Basic invariants
| Dimension: | $4$ |
| Group: | $\PGL(2,5)$ |
| Conductor: | \(5611284433\)\(\medspace = 1777^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.5611284433.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.1777.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 6.2.5611284433.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 8x^{4} + 45x^{3} - 211x^{2} + 293x - 121 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 a + 4 + \left(20 a + 9\right)\cdot 23 + \left(19 a + 19\right)\cdot 23^{2} + 5\cdot 23^{3} + \left(20 a + 10\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 + 19\cdot 23 + 20\cdot 23^{2} + 17\cdot 23^{3} + 19\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 a + 3 + \left(2 a + 16\right)\cdot 23 + \left(3 a + 15\right)\cdot 23^{2} + \left(22 a + 10\right)\cdot 23^{3} + \left(2 a + 3\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a + 8 + \left(20 a + 19\right)\cdot 23 + \left(19 a + 10\right)\cdot 23^{2} + \left(4 a + 16\right)\cdot 23^{3} + \left(13 a + 16\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 a + 5 + \left(2 a + 4\right)\cdot 23 + \left(3 a + 7\right)\cdot 23^{2} + \left(18 a + 6\right)\cdot 23^{3} + \left(9 a + 15\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 + 18\cdot 23^{2} + 11\cdot 23^{3} + 3\cdot 23^{4} +O(23^{5})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,5)(2,3)(4,6)$ | $-2$ |
| $15$ | $2$ | $(1,2)(3,5)$ | $0$ |
| $20$ | $3$ | $(1,5,3)(2,4,6)$ | $1$ |
| $30$ | $4$ | $(2,6,3,5)$ | $0$ |
| $24$ | $5$ | $(1,2,4,5,6)$ | $-1$ |
| $20$ | $6$ | $(1,6,5,2,3,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.