Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(16078125\)\(\medspace = 3 \cdot 5^{6} \cdot 7^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 5.1.16078125.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Projective image: | $S_5$ |
| Projective field: | Galois closure of 5.1.16078125.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$:
\( x^{2} + 149x + 6 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a + 59 + \left(126 a + 129\right)\cdot 151 + \left(54 a + 74\right)\cdot 151^{2} + \left(94 a + 93\right)\cdot 151^{3} + \left(81 a + 132\right)\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 87 + 18\cdot 151 + 91\cdot 151^{2} + 130\cdot 151^{3} + 20\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 82 a + 26 + \left(64 a + 97\right)\cdot 151 + \left(94 a + 127\right)\cdot 151^{2} + \left(103 a + 19\right)\cdot 151^{3} + \left(70 a + 30\right)\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 135 a + 91 + \left(24 a + 63\right)\cdot 151 + \left(96 a + 58\right)\cdot 151^{2} + \left(56 a + 76\right)\cdot 151^{3} + \left(69 a + 50\right)\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 69 a + 39 + \left(86 a + 144\right)\cdot 151 + \left(56 a + 100\right)\cdot 151^{2} + \left(47 a + 132\right)\cdot 151^{3} + \left(80 a + 67\right)\cdot 151^{4} +O(151^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 5 }$ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |