Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(13829636833225\)\(\medspace = 5^{2} \cdot 8209^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.205225.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.8209.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.5.205225.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 a + 32 + \left(26 a + 4\right)\cdot 37 + \left(6 a + 6\right)\cdot 37^{2} + \left(31 a + 8\right)\cdot 37^{3} + 35 a\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 a + 35 + \left(10 a + 25\right)\cdot 37 + \left(30 a + 6\right)\cdot 37^{2} + \left(5 a + 15\right)\cdot 37^{3} + \left(a + 1\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 + 21\cdot 37 + 21\cdot 37^{2} + 14\cdot 37^{3} + 27\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 + 9\cdot 37 + 26\cdot 37^{2} + 12\cdot 37^{3} + 30\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 + 12\cdot 37 + 13\cdot 37^{2} + 23\cdot 37^{3} + 14\cdot 37^{4} +O(37^{5})\)
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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.