Basic invariants
Dimension: | $15$ |
Group: | $S_7$ |
Conductor: | \(155\!\cdots\!849\)\(\medspace = 23^{10} \cdot 14369^{10} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.330487.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 42T411 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.330487.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} + x^{5} - x^{3} - x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 521 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 521 }$: \( x^{2} + 515x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 488 + 5\cdot 521 + 3\cdot 521^{2} + 208\cdot 521^{3} + 195\cdot 521^{4} +O(521^{5})\) |
$r_{ 2 }$ | $=$ | \( 169 a + 518 + \left(14 a + 437\right)\cdot 521 + \left(326 a + 282\right)\cdot 521^{2} + \left(308 a + 342\right)\cdot 521^{3} + \left(289 a + 361\right)\cdot 521^{4} +O(521^{5})\) |
$r_{ 3 }$ | $=$ | \( 155 + 361\cdot 521 + 468\cdot 521^{2} + 423\cdot 521^{3} + 373\cdot 521^{4} +O(521^{5})\) |
$r_{ 4 }$ | $=$ | \( 352 a + 490 + \left(506 a + 354\right)\cdot 521 + \left(194 a + 140\right)\cdot 521^{2} + \left(212 a + 305\right)\cdot 521^{3} + \left(231 a + 227\right)\cdot 521^{4} +O(521^{5})\) |
$r_{ 5 }$ | $=$ | \( 199 + 373\cdot 521 + 481\cdot 521^{2} + 3\cdot 521^{3} + 235\cdot 521^{4} +O(521^{5})\) |
$r_{ 6 }$ | $=$ | \( 327 + 10\cdot 521 + 31\cdot 521^{2} + 419\cdot 521^{3} + 143\cdot 521^{4} +O(521^{5})\) |
$r_{ 7 }$ | $=$ | \( 428 + 18\cdot 521 + 155\cdot 521^{2} + 381\cdot 521^{3} + 25\cdot 521^{4} +O(521^{5})\) |
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 |
$1$ | $1$ | $()$ | $15$ |
$21$ | $2$ | $(1,2)$ | $-5$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
$105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$70$ | $3$ | $(1,2,3)$ | $3$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$210$ | $4$ | $(1,2,3,4)$ | $-1$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.