Basic invariants
| Dimension: | $15$ |
| Group: | $S_7$ |
| Conductor: | \(110\!\cdots\!601\)\(\medspace = 73^{10} \cdot 5507^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.402011.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 42T411 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.402011.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} + 2x^{5} - 4x^{4} + 3x^{3} - 3x^{2} + 2x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 503 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 503 }$:
\( x^{2} + 498x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 35 + 315\cdot 503 + 173\cdot 503^{2} + 434\cdot 503^{3} + 350\cdot 503^{4} +O(503^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 191 a + 312 + \left(355 a + 370\right)\cdot 503 + \left(160 a + 383\right)\cdot 503^{2} + \left(127 a + 261\right)\cdot 503^{3} + \left(410 a + 84\right)\cdot 503^{4} +O(503^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 312 a + 261 + \left(147 a + 447\right)\cdot 503 + \left(342 a + 328\right)\cdot 503^{2} + \left(375 a + 234\right)\cdot 503^{3} + \left(92 a + 499\right)\cdot 503^{4} +O(503^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 461 + 246\cdot 503 + 411\cdot 503^{2} + 410\cdot 503^{3} + 20\cdot 503^{4} +O(503^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 210 a + 130 + \left(452 a + 464\right)\cdot 503 + \left(81 a + 428\right)\cdot 503^{2} + \left(68 a + 299\right)\cdot 503^{3} + \left(81 a + 307\right)\cdot 503^{4} +O(503^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 293 a + 174 + \left(50 a + 1\right)\cdot 503 + \left(421 a + 386\right)\cdot 503^{2} + \left(434 a + 55\right)\cdot 503^{3} + \left(421 a + 142\right)\cdot 503^{4} +O(503^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 137 + 166\cdot 503 + 402\cdot 503^{2} + 314\cdot 503^{3} + 103\cdot 503^{4} +O(503^{5})\)
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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.