Basic invariants
| Dimension: | $2$ |
| Group: | $D_{5}$ |
| Conductor: | \(1875\)\(\medspace = 3 \cdot 5^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.3515625.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{5}$ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $D_5$ |
| Projective stem field: | Galois closure of 5.1.3515625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} + 10x^{3} - 15x^{2} + 10x - 12 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 a + 5 + \left(10 a + 16\right)\cdot 17 + \left(2 a + 13\right)\cdot 17^{2} + \left(15 a + 6\right)\cdot 17^{3} + \left(14 a + 8\right)\cdot 17^{4} + \left(15 a + 12\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 + 10\cdot 17 + 12\cdot 17^{3} + 11\cdot 17^{4} + 8\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 9 + \left(8 a + 15\right)\cdot 17 + \left(16 a + 10\right)\cdot 17^{2} + 10 a\cdot 17^{3} + \left(6 a + 5\right)\cdot 17^{4} + \left(6 a + 8\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a + 10 + \left(6 a + 4\right)\cdot 17 + \left(14 a + 6\right)\cdot 17^{2} + \left(a + 2\right)\cdot 17^{3} + \left(2 a + 8\right)\cdot 17^{4} + \left(a + 13\right)\cdot 17^{5} +O(17^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 a + 12 + \left(8 a + 3\right)\cdot 17 + 2\cdot 17^{2} + \left(6 a + 12\right)\cdot 17^{3} + 10 a\cdot 17^{4} + \left(10 a + 8\right)\cdot 17^{5} +O(17^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $5$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $2$ | $5$ | $(1,2,4,3,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
| $2$ | $5$ | $(1,4,5,2,3)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.