Basic invariants
| Dimension: | $20$ |
| Group: | $S_7$ |
| Conductor: | \(181\!\cdots\!801\)\(\medspace = 19^{10} \cdot 11149^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.1.211831.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | even |
| Projective image: | $S_7$ |
| Projective field: | Galois closure of 7.1.211831.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 263 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 263 }$:
\( x^{2} + 261x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 261 + 63\cdot 263 + 35\cdot 263^{2} + 254\cdot 263^{3} + 54\cdot 263^{4} +O(263^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 228 a + 251 + \left(258 a + 231\right)\cdot 263 + \left(61 a + 97\right)\cdot 263^{2} + \left(184 a + 242\right)\cdot 263^{3} + \left(15 a + 215\right)\cdot 263^{4} +O(263^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 85 a + 177 + \left(48 a + 161\right)\cdot 263 + \left(5 a + 34\right)\cdot 263^{2} + \left(226 a + 27\right)\cdot 263^{3} + \left(260 a + 223\right)\cdot 263^{4} +O(263^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 184 + 198\cdot 263 + 69\cdot 263^{2} + 140\cdot 263^{3} +O(263^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 35 a + 181 + \left(4 a + 258\right)\cdot 263 + \left(201 a + 225\right)\cdot 263^{2} + \left(78 a + 22\right)\cdot 263^{3} + \left(247 a + 63\right)\cdot 263^{4} +O(263^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 178 a + 84 + \left(214 a + 173\right)\cdot 263 + \left(257 a + 259\right)\cdot 263^{2} + \left(36 a + 210\right)\cdot 263^{3} + \left(2 a + 255\right)\cdot 263^{4} +O(263^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 178 + 226\cdot 263 + 65\cdot 263^{2} + 154\cdot 263^{3} + 238\cdot 263^{4} +O(263^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $20$ |
| $21$ | $2$ | $(1,2)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-4$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |