Basic invariants
Dimension: | $35$ |
Group: | $S_7$ |
Conductor: | \(515\!\cdots\!296\)\(\medspace = 2^{105} \cdot 3^{61} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.161243136.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 70 |
Parity: | odd |
Determinant: | 1.24.2t1.b.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.161243136.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 2x^{6} + 6x^{5} - 10x^{4} + 14x^{3} - 12x^{2} + 4x + 4 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 389 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 389 }$: \( x^{2} + 379x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 46 + 318\cdot 389 + 188\cdot 389^{2} + 303\cdot 389^{3} + 325\cdot 389^{4} +O(389^{5})\)
$r_{ 2 }$ |
$=$ |
\( 226 + 81\cdot 389 + 2\cdot 389^{2} + 134\cdot 389^{3} + 3\cdot 389^{4} +O(389^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 78 a + 124 + \left(217 a + 258\right)\cdot 389 + \left(350 a + 367\right)\cdot 389^{2} + \left(324 a + 32\right)\cdot 389^{3} + \left(237 a + 251\right)\cdot 389^{4} +O(389^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 311 a + 126 + \left(171 a + 18\right)\cdot 389 + \left(38 a + 155\right)\cdot 389^{2} + \left(64 a + 208\right)\cdot 389^{3} + \left(151 a + 359\right)\cdot 389^{4} +O(389^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 339 a + 174 + \left(340 a + 227\right)\cdot 389 + \left(254 a + 363\right)\cdot 389^{2} + \left(49 a + 186\right)\cdot 389^{3} + \left(334 a + 140\right)\cdot 389^{4} +O(389^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 21 + 78\cdot 389 + 241\cdot 389^{2} + 261\cdot 389^{3} + 155\cdot 389^{4} +O(389^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 50 a + 63 + \left(48 a + 185\right)\cdot 389 + \left(134 a + 237\right)\cdot 389^{2} + \left(339 a + 39\right)\cdot 389^{3} + \left(54 a + 320\right)\cdot 389^{4} +O(389^{5})\)
| |
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.