Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(17923019113\)\(\medspace = 2617^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 5.1.2617.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PGL(2,5)$ |
| Parity: | even |
| Projective image: | $S_5$ |
| Projective field: | Galois closure of 5.1.2617.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 11 a + 13 + \left(16 a + 17\right)\cdot 29 + \left(12 a + 18\right)\cdot 29^{2} + \left(21 a + 28\right)\cdot 29^{3} + \left(4 a + 19\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 a + 10 + \left(12 a + 1\right)\cdot 29 + \left(16 a + 7\right)\cdot 29^{2} + \left(7 a + 7\right)\cdot 29^{3} + \left(24 a + 22\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 a + 20 + \left(10 a + 21\right)\cdot 29 + \left(27 a + 26\right)\cdot 29^{2} + \left(10 a + 10\right)\cdot 29^{3} + \left(14 a + 27\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a + 4 + \left(18 a + 26\right)\cdot 29 + \left(a + 7\right)\cdot 29^{2} + \left(18 a + 9\right)\cdot 29^{3} + \left(14 a + 1\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 + 20\cdot 29 + 26\cdot 29^{2} + 29^{3} + 16\cdot 29^{4} +O(29^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $-1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |