Basic invariants
Dimension: | $21$ |
Group: | $S_7$ |
Conductor: | \(132\!\cdots\!249\)\(\medspace = 23^{10} \cdot 709^{10} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.5.11561663.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 84 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.5.11561663.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} - 3x^{5} + 5x^{4} + 2x^{3} - 8x^{2} - 2x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 311 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 311 }$: \( x^{2} + 310x + 17 \)
Roots:
$r_{ 1 }$ | $=$ | \( 247 a + 33 + \left(16 a + 221\right)\cdot 311 + \left(23 a + 182\right)\cdot 311^{2} + \left(118 a + 249\right)\cdot 311^{3} + \left(206 a + 242\right)\cdot 311^{4} +O(311^{5})\) |
$r_{ 2 }$ | $=$ | \( 27 + 256\cdot 311 + 198\cdot 311^{2} + 92\cdot 311^{3} + 86\cdot 311^{4} +O(311^{5})\) |
$r_{ 3 }$ | $=$ | \( 105 + 264\cdot 311 + 16\cdot 311^{2} + 213\cdot 311^{3} + 152\cdot 311^{4} +O(311^{5})\) |
$r_{ 4 }$ | $=$ | \( 5 + 126\cdot 311 + 126\cdot 311^{2} + 251\cdot 311^{3} + 32\cdot 311^{4} +O(311^{5})\) |
$r_{ 5 }$ | $=$ | \( 259 a + 268 + \left(294 a + 174\right)\cdot 311 + \left(22 a + 245\right)\cdot 311^{2} + \left(90 a + 12\right)\cdot 311^{3} + \left(173 a + 2\right)\cdot 311^{4} +O(311^{5})\) |
$r_{ 6 }$ | $=$ | \( 64 a + 280 + \left(294 a + 301\right)\cdot 311 + \left(287 a + 188\right)\cdot 311^{2} + \left(192 a + 33\right)\cdot 311^{3} + \left(104 a + 20\right)\cdot 311^{4} +O(311^{5})\) |
$r_{ 7 }$ | $=$ | \( 52 a + 216 + \left(16 a + 210\right)\cdot 311 + \left(288 a + 284\right)\cdot 311^{2} + \left(220 a + 79\right)\cdot 311^{3} + \left(137 a + 85\right)\cdot 311^{4} +O(311^{5})\) |
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 |
$1$ | $1$ | $()$ | $21$ |
$21$ | $2$ | $(1,2)$ | $1$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
$105$ | $2$ | $(1,2)(3,4)$ | $1$ |
$70$ | $3$ | $(1,2,3)$ | $-3$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$210$ | $4$ | $(1,2,3,4)$ | $-1$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.