Basic invariants
Dimension: | $6$ |
Group: | $S_7$ |
Conductor: | \(426\!\cdots\!151\)\(\medspace = 19^{5} \cdot 11149^{5} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.211831.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 14T46 |
Parity: | odd |
Determinant: | 1.211831.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.211831.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + 2x^{5} - x^{4} + x^{2} - 2x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 263 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 263 }$: \( x^{2} + 261x + 5 \)
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 value |
$1$ | $1$ | $()$ | $6$ |
$21$ | $2$ | $(1,2)$ | $-4$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$105$ | $2$ | $(1,2)(3,4)$ | $2$ |
$70$ | $3$ | $(1,2,3)$ | $3$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$210$ | $4$ | $(1,2,3,4)$ | $-2$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.