Basic invariants
| Dimension: | $35$ |
| Group: | $S_7$ |
| Conductor: | \(272\!\cdots\!416\)\(\medspace = 2^{96} \cdot 3^{64} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.13436928.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 126 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.13436928.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 3x^{5} - x^{4} + 3x^{3} + 6x^{2} - x - 9 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 239 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 239 }$:
\( x^{2} + 237x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 223 a + 66 + \left(7 a + 162\right)\cdot 239 + \left(123 a + 187\right)\cdot 239^{2} + \left(56 a + 67\right)\cdot 239^{3} + \left(63 a + 144\right)\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a + 34 + \left(231 a + 194\right)\cdot 239 + \left(115 a + 186\right)\cdot 239^{2} + \left(182 a + 57\right)\cdot 239^{3} + \left(175 a + 214\right)\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 a + 80 + \left(85 a + 26\right)\cdot 239 + \left(229 a + 109\right)\cdot 239^{2} + \left(54 a + 210\right)\cdot 239^{3} + \left(162 a + 82\right)\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 228 a + 102 + \left(153 a + 185\right)\cdot 239 + \left(9 a + 4\right)\cdot 239^{2} + \left(184 a + 91\right)\cdot 239^{3} + \left(76 a + 113\right)\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 214 + 118\cdot 239 + 159\cdot 239^{2} + 141\cdot 239^{3} + 58\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 105 a + 125 + \left(46 a + 140\right)\cdot 239 + \left(101 a + 195\right)\cdot 239^{2} + \left(124 a + 119\right)\cdot 239^{3} + \left(185 a + 167\right)\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 134 a + 96 + \left(192 a + 128\right)\cdot 239 + \left(137 a + 112\right)\cdot 239^{2} + \left(114 a + 28\right)\cdot 239^{3} + \left(53 a + 175\right)\cdot 239^{4} +O(239^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $35$ |
| $21$ | $2$ | $(1,2)$ | $-5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.