# Properties

 Label 7.21956126976.16t713.a.a Dimension 7 Group $\PGL(2,7)$ Conductor $2^{8} \cdot 3^{6} \cdot 7^{6}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $7$ Group: $\PGL(2,7)$ Conductor: $21956126976= 2^{8} \cdot 3^{6} \cdot 7^{6}$ Artin number field: Splitting field of 8.2.153692888832.1 defined by $f= x^{8} - 4 x^{7} + 21 x^{4} - 18 x + 9$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 16T713 Parity: Even Determinant: 1.1.1t1.a.a Projective image: $SO(3,7)$ Projective field: Galois closure of 8.2.153692888832.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 40.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $x^{3} + 2 x + 27$
Roots:
 $r_{ 1 }$ $=$ $6 + 12\cdot 29 + 10\cdot 29^{2} + 5\cdot 29^{3} + 21\cdot 29^{4} + 23\cdot 29^{5} + 21\cdot 29^{6} + 25\cdot 29^{7} + 19\cdot 29^{8} + 25\cdot 29^{9} + 4\cdot 29^{10} + 7\cdot 29^{11} + 8\cdot 29^{12} + 13\cdot 29^{13} + 17\cdot 29^{14} + 17\cdot 29^{15} + 16\cdot 29^{16} + 20\cdot 29^{17} + 8\cdot 29^{18} + 16\cdot 29^{19} + 6\cdot 29^{20} + 24\cdot 29^{21} + 23\cdot 29^{22} + 25\cdot 29^{23} + 19\cdot 29^{24} + 5\cdot 29^{25} + 3\cdot 29^{26} + 19\cdot 29^{27} + 19\cdot 29^{28} + 10\cdot 29^{29} + 26\cdot 29^{30} + 23\cdot 29^{31} + 13\cdot 29^{32} + 27\cdot 29^{33} + 11\cdot 29^{34} + 8\cdot 29^{35} + 23\cdot 29^{36} + 8\cdot 29^{37} + 26\cdot 29^{38} + 6\cdot 29^{39} +O\left(29^{ 40 }\right)$ $r_{ 2 }$ $=$ $28 a^{2} + 3 a + \left(27 a^{2} + 6 a + 6\right)\cdot 29 + \left(5 a^{2} + 2 a + 28\right)\cdot 29^{2} + \left(10 a^{2} + 16 a + 24\right)\cdot 29^{3} + \left(21 a^{2} + 28 a + 26\right)\cdot 29^{4} + \left(9 a^{2} + 7 a + 7\right)\cdot 29^{5} + \left(26 a^{2} + 21 a + 9\right)\cdot 29^{6} + \left(24 a^{2} + 12\right)\cdot 29^{7} + \left(14 a^{2} + 3 a + 10\right)\cdot 29^{8} + \left(9 a^{2} + 13 a + 20\right)\cdot 29^{9} + \left(10 a^{2} + 4 a + 7\right)\cdot 29^{10} + \left(22 a^{2} + 26 a + 24\right)\cdot 29^{11} + \left(21 a + 22\right)\cdot 29^{12} + \left(15 a^{2} + 26 a + 23\right)\cdot 29^{13} + \left(15 a^{2} + 13 a + 9\right)\cdot 29^{14} + \left(26 a^{2} + 20 a + 5\right)\cdot 29^{15} + \left(a^{2} + 12 a + 4\right)\cdot 29^{16} + \left(12 a^{2} + 11 a + 18\right)\cdot 29^{17} + \left(6 a^{2} + 3 a + 18\right)\cdot 29^{18} + \left(14 a^{2} + 4 a + 15\right)\cdot 29^{19} + \left(27 a^{2} + 26 a + 9\right)\cdot 29^{20} + \left(14 a^{2} + 27 a + 23\right)\cdot 29^{21} + \left(12 a^{2} + 13 a + 8\right)\cdot 29^{22} + \left(2 a^{2} + 18 a + 10\right)\cdot 29^{23} + \left(13 a^{2} + 24 a + 28\right)\cdot 29^{24} + \left(14 a^{2} + 21 a + 5\right)\cdot 29^{25} + \left(18 a^{2} + a + 24\right)\cdot 29^{26} + \left(22 a^{2} + 13 a + 16\right)\cdot 29^{27} + \left(19 a^{2} + 6 a + 24\right)\cdot 29^{28} + \left(8 a^{2} + 28\right)\cdot 29^{29} + \left(11 a^{2} + 6 a + 1\right)\cdot 29^{30} + \left(8 a^{2} + 18 a + 21\right)\cdot 29^{31} + \left(19 a^{2} + 14 a + 18\right)\cdot 29^{32} + \left(8 a^{2} + 9 a + 23\right)\cdot 29^{33} + \left(4 a^{2} + 11 a + 8\right)\cdot 29^{34} + \left(25 a^{2} + 9 a + 16\right)\cdot 29^{35} + \left(10 a^{2} + 6 a + 12\right)\cdot 29^{36} + \left(14 a^{2} + 17 a + 17\right)\cdot 29^{37} + \left(10 a^{2} + 27 a + 16\right)\cdot 29^{38} + \left(26 a^{2} + 14 a + 21\right)\cdot 29^{39} +O\left(29^{ 40 }\right)$ $r_{ 3 }$ $=$ $17 a^{2} + 10 a + 27 + \left(18 a^{2} + a + 23\right)\cdot 29 + \left(14 a^{2} + 14 a + 7\right)\cdot 29^{2} + \left(8 a^{2} + 5 a + 11\right)\cdot 29^{3} + \left(25 a^{2} + 8 a + 2\right)\cdot 29^{4} + \left(18 a^{2} + 20 a + 22\right)\cdot 29^{5} + \left(11 a^{2} + 6 a + 24\right)\cdot 29^{6} + \left(11 a^{2} + 11 a + 4\right)\cdot 29^{7} + \left(21 a^{2} + 8 a + 12\right)\cdot 29^{8} + \left(9 a^{2} + 6 a + 3\right)\cdot 29^{9} + \left(21 a^{2} + 10 a + 11\right)\cdot 29^{10} + \left(7 a^{2} + 3 a + 1\right)\cdot 29^{11} + \left(13 a^{2} + 2 a + 7\right)\cdot 29^{12} + \left(11 a^{2} + 3 a + 2\right)\cdot 29^{13} + \left(26 a^{2} + 21 a + 7\right)\cdot 29^{14} + \left(18 a + 6\right)\cdot 29^{15} + \left(28 a^{2} + 18 a + 26\right)\cdot 29^{16} + \left(8 a^{2} + 13 a + 20\right)\cdot 29^{17} + \left(27 a^{2} + 16 a + 8\right)\cdot 29^{18} + \left(20 a^{2} + 2 a + 9\right)\cdot 29^{19} + \left(16 a^{2} + 12 a\right)\cdot 29^{20} + \left(23 a^{2} + 12 a + 26\right)\cdot 29^{21} + \left(7 a^{2} + 3 a + 27\right)\cdot 29^{22} + \left(6 a^{2} + 9 a + 3\right)\cdot 29^{23} + \left(21 a^{2} + 4 a + 19\right)\cdot 29^{24} + \left(21 a^{2} + 25 a + 28\right)\cdot 29^{25} + \left(24 a^{2} + 24 a + 9\right)\cdot 29^{26} + \left(7 a^{2} + 20 a + 21\right)\cdot 29^{27} + \left(28 a^{2} + 2 a + 11\right)\cdot 29^{28} + \left(13 a^{2} + 9 a + 3\right)\cdot 29^{29} + \left(3 a^{2} + 10 a + 3\right)\cdot 29^{30} + \left(17 a^{2} + 22 a + 22\right)\cdot 29^{31} + \left(13 a^{2} + 23 a + 10\right)\cdot 29^{32} + \left(a^{2} + 19 a + 15\right)\cdot 29^{33} + \left(21 a^{2} + 25 a + 20\right)\cdot 29^{34} + \left(23 a^{2} + 7 a + 16\right)\cdot 29^{35} + \left(18 a^{2} + 14 a + 25\right)\cdot 29^{36} + \left(24 a^{2} + 19 a + 24\right)\cdot 29^{37} + \left(19 a^{2} + 4 a + 7\right)\cdot 29^{38} + \left(16 a^{2} + 20 a + 20\right)\cdot 29^{39} +O\left(29^{ 40 }\right)$ $r_{ 4 }$ $=$ $2 a^{2} + 3 a + 4 + \left(2 a^{2} + 16 a + 10\right)\cdot 29 + \left(13 a^{2} + 9 a + 18\right)\cdot 29^{2} + \left(11 a^{2} + 13 a + 26\right)\cdot 29^{3} + \left(23 a^{2} + 19 a + 19\right)\cdot 29^{4} + \left(15 a^{2} + 5 a + 25\right)\cdot 29^{5} + \left(7 a^{2} + 9 a + 22\right)\cdot 29^{6} + \left(8 a^{2} + 11 a + 28\right)\cdot 29^{7} + \left(a^{2} + 8 a + 1\right)\cdot 29^{8} + \left(2 a^{2} + 14 a + 20\right)\cdot 29^{9} + \left(27 a^{2} + 12 a + 10\right)\cdot 29^{10} + \left(3 a^{2} + 25 a + 9\right)\cdot 29^{11} + \left(15 a^{2} + 26 a + 3\right)\cdot 29^{12} + \left(12 a^{2} + 22 a + 1\right)\cdot 29^{13} + \left(15 a^{2} + 6 a\right)\cdot 29^{14} + \left(27 a^{2} + 13 a + 26\right)\cdot 29^{15} + \left(3 a^{2} + 28 a + 6\right)\cdot 29^{16} + \left(21 a^{2} + 20 a + 1\right)\cdot 29^{17} + \left(13 a^{2} + 13 a + 9\right)\cdot 29^{18} + \left(14 a^{2} + 6\right)\cdot 29^{19} + \left(14 a^{2} + 3 a + 21\right)\cdot 29^{20} + \left(12 a^{2} + 9 a\right)\cdot 29^{21} + \left(2 a^{2} + 27 a + 5\right)\cdot 29^{22} + \left(25 a^{2} + 20 a + 21\right)\cdot 29^{23} + \left(3 a^{2} + 8 a + 25\right)\cdot 29^{24} + \left(26 a^{2} + 11\right)\cdot 29^{25} + \left(11 a^{2} + 22 a + 15\right)\cdot 29^{26} + \left(9 a^{2} + 18 a + 18\right)\cdot 29^{27} + \left(26 a^{2} + 12 a + 23\right)\cdot 29^{28} + \left(20 a^{2} + 19 a + 25\right)\cdot 29^{29} + \left(20 a^{2} + 27 a + 4\right)\cdot 29^{30} + \left(3 a^{2} + 10 a + 5\right)\cdot 29^{31} + \left(6 a^{2} + 2 a + 1\right)\cdot 29^{32} + \left(25 a^{2} + 7\right)\cdot 29^{33} + \left(9 a^{2} + 23 a + 16\right)\cdot 29^{34} + \left(13 a^{2} + 2 a\right)\cdot 29^{35} + \left(15 a^{2} + 3 a + 9\right)\cdot 29^{36} + \left(9 a^{2} + 9 a + 1\right)\cdot 29^{37} + \left(9 a^{2} + 2 a + 15\right)\cdot 29^{38} + \left(18 a^{2} + 13 a + 20\right)\cdot 29^{39} +O\left(29^{ 40 }\right)$ $r_{ 5 }$ $=$ $28 a^{2} + 23 a + \left(27 a^{2} + 6 a + 6\right)\cdot 29 + \left(9 a^{2} + 17 a + 14\right)\cdot 29^{2} + \left(7 a^{2} + 28 a + 11\right)\cdot 29^{3} + \left(13 a^{2} + 9 a + 6\right)\cdot 29^{4} + \left(3 a^{2} + 15 a + 9\right)\cdot 29^{5} + \left(24 a^{2} + 27 a + 6\right)\cdot 29^{6} + \left(24 a^{2} + 16 a + 12\right)\cdot 29^{7} + \left(12 a^{2} + 17 a + 17\right)\cdot 29^{8} + \left(17 a^{2} + a + 11\right)\cdot 29^{9} + \left(20 a^{2} + 12 a + 21\right)\cdot 29^{10} + \left(2 a^{2} + 6 a + 7\right)\cdot 29^{11} + \left(13 a^{2} + 9 a + 10\right)\cdot 29^{12} + \left(a^{2} + 8 a + 15\right)\cdot 29^{13} + \left(27 a^{2} + 8 a + 15\right)\cdot 29^{14} + \left(3 a^{2} + 24 a + 23\right)\cdot 29^{15} + \left(23 a^{2} + 16 a + 22\right)\cdot 29^{16} + \left(24 a^{2} + 25 a + 15\right)\cdot 29^{17} + \left(8 a^{2} + 11 a + 2\right)\cdot 29^{18} + \left(24 a + 26\right)\cdot 29^{19} + \left(16 a^{2} + 28 a + 3\right)\cdot 29^{20} + \left(a^{2} + 20 a + 15\right)\cdot 29^{21} + \left(14 a^{2} + 16 a + 20\right)\cdot 29^{22} + \left(a^{2} + 18 a + 18\right)\cdot 29^{23} + \left(12 a^{2} + 24 a + 7\right)\cdot 29^{24} + \left(17 a^{2} + 6 a\right)\cdot 29^{25} + \left(27 a^{2} + 5 a + 17\right)\cdot 29^{26} + \left(25 a^{2} + 26 a + 11\right)\cdot 29^{27} + \left(11 a^{2} + 9 a + 4\right)\cdot 29^{28} + \left(28 a^{2} + 9 a + 26\right)\cdot 29^{29} + \left(25 a^{2} + 24 a + 11\right)\cdot 29^{30} + \left(16 a^{2} + 28 a + 3\right)\cdot 29^{31} + \left(3 a^{2} + 11 a + 17\right)\cdot 29^{32} + \left(24 a^{2} + 19 a + 5\right)\cdot 29^{33} + \left(14 a^{2} + 23 a + 13\right)\cdot 29^{34} + \left(19 a^{2} + 16 a + 18\right)\cdot 29^{35} + \left(2 a^{2} + 19 a + 1\right)\cdot 29^{36} + \left(5 a^{2} + 2 a + 5\right)\cdot 29^{37} + \left(9 a^{2} + 28 a + 5\right)\cdot 29^{38} + \left(13 a^{2} + 4\right)\cdot 29^{39} +O\left(29^{ 40 }\right)$ $r_{ 6 }$ $=$ $16 a^{2} + 12 a + 16 + \left(26 a^{2} + a + 5\right)\cdot 29 + \left(25 a^{2} + 19 a + 13\right)\cdot 29^{2} + \left(10 a^{2} + 6 a + 14\right)\cdot 29^{3} + \left(12 a^{2} + 3 a + 4\right)\cdot 29^{4} + \left(19 a + 7\right)\cdot 29^{5} + \left(11 a^{2} + 19 a + 14\right)\cdot 29^{6} + \left(15 a^{2} + 16 a\right)\cdot 29^{7} + \left(20 a^{2} + 18 a + 11\right)\cdot 29^{8} + \left(25 a^{2} + 8 a + 5\right)\cdot 29^{9} + \left(16 a^{2} + 5 a + 5\right)\cdot 29^{10} + \left(10 a^{2} + 12 a + 5\right)\cdot 29^{11} + \left(12 a^{2} + 7 a + 25\right)\cdot 29^{12} + \left(15 a^{2} + 24 a + 26\right)\cdot 29^{13} + 14 a^{2}29^{14} + \left(11 a^{2} + 18 a + 1\right)\cdot 29^{15} + \left(27 a^{2} + 26 a + 6\right)\cdot 29^{16} + \left(22 a^{2} + 10 a + 20\right)\cdot 29^{17} + \left(22 a^{2} + 18 a + 2\right)\cdot 29^{18} + \left(2 a^{2} + 22 a + 14\right)\cdot 29^{19} + \left(16 a^{2} + 21 a + 28\right)\cdot 29^{20} + \left(18 a^{2} + a + 28\right)\cdot 29^{21} + \left(23 a^{2} + 11 a + 19\right)\cdot 29^{22} + \left(17 a^{2} + 4 a + 9\right)\cdot 29^{23} + \left(16 a^{2} + 27 a + 3\right)\cdot 29^{24} + \left(10 a^{2} + 21 a + 4\right)\cdot 29^{25} + \left(13 a^{2} + 12 a + 14\right)\cdot 29^{26} + \left(20 a^{2} + 7 a + 28\right)\cdot 29^{27} + \left(22 a^{2} + a + 13\right)\cdot 29^{28} + \left(19 a^{2} + 25 a + 1\right)\cdot 29^{29} + \left(28 a^{2} + a + 27\right)\cdot 29^{30} + \left(21 a^{2} + 10 a + 18\right)\cdot 29^{31} + \left(2 a^{2} + 18 a + 15\right)\cdot 29^{32} + 12 a^{2}29^{33} + \left(27 a^{2} + 17 a\right)\cdot 29^{34} + \left(26 a^{2} + 21 a + 21\right)\cdot 29^{35} + \left(14 a^{2} + 10\right)\cdot 29^{36} + \left(4 a^{2} + 19 a + 17\right)\cdot 29^{37} + \left(26 a^{2} + 27 a + 6\right)\cdot 29^{38} + \left(4 a^{2} + a + 14\right)\cdot 29^{39} +O\left(29^{ 40 }\right)$ $r_{ 7 }$ $=$ $25 a^{2} + 7 a + 28 + \left(12 a^{2} + 26 a + 25\right)\cdot 29 + \left(17 a^{2} + 24 a + 1\right)\cdot 29^{2} + \left(9 a^{2} + 16 a + 3\right)\cdot 29^{3} + \left(20 a^{2} + 17 a + 15\right)\cdot 29^{4} + \left(9 a^{2} + 18 a + 19\right)\cdot 29^{5} + \left(6 a^{2} + 2 a + 17\right)\cdot 29^{6} + \left(2 a^{2} + a + 21\right)\cdot 29^{7} + \left(16 a^{2} + 2 a + 14\right)\cdot 29^{8} + \left(22 a^{2} + 14 a + 20\right)\cdot 29^{9} + \left(19 a^{2} + 13 a + 18\right)\cdot 29^{10} + \left(10 a^{2} + 13 a + 24\right)\cdot 29^{11} + \left(3 a^{2} + 19 a + 22\right)\cdot 29^{12} + \left(2 a^{2} + a + 18\right)\cdot 29^{13} + \left(17 a^{2} + 7 a + 23\right)\cdot 29^{14} + \left(16 a^{2} + 21 a + 7\right)\cdot 29^{15} + \left(2 a^{2} + 12 a + 21\right)\cdot 29^{16} + \left(26 a^{2} + 4 a + 14\right)\cdot 29^{17} + \left(7 a^{2} + 23 a + 21\right)\cdot 29^{18} + \left(5 a^{2} + 3 a + 7\right)\cdot 29^{19} + \left(25 a^{2} + 24 a + 21\right)\cdot 29^{20} + \left(15 a^{2} + 14 a + 15\right)\cdot 29^{21} + \left(26 a^{2} + 14 a + 4\right)\cdot 29^{22} + \left(4 a^{2} + 15 a + 2\right)\cdot 29^{23} + \left(20 a^{2} + 26 a + 8\right)\cdot 29^{24} + \left(25 a^{2} + 10 a + 24\right)\cdot 29^{25} + \left(19 a^{2} + 20 a + 22\right)\cdot 29^{26} + 11\cdot 29^{27} + \left(7 a^{2} + 25 a + 12\right)\cdot 29^{28} + \left(24 a^{2} + 23 a + 7\right)\cdot 29^{29} + \left(25 a^{2} + 16 a + 23\right)\cdot 29^{30} + \left(18 a^{2} + 25 a + 14\right)\cdot 29^{31} + \left(12 a^{2} + 15 a + 9\right)\cdot 29^{32} + \left(15 a^{2} + 8 a + 24\right)\cdot 29^{33} + \left(9 a^{2} + 15 a + 14\right)\cdot 29^{34} + \left(7 a^{2} + 28 a + 4\right)\cdot 29^{35} + \left(24 a^{2} + 13 a + 23\right)\cdot 29^{36} + \left(28 a^{2} + 19 a + 20\right)\cdot 29^{37} + \left(11 a^{2} + 25 a + 16\right)\cdot 29^{38} + \left(7 a^{2} + 6 a + 17\right)\cdot 29^{39} +O\left(29^{ 40 }\right)$ $r_{ 8 }$ $=$ $10 + 26\cdot 29 + 21\cdot 29^{2} + 18\cdot 29^{3} + 19\cdot 29^{4} + 28\cdot 29^{6} + 9\cdot 29^{7} + 28\cdot 29^{8} + 8\cdot 29^{9} + 7\cdot 29^{10} + 7\cdot 29^{11} + 16\cdot 29^{12} + 14\cdot 29^{13} + 12\cdot 29^{14} + 28\cdot 29^{15} + 11\cdot 29^{16} + 4\cdot 29^{17} + 15\cdot 29^{18} + 20\cdot 29^{19} + 24\cdot 29^{20} + 10\cdot 29^{21} + 5\cdot 29^{22} + 24\cdot 29^{23} + 3\cdot 29^{24} + 6\cdot 29^{25} + 9\cdot 29^{26} + 17\cdot 29^{27} + 5\cdot 29^{28} + 12\cdot 29^{29} + 17\cdot 29^{30} + 6\cdot 29^{31} + 12\cdot 29^{33} + 29^{34} + 29^{35} + 10\cdot 29^{36} + 20\cdot 29^{37} + 21\cdot 29^{38} + 10\cdot 29^{39} +O\left(29^{ 40 }\right)$

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

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

### Character values on conjugacy classes

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