Basic invariants
| Dimension: | $14$ |
| Group: | $S_7$ |
| Conductor: | \(269\!\cdots\!849\)\(\medspace = 191^{10} \cdot 601^{10} \cdot 607^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.69678137.1 |
| 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 7.7.69678137.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 3x^{6} - 3x^{5} + 13x^{4} - 2x^{3} - 9x^{2} + x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 313 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 313 }$:
\( x^{2} + 310x + 10 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 195 a + 251 + \left(136 a + 74\right)\cdot 313 + \left(259 a + 150\right)\cdot 313^{2} + \left(146 a + 303\right)\cdot 313^{3} + \left(38 a + 232\right)\cdot 313^{4} +O(313^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 264 + 13\cdot 313 + 308\cdot 313^{2} + 50\cdot 313^{3} + 44\cdot 313^{4} +O(313^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 30 a + 33 + \left(4 a + 93\right)\cdot 313 + \left(17 a + 176\right)\cdot 313^{2} + \left(167 a + 280\right)\cdot 313^{3} + \left(77 a + 10\right)\cdot 313^{4} +O(313^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 249 a + 283 + \left(151 a + 92\right)\cdot 313 + \left(179 a + 77\right)\cdot 313^{2} + \left(178 a + 288\right)\cdot 313^{3} + \left(110 a + 109\right)\cdot 313^{4} +O(313^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 118 a + 210 + \left(176 a + 289\right)\cdot 313 + \left(53 a + 165\right)\cdot 313^{2} + \left(166 a + 171\right)\cdot 313^{3} + \left(274 a + 201\right)\cdot 313^{4} +O(313^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 64 a + 91 + \left(161 a + 299\right)\cdot 313 + \left(133 a + 150\right)\cdot 313^{2} + \left(134 a + 18\right)\cdot 313^{3} + \left(202 a + 263\right)\cdot 313^{4} +O(313^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 283 a + 123 + \left(308 a + 75\right)\cdot 313 + \left(295 a + 223\right)\cdot 313^{2} + \left(145 a + 138\right)\cdot 313^{3} + \left(235 a + 76\right)\cdot 313^{4} +O(313^{5})\)
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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.