Basic invariants
| Dimension: | $14$ |
| Group: | $S_7$ |
| Conductor: | \(205\!\cdots\!721\)\(\medspace = 3^{6} \cdot 37^{10} \cdot 2381^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.3.792873.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 42T413 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.3.792873.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} + 2x^{5} + x^{4} - 2x^{3} + 2x^{2} - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$:
\( x^{2} + 69x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 21\cdot 71 + 57\cdot 71^{2} + 50\cdot 71^{3} + 52\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 61 a + 55 + \left(31 a + 11\right)\cdot 71 + \left(70 a + 54\right)\cdot 71^{2} + \left(6 a + 64\right)\cdot 71^{3} + \left(51 a + 3\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 38 + 71 + 38\cdot 71^{2} + 23\cdot 71^{3} + 46\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a + 35 + \left(39 a + 14\right)\cdot 71 + 21\cdot 71^{2} + \left(64 a + 8\right)\cdot 71^{3} + \left(19 a + 28\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 32 a + 33 + \left(17 a + 12\right)\cdot 71 + \left(25 a + 43\right)\cdot 71^{2} + \left(16 a + 61\right)\cdot 71^{3} + \left(39 a + 10\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 26 + 65\cdot 71 + 64\cdot 71^{2} + 5\cdot 71^{3} + 69\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 39 a + 26 + \left(53 a + 15\right)\cdot 71 + \left(45 a + 5\right)\cdot 71^{2} + \left(54 a + 69\right)\cdot 71^{3} + \left(31 a + 1\right)\cdot 71^{4} +O(71^{5})\)
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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 |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $-6$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.