Basic invariants
Dimension: | $35$ |
Group: | $S_7$ |
Conductor: | \(176\!\cdots\!801\)\(\medspace = 289987^{20} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.289987.1 |
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 7.1.289987.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + 2x^{5} - 2x^{4} + 2x^{3} - x^{2} + x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 491 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 491 }$: \( x^{2} + 487x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 292 a + 4 + \left(91 a + 406\right)\cdot 491 + \left(249 a + 160\right)\cdot 491^{2} + \left(305 a + 160\right)\cdot 491^{3} + \left(243 a + 302\right)\cdot 491^{4} +O(491^{5})\)
$r_{ 2 }$ |
$=$ |
\( 199 a + 190 + \left(399 a + 480\right)\cdot 491 + \left(241 a + 83\right)\cdot 491^{2} + \left(185 a + 151\right)\cdot 491^{3} + \left(247 a + 480\right)\cdot 491^{4} +O(491^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 460 a + 14 + \left(391 a + 208\right)\cdot 491 + \left(358 a + 435\right)\cdot 491^{2} + \left(93 a + 232\right)\cdot 491^{3} + \left(389 a + 22\right)\cdot 491^{4} +O(491^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 12 a + 130 + \left(293 a + 191\right)\cdot 491 + \left(29 a + 486\right)\cdot 491^{2} + \left(120 a + 31\right)\cdot 491^{3} + \left(249 a + 33\right)\cdot 491^{4} +O(491^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 479 a + 178 + \left(197 a + 369\right)\cdot 491 + \left(461 a + 311\right)\cdot 491^{2} + \left(370 a + 482\right)\cdot 491^{3} + \left(241 a + 418\right)\cdot 491^{4} +O(491^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 86 + 466\cdot 491 + 479\cdot 491^{2} + 164\cdot 491^{3} + 203\cdot 491^{4} +O(491^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 31 a + 381 + \left(99 a + 333\right)\cdot 491 + \left(132 a + 5\right)\cdot 491^{2} + \left(397 a + 249\right)\cdot 491^{3} + \left(101 a + 12\right)\cdot 491^{4} +O(491^{5})\)
| |
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.