Basic invariants
| Dimension: | $2$ |
| Group: | $D_{5}$ |
| Conductor: | \(875\)\(\medspace = 5^{3} \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.765625.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{5}$ |
| Parity: | odd |
| Determinant: | 1.35.2t1.a.a |
| Projective image: | $D_5$ |
| Projective stem field: | Galois closure of 5.1.765625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 5x^{2} + 10x - 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a + 4 + \left(9 a + 6\right)\cdot 19 + \left(10 a + 2\right)\cdot 19^{2} + \left(18 a + 18\right)\cdot 19^{3} + \left(9 a + 8\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 17 + \left(9 a + 2\right)\cdot 19 + \left(8 a + 3\right)\cdot 19^{2} + 7\cdot 19^{3} + 9 a\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 10 + \left(14 a + 16\right)\cdot 19 + \left(9 a + 8\right)\cdot 19^{2} + \left(3 a + 14\right)\cdot 19^{3} + \left(3 a + 16\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 + 17\cdot 19 + 9\cdot 19^{3} + 14\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 9 + \left(4 a + 13\right)\cdot 19 + \left(9 a + 3\right)\cdot 19^{2} + \left(15 a + 8\right)\cdot 19^{3} + \left(15 a + 16\right)\cdot 19^{4} +O(19^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $5$ | $2$ | $(1,3)(2,4)$ | $0$ | ✓ |
| $2$ | $5$ | $(1,5,3,2,4)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ | |
| $2$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |