot 29^{2} + \left(5 a^{2} + 4 a + 5\right)\cdot 29^{3} + \left(a^{2} + 4 a + 11\right)\cdot 29^{4} + \left(17 a^{2} + 20 a + 3\right)\cdot 29^{5} + \left(11 a^{2} + 7 a + 7\right)\cdot 29^{6} + \left(5 a^{2} + 13 a + 2\right)\cdot 29^{7} + \left(2 a^{2} + 26 a + 24\right)\cdot 29^{8} + \left(27 a^{2} + 8 a + 6\right)\cdot 29^{9} + \left(a^{2} + 17\right)\cdot 29^{10} + \left(20 a^{2} + 24 a + 22\right)\cdot 29^{11} + \left(a^{2} + 23 a + 25\right)\cdot 29^{12} + \left(14 a^{2} + 18 a + 25\right)\cdot 29^{13} + \left(7 a^{2} + 18 a + 12\right)\cdot 29^{14} + \left(7 a^{2} + 13 a + 20\right)\cdot 29^{15} + \left(16 a^{2} + 6 a + 21\right)\cdot 29^{16} + \left(24 a^{2} + 6 a + 24\right)\cdot 29^{17} + \left(2 a^{2} + 26 a + 28\right)\cdot 29^{18} + \left(5 a^{2} + 5 a\right)\cdot 29^{19} + \left(14 a^{2} + 13 a + 26\right)\cdot 29^{20} + \left(23 a^{2} + 2 a + 8\right)\cdot 29^{21} + \left(10 a^{2} + 20 a + 14\right)\cdot 29^{22} + \left(16 a^{2} + 11 a + 23\right)\cdot 29^{23} + \left(a^{2} + 16 a + 25\right)\cdot 29^{24} + \left(3 a^{2} + 2 a + 28\right)\cdot 29^{25} + \left(9 a^{2} + 22 a + 24\right)\cdot 29^{26} + \left(10 a^{2} + 13 a + 28\right)\cdot 29^{27} + \left(26 a^{2} + 5 a + 8\right)\cdot 29^{28} + \left(18 a^{2} + 10 a + 11\right)\cdot 29^{29} + \left(2 a^{2} + 2 a + 11\right)\cdot 29^{30} + \left(28 a^{2} + a + 19\right)\cdot 29^{31} + \left(4 a^{2} + 20 a + 6\right)\cdot 29^{32} + \left(3 a^{2} + 12 a + 28\right)\cdot 29^{33} + \left(27 a^{2} + 19\right)\cdot 29^{34} + \left(a^{2} + 4 a + 8\right)\cdot 29^{35} +O\left(29^{ 36 }\right)$ $r_{ 7 }$ $=$ $23 + 15\cdot 29^{2} + 7\cdot 29^{3} + 9\cdot 29^{4} + 20\cdot 29^{6} + 24\cdot 29^{7} + 20\cdot 29^{8} + 12\cdot 29^{9} + 29^{10} + 21\cdot 29^{12} + 2\cdot 29^{13} + 29^{14} + 10\cdot 29^{15} + 10\cdot 29^{16} + 9\cdot 29^{17} + 22\cdot 29^{18} + 17\cdot 29^{19} + 26\cdot 29^{20} + 27\cdot 29^{21} + 26\cdot 29^{22} + 8\cdot 29^{23} + 16\cdot 29^{24} + 15\cdot 29^{25} + 4\cdot 29^{26} + 27\cdot 29^{27} + 9\cdot 29^{28} + 19\cdot 29^{29} + 28\cdot 29^{30} + 29^{31} + 5\cdot 29^{32} + 12\cdot 29^{33} + 8\cdot 29^{34} + 15\cdot 29^{35} +O\left(29^{ 36 }\right)$ $r_{ 8 }$ $=$ $4 a^{2} + 16 a + 15 + \left(2 a^{2} + 12 a + 25\right)\cdot 29 + \left(a^{2} + 11 a + 7\right)\cdot 29^{2} + \left(19 a^{2} + 13 a + 14\right)\cdot 29^{3} + \left(5 a^{2} + 21 a + 7\right)\cdot 29^{4} + \left(18 a^{2} + 23 a + 24\right)\cdot 29^{5} + \left(4 a^{2} + 26 a + 26\right)\cdot 29^{6} + \left(17 a^{2} + 7 a + 17\right)\cdot 29^{7} + \left(8 a^{2} + 18 a + 3\right)\cdot 29^{8} + \left(13 a^{2} + 10 a + 27\right)\cdot 29^{9} + \left(23 a^{2} + 9 a + 16\right)\cdot 29^{10} + \left(15 a^{2} + 15 a + 26\right)\cdot 29^{11} + \left(9 a^{2} + 14 a + 16\right)\cdot 29^{12} + \left(26 a^{2} + 13 a + 3\right)\cdot 29^{13} + \left(28 a^{2} + a + 22\right)\cdot 29^{14} + \left(9 a^{2} + 17 a + 4\right)\cdot 29^{15} + \left(22 a^{2} + 27 a + 20\right)\cdot 29^{16} + \left(5 a^{2} + 22 a + 28\right)\cdot 29^{17} + \left(17 a^{2} + 13 a + 18\right)\cdot 29^{18} + \left(19 a^{2} + 27 a + 10\right)\cdot 29^{19} + \left(21 a^{2} + 14 a + 26\right)\cdot 29^{20} + \left(13 a^{2} + 13 a + 24\right)\cdot 29^{21} + \left(18 a^{2} + 16 a + 14\right)\cdot 29^{22} + \left(8 a^{2} + 14 a + 3\right)\cdot 29^{23} + \left(19 a^{2} + 25 a + 1\right)\cdot 29^{24} + \left(3 a^{2} + 8 a + 20\right)\cdot 29^{25} + \left(21 a^{2} + 27 a + 21\right)\cdot 29^{26} + \left(a^{2} + 15 a + 7\right)\cdot 29^{27} + \left(21 a^{2} + 23 a + 21\right)\cdot 29^{28} + \left(7 a^{2} + 18 a + 15\right)\cdot 29^{29} + \left(9 a^{2} + 10 a + 10\right)\cdot 29^{30} + \left(8 a^{2} + 2 a + 12\right)\cdot 29^{31} + \left(18 a^{2} + 7 a + 24\right)\cdot 29^{32} + \left(4 a^{2} + 24 a + 10\right)\cdot 29^{33} + \left(4 a^{2} + 13 a + 18\right)\cdot 29^{34} + \left(17 a^{2} + 17 a + 9\right)\cdot 29^{35} +O\left(29^{ 36 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $7$ $21$ $2$ $(1,3)(2,7)(4,5)(6,8)$ $-1$ $28$ $2$ $(1,7)(2,5)(4,8)$ $1$ $56$ $3$ $(1,4,5)(2,7,8)$ $1$ $42$ $4$ $(1,4,5,8)(2,6,3,7)$ $-1$ $56$ $6$ $(1,2,4,7,5,8)$ $1$ $48$ $7$ $(1,2,3,4,5,6,8)$ $0$ $42$ $8$ $(1,7,4,2,5,6,8,3)$ $-1$ $42$ $8$ $(1,2,8,7,5,3,4,6)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.