Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(186037\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.186037.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.186037.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.5.186037.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 6x^{3} + 2x^{2} + 5x - 2 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 21\cdot 31 + 14\cdot 31^{2} + 28\cdot 31^{3} + 25\cdot 31^{4} +O(31^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 19 + 24\cdot 31^{2} + 14\cdot 31^{3} + 7\cdot 31^{4} +O(31^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 11 a + 23 + \left(12 a + 19\right)\cdot 31 + \left(15 a + 3\right)\cdot 31^{2} + \left(11 a + 25\right)\cdot 31^{3} + \left(6 a + 24\right)\cdot 31^{4} +O(31^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 4 + 18\cdot 31 + 28\cdot 31^{2} + 22\cdot 31^{3} + 8\cdot 31^{4} +O(31^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 20 a + 14 + \left(18 a + 2\right)\cdot 31 + \left(15 a + 22\right)\cdot 31^{2} + \left(19 a + 1\right)\cdot 31^{3} + \left(24 a + 26\right)\cdot 31^{4} +O(31^{5})\)
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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 value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.