Basic invariants
| Dimension: | $15$ |
| Group: | $S_7$ |
| Conductor: | \(353\!\cdots\!416\)\(\medspace = 2^{34} \cdot 3^{30} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.60466176.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 42T412 |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.60466176.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 3x^{5} - 2x^{4} + 3x^{3} + 12x^{2} + 19x + 6 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 119 + 78\cdot 313 + 158\cdot 313^{2} + 256\cdot 313^{3} + 278\cdot 313^{4} +O(313^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 68 a + 262 + \left(148 a + 309\right)\cdot 313 + \left(207 a + 252\right)\cdot 313^{2} + \left(199 a + 152\right)\cdot 313^{3} + \left(240 a + 83\right)\cdot 313^{4} +O(313^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 219 + 150\cdot 313 + 203\cdot 313^{2} + 244\cdot 313^{3} + 119\cdot 313^{4} +O(313^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 51 a + 281 + \left(203 a + 81\right)\cdot 313 + \left(66 a + 308\right)\cdot 313^{2} + \left(145 a + 67\right)\cdot 313^{3} + \left(177 a + 84\right)\cdot 313^{4} +O(313^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 245 a + 153 + \left(164 a + 60\right)\cdot 313 + \left(105 a + 101\right)\cdot 313^{2} + \left(113 a + 231\right)\cdot 313^{3} + \left(72 a + 292\right)\cdot 313^{4} +O(313^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 97 + 243\cdot 313 + 235\cdot 313^{2} + 174\cdot 313^{3} + 234\cdot 313^{4} +O(313^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 262 a + 121 + \left(109 a + 14\right)\cdot 313 + \left(246 a + 305\right)\cdot 313^{2} + \left(167 a + 123\right)\cdot 313^{3} + \left(135 a + 158\right)\cdot 313^{4} +O(313^{5})\)
|
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.