Basic invariants
| Dimension: | $2$ |
| Group: | $D_{7}$ |
| Conductor: | \(1352\)\(\medspace = 2^{3} \cdot 13^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.1.2471326208.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{7}$ |
| Parity: | odd |
| Projective image: | $D_7$ |
| Projective field: | Galois closure of 7.1.2471326208.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 22 a + 3 + \left(10 a + 19\right)\cdot 23 + \left(3 a + 14\right)\cdot 23^{2} + \left(12 a + 5\right)\cdot 23^{3} + \left(6 a + 16\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( a + 1 + \left(12 a + 19\right)\cdot 23 + \left(19 a + 10\right)\cdot 23^{2} + \left(10 a + 3\right)\cdot 23^{3} + \left(16 a + 17\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 14 + \left(11 a + 10\right)\cdot 23 + \left(13 a + 11\right)\cdot 23^{2} + \left(10 a + 5\right)\cdot 23^{3} + \left(9 a + 15\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 + 19\cdot 23 + 5\cdot 23^{2} + 9\cdot 23^{3} + 21\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 a + 3 + \left(11 a + 4\right)\cdot 23 + \left(9 a + 4\right)\cdot 23^{2} + \left(12 a + 13\right)\cdot 23^{3} + 13 a\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 a + 3 + \left(18 a + 21\right)\cdot 23 + \left(15 a + 15\right)\cdot 23^{2} + \left(5 a + 6\right)\cdot 23^{3} + \left(19 a + 17\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 9 a + 8 + \left(4 a + 21\right)\cdot 23 + \left(7 a + 5\right)\cdot 23^{2} + \left(17 a + 2\right)\cdot 23^{3} + \left(3 a + 4\right)\cdot 23^{4} +O(23^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 7 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 7 }$ | Character values | ||
| $c1$ | $c2$ | $c3$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ | $2$ |
| $7$ | $2$ | $(1,5)(2,4)(3,7)$ | $0$ | $0$ | $0$ |
| $2$ | $7$ | $(1,4,2,5,7,6,3)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
| $2$ | $7$ | $(1,2,7,3,4,5,6)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
| $2$ | $7$ | $(1,5,3,2,6,4,7)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |