Basic invariants
| Dimension: | $21$ |
| Group: | $S_7$ |
| Conductor: | \(179\!\cdots\!576\)\(\medspace = 2^{30} \cdot 8363^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.535232.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 84 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.535232.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} + 3x^{5} - 4x^{4} + 3x^{3} - 4x^{2} + 2x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 337 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 337 }$:
\( x^{2} + 332x + 10 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 320 a + 196 + \left(81 a + 78\right)\cdot 337 + \left(102 a + 288\right)\cdot 337^{2} + \left(222 a + 60\right)\cdot 337^{3} + \left(322 a + 212\right)\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 114 a + 195 + \left(129 a + 208\right)\cdot 337 + \left(292 a + 305\right)\cdot 337^{2} + \left(144 a + 58\right)\cdot 337^{3} + \left(256 a + 57\right)\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 a + 111 + \left(255 a + 168\right)\cdot 337 + \left(234 a + 43\right)\cdot 337^{2} + \left(114 a + 59\right)\cdot 337^{3} + \left(14 a + 255\right)\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 293 + 63\cdot 337 + 144\cdot 337^{2} + 331\cdot 337^{3} + 317\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 223 a + 91 + \left(207 a + 67\right)\cdot 337 + \left(44 a + 290\right)\cdot 337^{2} + \left(192 a + 153\right)\cdot 337^{3} + \left(80 a + 183\right)\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 208 a + 217 + \left(294 a + 253\right)\cdot 337 + \left(115 a + 332\right)\cdot 337^{2} + \left(43 a + 122\right)\cdot 337^{3} + \left(248 a + 236\right)\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 129 a + 246 + \left(42 a + 170\right)\cdot 337 + \left(221 a + 280\right)\cdot 337^{2} + \left(293 a + 223\right)\cdot 337^{3} + \left(88 a + 85\right)\cdot 337^{4} +O(337^{5})\)
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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 |
| $1$ | $1$ | $()$ | $21$ |
| $21$ | $2$ | $(1,2)$ | $1$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $70$ | $3$ | $(1,2,3)$ | $-3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.