Basic invariants
| Dimension: | $15$ |
| Group: | $S_7$ |
| Conductor: | \(469\!\cdots\!623\)\(\medspace = 7^{5} \cdot 29^{8} \cdot 89^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.523943.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 42T412 |
| Parity: | odd |
| Determinant: | 1.623.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.523943.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} + 2x^{5} - x^{4} - x^{3} + 3x^{2} - 2x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$:
\( x^{2} + 138x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 a + 10 + \left(87 a + 51\right)\cdot 139 + 42 a\cdot 139^{2} + \left(68 a + 50\right)\cdot 139^{3} + \left(95 a + 28\right)\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 125 a + 12 + \left(130 a + 20\right)\cdot 139 + \left(3 a + 133\right)\cdot 139^{2} + \left(85 a + 93\right)\cdot 139^{3} + \left(71 a + 116\right)\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 + 32\cdot 139 + 95\cdot 139^{2} + 101\cdot 139^{3} + 90\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a + 22 + \left(7 a + 90\right)\cdot 139 + \left(57 a + 18\right)\cdot 139^{2} + \left(45 a + 105\right)\cdot 139^{3} + \left(106 a + 119\right)\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 a + 137 + \left(8 a + 25\right)\cdot 139 + \left(135 a + 6\right)\cdot 139^{2} + \left(53 a + 36\right)\cdot 139^{3} + \left(67 a + 103\right)\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 110 a + 39 + \left(51 a + 109\right)\cdot 139 + \left(96 a + 94\right)\cdot 139^{2} + \left(70 a + 75\right)\cdot 139^{3} + \left(43 a + 55\right)\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 130 a + 31 + \left(131 a + 88\right)\cdot 139 + \left(81 a + 68\right)\cdot 139^{2} + \left(93 a + 93\right)\cdot 139^{3} + \left(32 a + 41\right)\cdot 139^{4} +O(139^{5})\)
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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 |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.