Basic invariants
| Dimension: | $15$ |
| Group: | $S_7$ |
| Conductor: | \(846\!\cdots\!851\)\(\medspace = 242971^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.242971.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 42T412 |
| Parity: | odd |
| Determinant: | 1.242971.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.242971.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} - x^{5} + 2x^{4} - 2x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$:
\( x^{2} + 78x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + 17 + \left(61 a + 42\right)\cdot 79 + \left(77 a + 70\right)\cdot 79^{2} + \left(77 a + 52\right)\cdot 79^{3} + \left(39 a + 45\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 a + 5 + \left(62 a + 4\right)\cdot 79 + 43\cdot 79^{2} + \left(9 a + 51\right)\cdot 79^{3} + \left(61 a + 18\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 a + 31 + 77\cdot 79 + \left(46 a + 28\right)\cdot 79^{2} + \left(46 a + 8\right)\cdot 79^{3} + \left(51 a + 20\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 63 a + 47 + \left(78 a + 61\right)\cdot 79 + \left(32 a + 74\right)\cdot 79^{2} + \left(32 a + 8\right)\cdot 79^{3} + \left(27 a + 25\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 56 a + 28 + \left(16 a + 43\right)\cdot 79 + \left(78 a + 60\right)\cdot 79^{2} + \left(69 a + 59\right)\cdot 79^{3} + \left(17 a + 70\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 75 a + 21 + \left(17 a + 20\right)\cdot 79 + \left(a + 8\right)\cdot 79^{2} + \left(a + 53\right)\cdot 79^{3} + \left(39 a + 7\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 10 + 67\cdot 79 + 29\cdot 79^{2} + 2\cdot 79^{3} + 49\cdot 79^{4} +O(79^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.