Basic invariants
| Dimension: | $5$ |
| Group: | $\PGL(2,5)$ |
| Conductor: | \(7102225\)\(\medspace = 5^{2} \cdot 13^{2} \cdot 41^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.18927429625.2 |
| 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 6.2.18927429625.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 17x^{4} + 117x^{3} - 188x^{2} + 23x + 594 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 + 36\cdot 53 + 29\cdot 53^{2} + 11\cdot 53^{3} + 11\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 a + 23 + \left(40 a + 13\right)\cdot 53 + \left(14 a + 30\right)\cdot 53^{2} + \left(47 a + 43\right)\cdot 53^{3} + \left(24 a + 51\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 34 a + 46 + \left(12 a + 49\right)\cdot 53 + \left(38 a + 48\right)\cdot 53^{2} + \left(5 a + 5\right)\cdot 53^{3} + \left(28 a + 51\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 39 a + \left(32 a + 18\right)\cdot 53 + \left(17 a + 40\right)\cdot 53^{2} + \left(15 a + 6\right)\cdot 53^{3} + \left(49 a + 48\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 a + 50 + \left(20 a + 3\right)\cdot 53 + \left(35 a + 25\right)\cdot 53^{2} + \left(37 a + 50\right)\cdot 53^{3} + \left(3 a + 17\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 25 + 37\cdot 53 + 37\cdot 53^{2} + 40\cdot 53^{3} + 31\cdot 53^{4} +O(53^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,3)(2,4)(5,6)$ | $1$ |
| $15$ | $2$ | $(1,5)(2,4)$ | $1$ |
| $20$ | $3$ | $(1,3,4)(2,5,6)$ | $-1$ |
| $30$ | $4$ | $(1,4,5,2)$ | $-1$ |
| $24$ | $5$ | $(1,6,4,3,5)$ | $0$ |
| $20$ | $6$ | $(1,6,3,2,4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.