Basic invariants
| Dimension: | $15$ |
| Group: | $S_7$ |
| Conductor: | \(333\!\cdots\!351\)\(\medspace = 17^{5} \cdot 11863^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.201671.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 42T412 |
| Parity: | odd |
| Determinant: | 1.201671.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.201671.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} + 2x^{5} - 2x^{4} + 2x^{3} - 2x^{2} + 2x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 277 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 277 }$:
\( x^{2} + 274x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 168 a + 90 + \left(208 a + 2\right)\cdot 277 + \left(236 a + 96\right)\cdot 277^{2} + \left(152 a + 64\right)\cdot 277^{3} + \left(129 a + 87\right)\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 177 a + 107 + \left(86 a + 169\right)\cdot 277 + \left(32 a + 270\right)\cdot 277^{2} + \left(165 a + 183\right)\cdot 277^{3} + \left(63 a + 168\right)\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 167 a + 114 + \left(265 a + 45\right)\cdot 277 + \left(177 a + 249\right)\cdot 277^{2} + \left(125 a + 27\right)\cdot 277^{3} + \left(57 a + 75\right)\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 100 a + 84 + \left(190 a + 252\right)\cdot 277 + \left(244 a + 3\right)\cdot 277^{2} + \left(111 a + 93\right)\cdot 277^{3} + \left(213 a + 194\right)\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 59 + 57\cdot 277 + 204\cdot 277^{2} + 225\cdot 277^{3} + 137\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 109 a + 40 + \left(68 a + 183\right)\cdot 277 + \left(40 a + 43\right)\cdot 277^{2} + \left(124 a + 9\right)\cdot 277^{3} + \left(147 a + 46\right)\cdot 277^{4} +O(277^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 110 a + 61 + \left(11 a + 121\right)\cdot 277 + \left(99 a + 240\right)\cdot 277^{2} + \left(151 a + 226\right)\cdot 277^{3} + \left(219 a + 121\right)\cdot 277^{4} +O(277^{5})\)
|
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.