Basic invariants
| Dimension: | $2$ |
| Group: | $S_3 \times C_5$ |
| Conductor: | \(3751\)\(\medspace = 11^{2} \cdot 31 \) |
| Artin number field: | Galois closure of 15.5.89850539899830393871.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $S_3 \times C_5$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.31.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{5} + x + 42 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( a^{4} + 21 a^{3} + 29 a^{2} + 37 a + 24 + \left(25 a^{4} + 44 a^{3} + 24 a^{2} + 21 a + 38\right)\cdot 47 + \left(6 a^{4} + 4 a^{3} + 2 a^{2} + 17 a + 3\right)\cdot 47^{2} + \left(21 a^{4} + 22 a^{3} + 13 a^{2} + 36 a + 23\right)\cdot 47^{3} + \left(31 a^{4} + 16 a^{3} + 21 a^{2} + 46 a + 46\right)\cdot 47^{4} + \left(4 a^{4} + 42 a^{3} + 37 a^{2} + 10 a + 26\right)\cdot 47^{5} + \left(36 a^{4} + 10 a^{3} + 16 a^{2} + 9 a + 27\right)\cdot 47^{6} + \left(30 a^{4} + 11 a^{3} + 18 a^{2} + 42 a + 11\right)\cdot 47^{7} + \left(43 a^{4} + 31 a^{3} + 38 a^{2} + 42 a + 31\right)\cdot 47^{8} +O(47^{9})\)
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| $r_{ 2 }$ | $=$ |
\( 7 a^{4} + 6 a^{3} + 15 a^{2} + 24 a + 27 + \left(11 a^{4} + 16 a^{3} + 22 a^{2} + 6 a + 46\right)\cdot 47 + \left(42 a^{4} + 26 a^{3} + 12 a^{2} + 11 a + 37\right)\cdot 47^{2} + \left(4 a^{4} + 9 a^{3} + 38 a^{2} + 30 a + 26\right)\cdot 47^{3} + \left(32 a^{4} + 24 a^{3} + 23 a^{2} + 41 a + 29\right)\cdot 47^{4} + \left(24 a^{4} + 14 a^{3} + 41 a^{2} + 45 a + 8\right)\cdot 47^{5} + \left(3 a^{4} + 5 a^{3} + 32 a^{2} + 32 a + 32\right)\cdot 47^{6} + \left(31 a^{4} + 24 a^{3} + 19 a^{2} + 24 a + 44\right)\cdot 47^{7} + \left(3 a^{4} + 17 a^{3} + 42 a^{2} + 8 a + 40\right)\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 a^{4} + 41 a^{3} + 35 a^{2} + a + 37 + \left(9 a^{4} + 42 a^{3} + 33 a^{2} + 25 a + 35\right)\cdot 47 + \left(29 a^{4} + 11 a^{3} + 38 a^{2} + 29 a + 1\right)\cdot 47^{2} + \left(17 a^{4} + 12 a^{3} + 43 a^{2} + 21 a + 32\right)\cdot 47^{3} + \left(30 a^{4} + 6 a^{3} + 8 a^{2} + 40 a + 36\right)\cdot 47^{4} + \left(43 a^{4} + 37 a^{3} + 11 a^{2} + 33 a + 22\right)\cdot 47^{5} + \left(21 a^{4} + 16 a^{3} + 14 a^{2} + 16 a + 36\right)\cdot 47^{6} + \left(26 a^{4} + 22 a^{3} + 35 a^{2} + 10 a + 23\right)\cdot 47^{7} + \left(8 a^{4} + 18 a^{3} + 43 a^{2} + 2 a + 19\right)\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 a^{4} + 24 a^{3} + 10 a^{2} + 23 a + 11 + \left(9 a^{4} + 3 a^{3} + 24 a^{2} + 29 a + 7\right)\cdot 47 + \left(10 a^{4} + 32 a^{3} + 4 a + 16\right)\cdot 47^{2} + \left(33 a^{4} + 28 a^{3} + 23 a^{2} + 32 a + 23\right)\cdot 47^{3} + \left(15 a^{4} + 33 a^{3} + 26 a^{2} + 9 a + 24\right)\cdot 47^{4} + \left(39 a^{4} + 40 a^{2} + 30 a + 26\right)\cdot 47^{5} + \left(32 a^{4} + 18 a^{3} + 22 a^{2} + 16 a + 15\right)\cdot 47^{6} + \left(11 a^{4} + 44 a^{3} + 45 a^{2} + 34 a + 43\right)\cdot 47^{7} + \left(20 a^{4} + 2 a^{3} + 30 a^{2} + 7 a + 21\right)\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a^{4} + 23 a^{3} + 21 a^{2} + 37 a + 39 + \left(38 a^{4} + 33 a^{3} + 18 a^{2} + 39 a + 30\right)\cdot 47 + \left(24 a^{4} + 16 a^{3} + 43 a^{2} + 26 a + 14\right)\cdot 47^{2} + \left(30 a^{4} + 29 a^{3} + 35 a^{2} + 40 a\right)\cdot 47^{3} + \left(a^{4} + 43 a^{3} + 41 a^{2} + 38 a + 24\right)\cdot 47^{4} + \left(a^{3} + 38 a^{2} + 6 a + 26\right)\cdot 47^{5} + \left(25 a^{4} + 16 a^{3} + 29 a^{2} + 26 a + 30\right)\cdot 47^{6} + \left(39 a^{4} + 19 a^{3} + 24 a^{2} + 41 a + 32\right)\cdot 47^{7} + \left(23 a^{4} + 40 a^{3} + 9 a^{2} + 33 a\right)\cdot 47^{8} +O(47^{9})\)
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| $r_{ 6 }$ | $=$ |
\( 24 a^{4} + 23 a^{3} + 19 a^{2} + 11 a + 31 + \left(6 a^{4} + 21 a^{3} + 41 a^{2} + 3 a + 33\right)\cdot 47 + \left(34 a^{4} + 4 a^{3} + 9 a^{2} + 35 a + 5\right)\cdot 47^{2} + \left(33 a^{4} + 13 a^{3} + 2 a^{2} + 2 a + 26\right)\cdot 47^{3} + \left(9 a^{4} + 33 a^{3} + 29 a + 29\right)\cdot 47^{4} + \left(4 a^{4} + 22 a^{3} + 45 a^{2} + 15 a\right)\cdot 47^{5} + \left(40 a^{4} + 11 a^{3} + 31 a^{2} + 36 a + 4\right)\cdot 47^{6} + \left(17 a^{4} + 45 a^{3} + 27 a^{2} + 33 a + 45\right)\cdot 47^{7} + \left(4 a^{4} + 25 a^{3} + 41 a^{2} + a + 34\right)\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 26 a^{4} + 21 a^{3} + a^{2} + 8 a + 14 + \left(20 a^{4} + 25 a^{3} + 33 a^{2} + 34 a + 35\right)\cdot 47 + \left(21 a^{4} + 16 a^{3} + 26 a^{2} + 16 a + 30\right)\cdot 47^{2} + \left(40 a^{4} + 35 a^{3} + 29 a^{2} + 9 a + 17\right)\cdot 47^{3} + \left(32 a^{4} + 16 a^{3} + 25 a^{2} + 22 a + 11\right)\cdot 47^{4} + \left(43 a^{4} + 35 a^{3} + 27 a^{2} + 3 a + 33\right)\cdot 47^{5} + \left(36 a^{4} + 28 a^{3} + 28 a + 30\right)\cdot 47^{6} + \left(20 a^{4} + 35 a^{3} + 27 a + 17\right)\cdot 47^{7} + \left(36 a^{4} + 4 a^{3} + 20 a^{2} + 39 a + 1\right)\cdot 47^{8} +O(47^{9})\)
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| $r_{ 8 }$ | $=$ |
\( 28 a^{4} + 43 a^{3} + 14 a^{2} + 4 a + 6 + \left(8 a^{4} + 6 a^{3} + 36 a^{2} + 11 a + 7\right)\cdot 47 + \left(22 a^{4} + 8 a^{3} + 2 a^{2} + 34 a + 43\right)\cdot 47^{2} + \left(40 a^{4} + 36 a^{3} + 13 a^{2} + 10 a + 40\right)\cdot 47^{3} + \left(45 a^{4} + 36 a^{3} + 14 a^{2} + 40 a + 20\right)\cdot 47^{4} + \left(39 a^{4} + 34 a^{3} + 46 a^{2} + 11 a + 10\right)\cdot 47^{5} + \left(31 a^{4} + 25 a^{3} + 18 a^{2} + 18 a + 16\right)\cdot 47^{6} + \left(17 a^{4} + 26 a^{3} + a^{2} + 3 a + 7\right)\cdot 47^{7} + \left(42 a^{4} + 30 a^{3} + 29 a^{2} + 35 a + 37\right)\cdot 47^{8} +O(47^{9})\)
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| $r_{ 9 }$ | $=$ |
\( 30 a^{4} + 10 a^{3} + 3 a^{2} + 12 a + 19 + \left(43 a^{4} + 24 a^{3} + 46 a^{2} + 18 a + 6\right)\cdot 47 + \left(11 a^{4} + 23 a^{3} + 39 a^{2} + 43 a + 8\right)\cdot 47^{2} + \left(35 a^{4} + 18 a^{2} + 21 a + 6\right)\cdot 47^{3} + \left(20 a^{4} + 39 a^{3} + 30 a^{2} + 29 a + 19\right)\cdot 47^{4} + \left(11 a^{4} + 35 a^{3} + 31 a^{2} + 44 a + 32\right)\cdot 47^{5} + \left(29 a^{4} + 21 a^{3} + 32 a^{2} + 2 a + 12\right)\cdot 47^{6} + \left(7 a^{4} + 39 a^{3} + a^{2} + 33 a + 21\right)\cdot 47^{7} + \left(31 a^{4} + 32 a^{3} + 24 a^{2} + 2\right)\cdot 47^{8} +O(47^{9})\)
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| $r_{ 10 }$ | $=$ |
\( 39 a^{4} + 20 a^{3} + 3 a^{2} + 33 a + 43 + \left(10 a^{4} + 33 a^{3} + 18 a + 8\right)\cdot 47 + \left(45 a^{4} + 15 a^{3} + 32 a^{2} + 18 a + 5\right)\cdot 47^{2} + \left(20 a^{4} + 15 a^{3} + 42 a^{2} + 27 a + 44\right)\cdot 47^{3} + \left(30 a^{4} + 6 a^{3} + a^{2} + 5 a + 17\right)\cdot 47^{4} + \left(17 a^{4} + 37 a^{3} + 15 a^{2} + 37 a + 11\right)\cdot 47^{5} + \left(7 a^{4} + 30 a^{3} + 44 a^{2} + 4 a + 34\right)\cdot 47^{6} + \left(32 a^{4} + 11 a^{3} + 8 a^{2} + 27 a + 37\right)\cdot 47^{7} + \left(46 a^{4} + 45 a^{3} + 13 a^{2} + 42 a + 21\right)\cdot 47^{8} +O(47^{9})\)
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| $r_{ 11 }$ | $=$ |
\( 40 a^{4} + 17 a^{3} + 34 a^{2} + 5 a + 44 + \left(41 a^{4} + 29 a^{3} + 33 a^{2} + 7 a + 42\right)\cdot 47 + \left(37 a^{4} + 27 a^{3} + 14 a^{2} + 31 a + 43\right)\cdot 47^{2} + \left(44 a^{4} + 37 a^{3} + 26 a^{2} + 9 a + 39\right)\cdot 47^{3} + \left(41 a^{4} + 32 a^{3} + 43 a^{2} + 41 a + 46\right)\cdot 47^{4} + \left(10 a^{4} + 30 a^{3} + 25 a^{2} + 32 a + 6\right)\cdot 47^{5} + \left(46 a^{4} + 40 a^{3} + 25 a^{2} + 41 a + 38\right)\cdot 47^{6} + \left(31 a^{4} + 38 a^{3} + 2 a^{2} + 37 a + 7\right)\cdot 47^{7} + \left(45 a^{4} + 17 a^{3} + 35 a^{2} + 7 a + 18\right)\cdot 47^{8} +O(47^{9})\)
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| $r_{ 12 }$ | $=$ |
\( 42 a^{4} + 14 a^{3} + 23 a^{2} + 45 a + 36 + \left(11 a^{4} + 36 a^{3} + 29 a^{2} + 35 a + 9\right)\cdot 47 + \left(10 a^{4} + 6 a^{3} + 10 a^{2} + 23 a + 24\right)\cdot 47^{2} + \left(28 a^{4} + 17 a^{3} + 39 a^{2} + 31 a + 40\right)\cdot 47^{3} + \left(24 a^{4} + 11 a^{3} + 21 a^{2} + 25 a + 3\right)\cdot 47^{4} + \left(35 a^{4} + 9 a^{3} + 23 a^{2} + 42 a + 35\right)\cdot 47^{5} + \left(39 a^{4} + 9 a^{3} + 31 a^{2} + 17 a + 3\right)\cdot 47^{6} + \left(46 a^{4} + 35 a^{3} + 20 a^{2} + 19 a + 40\right)\cdot 47^{7} + \left(38 a^{4} + 20 a^{3} + 13 a^{2} + 12 a + 43\right)\cdot 47^{8} +O(47^{9})\)
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| $r_{ 13 }$ | $=$ |
\( 44 a^{4} + 3 a^{3} + 27 a^{2} + 28 a + 2 + \left(19 a^{4} + 19 a^{2} + 9 a + 25\right)\cdot 47 + \left(38 a^{4} + 26 a^{3} + 10 a^{2} + 42 a + 10\right)\cdot 47^{2} + \left(19 a^{4} + 45 a^{3} + 15 a^{2} + 34 a + 3\right)\cdot 47^{3} + \left(4 a^{4} + 43 a^{3} + 21 a^{2} + 42 a + 6\right)\cdot 47^{4} + \left(46 a^{4} + 35 a^{3} + 21 a^{2} + 27 a + 13\right)\cdot 47^{5} + \left(16 a^{4} + 6 a^{3} + 14 a^{2} + 29 a + 12\right)\cdot 47^{6} + \left(8 a^{4} + 13 a^{3} + 19 a^{2} + 32 a + 31\right)\cdot 47^{7} + \left(6 a^{4} + 16 a^{3} + 32 a^{2} + 5 a + 10\right)\cdot 47^{8} +O(47^{9})\)
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| $r_{ 14 }$ | $=$ |
\( 46 a^{4} + 27 a^{3} + 23 a^{2} + 20 a + 30 + \left(28 a^{4} + 36 a^{3} + 33 a^{2} + 6 a + 32\right)\cdot 47 + \left(14 a^{4} + 6 a^{3} + 43 a^{2} + 8 a + 34\right)\cdot 47^{2} + \left(20 a^{4} + 29 a^{3} + 10 a^{2} + 4 a + 29\right)\cdot 47^{3} + \left(32 a^{4} + 23 a^{3} + 6 a^{2} + 44 a + 1\right)\cdot 47^{4} + \left(14 a^{4} + 11 a^{3} + 7 a^{2} + 4 a + 10\right)\cdot 47^{5} + \left(29 a^{4} + 3 a^{3} + 5 a^{2} + 12 a + 15\right)\cdot 47^{6} + \left(17 a^{4} + 23 a^{3} + 9 a + 43\right)\cdot 47^{7} + \left(31 a^{4} + 13 a^{3} + 34 a^{2} + 4 a + 34\right)\cdot 47^{8} +O(47^{9})\)
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| $r_{ 15 }$ | $=$ |
\( 46 a^{4} + 36 a^{3} + 25 a^{2} + 41 a + 13 + \left(42 a^{4} + 21 a^{3} + 26 a^{2} + 14 a + 15\right)\cdot 47 + \left(26 a^{4} + 7 a^{3} + 40 a^{2} + 33 a + 1\right)\cdot 47^{2} + \left(31 a^{4} + 44 a^{3} + 23 a^{2} + 15 a + 22\right)\cdot 47^{3} + \left(21 a^{4} + 7 a^{3} + 41 a^{2} + 12 a + 10\right)\cdot 47^{4} + \left(39 a^{4} + 26 a^{3} + 9 a^{2} + 27 a + 17\right)\cdot 47^{5} + \left(25 a^{4} + 36 a^{3} + 7 a^{2} + 35 a + 19\right)\cdot 47^{6} + \left(35 a^{4} + 32 a^{3} + 9 a^{2} + 45 a + 15\right)\cdot 47^{7} + \left(39 a^{4} + 10 a^{3} + 15 a^{2} + 36 a + 9\right)\cdot 47^{8} +O(47^{9})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 15 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 15 }$ | Character values | |||
| $c1$ | $c2$ | $c3$ | $c4$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ | $2$ | $2$ |
| $3$ | $2$ | $(1,2)(4,14)(5,9)(7,13)(11,15)$ | $0$ | $0$ | $0$ | $0$ |
| $2$ | $3$ | $(1,2,10)(3,15,11)(4,14,8)(5,12,9)(6,13,7)$ | $-1$ | $-1$ | $-1$ | $-1$ |
| $1$ | $5$ | $(1,13,15,9,4)(2,7,11,5,14)(3,12,8,10,6)$ | $2 \zeta_{5}^{2}$ | $2 \zeta_{5}$ | $2 \zeta_{5}^{3}$ | $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ |
| $1$ | $5$ | $(1,15,4,13,9)(2,11,14,7,5)(3,8,6,12,10)$ | $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ | $2 \zeta_{5}^{2}$ | $2 \zeta_{5}$ | $2 \zeta_{5}^{3}$ |
| $1$ | $5$ | $(1,9,13,4,15)(2,5,7,14,11)(3,10,12,6,8)$ | $2 \zeta_{5}$ | $2 \zeta_{5}^{3}$ | $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ | $2 \zeta_{5}^{2}$ |
| $1$ | $5$ | $(1,4,9,15,13)(2,14,5,11,7)(3,6,10,8,12)$ | $2 \zeta_{5}^{3}$ | $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ | $2 \zeta_{5}^{2}$ | $2 \zeta_{5}$ |
| $3$ | $10$ | $(1,5,13,14,15,2,9,7,4,11)(3,10,12,6,8)$ | $0$ | $0$ | $0$ | $0$ |
| $3$ | $10$ | $(1,14,9,11,13,2,4,5,15,7)(3,6,10,8,12)$ | $0$ | $0$ | $0$ | $0$ |
| $3$ | $10$ | $(1,7,15,5,4,2,13,11,9,14)(3,12,8,10,6)$ | $0$ | $0$ | $0$ | $0$ |
| $3$ | $10$ | $(1,11,4,7,9,2,15,14,13,5)(3,8,6,12,10)$ | $0$ | $0$ | $0$ | $0$ |
| $2$ | $15$ | $(1,12,7,4,3,2,9,6,14,15,10,5,13,8,11)$ | $-\zeta_{5}$ | $-\zeta_{5}^{3}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{5}^{2}$ |
| $2$ | $15$ | $(1,7,3,9,14,10,13,11,12,4,2,6,15,5,8)$ | $-\zeta_{5}^{2}$ | $-\zeta_{5}$ | $-\zeta_{5}^{3}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
| $2$ | $15$ | $(1,3,14,13,12,2,15,8,7,9,10,11,4,6,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{5}^{2}$ | $-\zeta_{5}$ | $-\zeta_{5}^{3}$ |
| $2$ | $15$ | $(1,14,12,15,7,10,4,5,3,13,2,8,9,11,6)$ | $-\zeta_{5}^{3}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{5}^{2}$ | $-\zeta_{5}$ |