Basic invariants
Dimension: | $9$ |
Group: | $S_6$ |
Conductor: | \(30176086429637\)\(\medspace = 163^{3} \cdot 191^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.31133.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_{6}$ |
Parity: | even |
Determinant: | 1.31133.2t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.2.31133.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{4} - x^{3} + 2x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: \( x^{2} + 96x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 10 + 46\cdot 97 + 94\cdot 97^{2} + 64\cdot 97^{3} + 28\cdot 97^{4} +O(97^{5})\) |
$r_{ 2 }$ | $=$ | \( 10 a + 33 + \left(28 a + 6\right)\cdot 97 + \left(12 a + 23\right)\cdot 97^{2} + \left(83 a + 45\right)\cdot 97^{3} + \left(11 a + 77\right)\cdot 97^{4} +O(97^{5})\) |
$r_{ 3 }$ | $=$ | \( 21 a + 27 + \left(5 a + 2\right)\cdot 97 + \left(77 a + 2\right)\cdot 97^{2} + \left(81 a + 72\right)\cdot 97^{3} + \left(32 a + 35\right)\cdot 97^{4} +O(97^{5})\) |
$r_{ 4 }$ | $=$ | \( 33 + 31\cdot 97 + 90\cdot 97^{2} + 12\cdot 97^{3} + 59\cdot 97^{4} +O(97^{5})\) |
$r_{ 5 }$ | $=$ | \( 76 a + 48 + \left(91 a + 83\right)\cdot 97 + \left(19 a + 73\right)\cdot 97^{2} + \left(15 a + 76\right)\cdot 97^{3} + \left(64 a + 83\right)\cdot 97^{4} +O(97^{5})\) |
$r_{ 6 }$ | $=$ | \( 87 a + 43 + \left(68 a + 24\right)\cdot 97 + \left(84 a + 7\right)\cdot 97^{2} + \left(13 a + 19\right)\cdot 97^{3} + \left(85 a + 6\right)\cdot 97^{4} +O(97^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
$1$ | $1$ | $()$ | $9$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
$15$ | $2$ | $(1,2)$ | $3$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$40$ | $3$ | $(1,2,3)$ | $0$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
$90$ | $4$ | $(1,2,3,4)$ | $-1$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.