Basic invariants
| Dimension: | $2$ |
| Group: | $D_{5}$ |
| Conductor: | \(2299\)\(\medspace = 11^{2} \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 5.1.5285401.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{5}$ |
| Parity: | odd |
| Projective image: | $D_5$ |
| Projective field: | Galois closure of 5.1.5285401.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a + 10 + \left(6 a + 7\right)\cdot 13 + \left(11 a + 7\right)\cdot 13^{2} + \left(7 a + 12\right)\cdot 13^{3} + 6 a\cdot 13^{4} + 10 a\cdot 13^{5} + \left(7 a + 8\right)\cdot 13^{6} +O(13^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + \left(3 a + 5\right)\cdot 13 + \left(3 a + 3\right)\cdot 13^{2} + \left(10 a + 6\right)\cdot 13^{3} + \left(3 a + 11\right)\cdot 13^{4} + \left(7 a + 3\right)\cdot 13^{5} + \left(3 a + 12\right)\cdot 13^{6} +O(13^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a + 9 + \left(9 a + 12\right)\cdot 13 + \left(9 a + 2\right)\cdot 13^{2} + 2 a\cdot 13^{3} + \left(9 a + 5\right)\cdot 13^{4} + \left(5 a + 7\right)\cdot 13^{5} + \left(9 a + 8\right)\cdot 13^{6} +O(13^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 + 6\cdot 13 + 12\cdot 13^{2} + 10\cdot 13^{3} + 8\cdot 13^{4} + 10\cdot 13^{5} + 4\cdot 13^{6} +O(13^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 a + 4 + \left(6 a + 7\right)\cdot 13 + \left(a + 12\right)\cdot 13^{2} + \left(5 a + 8\right)\cdot 13^{3} + \left(6 a + 12\right)\cdot 13^{4} + \left(2 a + 3\right)\cdot 13^{5} + \left(5 a + 5\right)\cdot 13^{6} +O(13^{7})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $5$ | $2$ | $(1,5)(2,3)$ | $0$ | $0$ |
| $2$ | $5$ | $(1,2,3,5,4)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
| $2$ | $5$ | $(1,3,4,2,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |