Basic invariants
| Dimension: | $14$ |
| Group: | $S_7$ |
| Conductor: | \(377\!\cdots\!625\)\(\medspace = 5^{4} \cdot 13^{4} \cdot 1205753^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.78373945.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 21T38 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.7.78373945.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} - 7x^{5} + 3x^{4} + 13x^{3} + x^{2} - 4x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 211 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 211 }$:
\( x^{2} + 207x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 185 a + 115 + \left(147 a + 159\right)\cdot 211 + \left(157 a + 90\right)\cdot 211^{2} + \left(134 a + 22\right)\cdot 211^{3} + \left(91 a + 49\right)\cdot 211^{4} +O(211^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 35 + 122\cdot 211 + 141\cdot 211^{2} + 115\cdot 211^{3} + 183\cdot 211^{4} +O(211^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 26 a + 11 + \left(63 a + 144\right)\cdot 211 + \left(53 a + 151\right)\cdot 211^{2} + \left(76 a + 192\right)\cdot 211^{3} + \left(119 a + 69\right)\cdot 211^{4} +O(211^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 201 a + 132 + \left(84 a + 8\right)\cdot 211 + \left(27 a + 80\right)\cdot 211^{2} + 102 a\cdot 211^{3} + \left(172 a + 25\right)\cdot 211^{4} +O(211^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 195 + 165\cdot 211 + 176\cdot 211^{2} + 142\cdot 211^{3} + 202\cdot 211^{4} +O(211^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 54 + 96\cdot 211 + 98\cdot 211^{2} + 199\cdot 211^{3} + 122\cdot 211^{4} +O(211^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 10 a + 92 + \left(126 a + 147\right)\cdot 211 + \left(183 a + 104\right)\cdot 211^{2} + \left(108 a + 170\right)\cdot 211^{3} + \left(38 a + 190\right)\cdot 211^{4} +O(211^{5})\)
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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.