Basic invariants
| Dimension: | $2$ |
| Group: | $D_{5}$ |
| Conductor: | \(375\)\(\medspace = 3 \cdot 5^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.140625.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{5}$ |
| Parity: | odd |
| Determinant: | 1.15.2t1.a.a |
| Projective image: | $D_5$ |
| Projective stem field: | Galois closure of 5.1.140625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 5x^{2} - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$:
\( x^{2} + 6x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 4\cdot 7 + 3\cdot 7^{4} + 3\cdot 7^{5} +O(7^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( a + 4 + \left(5 a + 2\right)\cdot 7 + 4\cdot 7^{2} + \left(2 a + 6\right)\cdot 7^{3} + \left(5 a + 6\right)\cdot 7^{4} + \left(4 a + 1\right)\cdot 7^{5} +O(7^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( a + 5 + \left(a + 3\right)\cdot 7 + \left(3 a + 3\right)\cdot 7^{2} + \left(3 a + 6\right)\cdot 7^{3} + 2 a\cdot 7^{4} + 2 a\cdot 7^{5} +O(7^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + 5 + \left(a + 6\right)\cdot 7 + \left(6 a + 6\right)\cdot 7^{2} + 4 a\cdot 7^{3} + \left(a + 3\right)\cdot 7^{4} + \left(2 a + 1\right)\cdot 7^{5} +O(7^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 a + 6 + \left(5 a + 3\right)\cdot 7 + \left(3 a + 5\right)\cdot 7^{2} + \left(3 a + 6\right)\cdot 7^{3} + \left(4 a + 6\right)\cdot 7^{4} + \left(4 a + 6\right)\cdot 7^{5} +O(7^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $5$ | $2$ | $(1,4)(2,5)$ | $0$ |
| $2$ | $5$ | $(1,5,2,4,3)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
| $2$ | $5$ | $(1,2,3,5,4)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.