Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(494725990429\)\(\medspace = 11^{3} \cdot 719^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.7909.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PGL(2,5)$ |
| Parity: | even |
| Determinant: | 1.7909.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.7909.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} + 2x^{3} - 2x^{2} + x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$:
\( x^{2} + 69x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 49 + 54\cdot 71 + 6\cdot 71^{2} + 21\cdot 71^{3} + 27\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 57 a + 35 + \left(47 a + 50\right)\cdot 71 + \left(37 a + 37\right)\cdot 71^{2} + \left(62 a + 29\right)\cdot 71^{3} + \left(42 a + 10\right)\cdot 71^{4} +O(71^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 37 + 49\cdot 71 + 3\cdot 71^{2} + 32\cdot 71^{3} + 53\cdot 71^{4} +O(71^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 16 + 40\cdot 71 + 28\cdot 71^{2} + 13\cdot 71^{3} + 17\cdot 71^{4} +O(71^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 14 a + 7 + \left(23 a + 18\right)\cdot 71 + \left(33 a + 65\right)\cdot 71^{2} + \left(8 a + 45\right)\cdot 71^{3} + \left(28 a + 33\right)\cdot 71^{4} +O(71^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $5$ | |
| $10$ | $2$ | $(1,2)$ | $-1$ | |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ | ✓ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $1$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |