Basic invariants
| Dimension: | $2$ |
| Group: | $D_{7}$ |
| Conductor: | \(2279\)\(\medspace = 43 \cdot 53 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.11836763639.1 |
| Galois orbit size: | $3$ |
| Smallest permutation container: | $D_{7}$ |
| Parity: | odd |
| Determinant: | 1.2279.2t1.a.a |
| Projective image: | $D_7$ |
| Projective stem field: | Galois closure of 7.1.11836763639.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 10x^{5} - 2x^{4} + 44x^{3} - 10x^{2} + 43x - 43 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 24 a + 22 + \left(4 a + 17\right)\cdot 29 + \left(14 a + 27\right)\cdot 29^{2} + \left(16 a + 26\right)\cdot 29^{3} + \left(22 a + 17\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + 26 + \left(24 a + 17\right)\cdot 29 + \left(14 a + 6\right)\cdot 29^{2} + \left(12 a + 8\right)\cdot 29^{3} + \left(6 a + 27\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 a + 28 + \left(27 a + 24\right)\cdot 29 + \left(12 a + 4\right)\cdot 29^{2} + \left(10 a + 6\right)\cdot 29^{3} + \left(12 a + 7\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 + 26\cdot 29 + 4\cdot 29^{2} + 3\cdot 29^{3} + 10\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 18 + \left(a + 21\right)\cdot 29 + \left(16 a + 12\right)\cdot 29^{2} + \left(18 a + 16\right)\cdot 29^{3} + 16 a\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 a + 6 + \left(18 a + 25\right)\cdot 29 + \left(a + 5\right)\cdot 29^{2} + \left(3 a + 6\right)\cdot 29^{3} + \left(20 a + 21\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 8 a + 24 + \left(10 a + 10\right)\cdot 29 + \left(27 a + 24\right)\cdot 29^{2} + \left(25 a + 19\right)\cdot 29^{3} + \left(8 a + 2\right)\cdot 29^{4} +O(29^{5})\)
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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 value |
| $1$ | $1$ | $()$ | $2$ |
| $7$ | $2$ | $(1,7)(2,4)(3,6)$ | $0$ |
| $2$ | $7$ | $(1,7,6,2,5,4,3)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
| $2$ | $7$ | $(1,6,5,3,7,2,4)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
| $2$ | $7$ | $(1,2,3,6,4,7,5)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.