Properties

Label 7.3e6_5e6_7e6.16t713.1c1
Dimension 7
Group $\PGL(2,7)$
Conductor $ 3^{6} \cdot 5^{6} \cdot 7^{6}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$7$
Group:$\PGL(2,7)$
Conductor:$1340095640625= 3^{6} \cdot 5^{6} \cdot 7^{6} $
Artin number field: Splitting field of $f= x^{8} - x^{7} + 7 x^{6} - 28 x^{5} + 70 x^{4} - 112 x^{3} + 112 x^{2} - 64 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 16T713
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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