Basic invariants
| Dimension: | $20$ |
| Group: | $S_7$ |
| Conductor: | \(834\!\cdots\!649\)\(\medspace = 1489^{10} \cdot 20857^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.31056073.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.7.31056073.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 3x^{6} - 3x^{5} + 11x^{4} + x^{3} - 9x^{2} + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 307 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 307 }$:
\( x^{2} + 306x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 89 + 70\cdot 307 + 100\cdot 307^{2} + 232\cdot 307^{3} + 94\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 66 a + 59 + \left(44 a + 185\right)\cdot 307 + \left(250 a + 154\right)\cdot 307^{2} + \left(211 a + 116\right)\cdot 307^{3} + \left(240 a + 254\right)\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 241 a + 125 + \left(262 a + 163\right)\cdot 307 + \left(56 a + 53\right)\cdot 307^{2} + \left(95 a + 78\right)\cdot 307^{3} + \left(66 a + 283\right)\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 23 + 57\cdot 307 + 270\cdot 307^{2} + 287\cdot 307^{3} + 23\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 230 a + 259 + \left(16 a + 249\right)\cdot 307 + \left(19 a + 271\right)\cdot 307^{2} + \left(173 a + 292\right)\cdot 307^{3} + \left(143 a + 75\right)\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 77 a + 182 + \left(290 a + 36\right)\cdot 307 + \left(287 a + 274\right)\cdot 307^{2} + \left(133 a + 139\right)\cdot 307^{3} + \left(163 a + 46\right)\cdot 307^{4} +O(307^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 187 + 158\cdot 307 + 103\cdot 307^{2} + 80\cdot 307^{3} + 142\cdot 307^{4} +O(307^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $20$ |
| $21$ | $2$ | $(1,2)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-4$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.