Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(10916416\)\(\medspace = 2^{6} \cdot 7^{2} \cdot 59^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.682276.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.682276.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 5x^{3} + 6x + 5 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 25 a + 34 + \left(63 a + 70\right)\cdot 73 + \left(38 a + 47\right)\cdot 73^{2} + \left(26 a + 26\right)\cdot 73^{3} + \left(40 a + 52\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 48 a + 36 + \left(9 a + 16\right)\cdot 73 + \left(34 a + 28\right)\cdot 73^{2} + \left(46 a + 67\right)\cdot 73^{3} + 32 a\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 + 63\cdot 73 + 50\cdot 73^{3} + 25\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 49 a + 25 + 14 a\cdot 73 + \left(21 a + 10\right)\cdot 73^{2} + \left(63 a + 26\right)\cdot 73^{3} + \left(a + 62\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 24 a + 26 + \left(58 a + 68\right)\cdot 73 + \left(51 a + 58\right)\cdot 73^{2} + \left(9 a + 48\right)\cdot 73^{3} + \left(71 a + 4\right)\cdot 73^{4} +O(73^{5})\)
|
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 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.