Basic invariants
| Dimension: | $5$ |
| Group: | $\PGL(2,5)$ |
| Conductor: | \(20349121\)\(\medspace = 13^{2} \cdot 347^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.91794884831.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 6.0.91794884831.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 27x^{3} + 339x^{2} + 631x + 1044 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 4 a + 17 + \left(19 a + 34\right)\cdot 37 + \left(23 a + 16\right)\cdot 37^{2} + \left(8 a + 23\right)\cdot 37^{3} + \left(21 a + 24\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 + 35\cdot 37 + 35\cdot 37^{2} + 5\cdot 37^{3} + 11\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 4 + \left(5 a + 33\right)\cdot 37 + 13\cdot 37^{2} + \left(25 a + 15\right)\cdot 37^{3} + \left(8 a + 30\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 + 12\cdot 37 + 17\cdot 37^{2} + 27\cdot 37^{3} + 14\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 a + 2 + 31 a\cdot 37 + \left(36 a + 9\right)\cdot 37^{2} + \left(11 a + 4\right)\cdot 37^{3} + \left(28 a + 3\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 a + 33 + \left(17 a + 32\right)\cdot 37 + \left(13 a + 17\right)\cdot 37^{2} + \left(28 a + 34\right)\cdot 37^{3} + \left(15 a + 26\right)\cdot 37^{4} +O(37^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,5)(2,4)(3,6)$ | $1$ |
| $15$ | $2$ | $(1,4)(3,6)$ | $1$ |
| $20$ | $3$ | $(1,6,5)(2,4,3)$ | $-1$ |
| $30$ | $4$ | $(1,3,4,6)$ | $-1$ |
| $24$ | $5$ | $(1,2,5,4,6)$ | $0$ |
| $20$ | $6$ | $(1,3,6,2,5,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.