Basic invariants
Dimension: | $5$ |
Group: | $S_6$ |
Conductor: | \(50173\)\(\medspace = 131 \cdot 383 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.50173.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_6$ |
Parity: | even |
Determinant: | 1.50173.2t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.2.50173.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + x^{4} + 2x^{3} - 4x^{2} + 3x - 1 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 5 + 91\cdot 97 + 75\cdot 97^{2} + 36\cdot 97^{4} +O(97^{5})\)
$r_{ 2 }$ |
$=$ |
\( 66 a + 3 + \left(28 a + 43\right)\cdot 97 + \left(16 a + 75\right)\cdot 97^{2} + \left(70 a + 66\right)\cdot 97^{3} + \left(26 a + 34\right)\cdot 97^{4} +O(97^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 31 + 12\cdot 97 + 72\cdot 97^{2} + 47\cdot 97^{3} + 92\cdot 97^{4} +O(97^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 87 a + 31 a\cdot 97 + \left(a + 66\right)\cdot 97^{2} + \left(54 a + 49\right)\cdot 97^{3} + \left(71 a + 59\right)\cdot 97^{4} +O(97^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 10 a + 87 + \left(65 a + 41\right)\cdot 97 + \left(95 a + 35\right)\cdot 97^{2} + \left(42 a + 5\right)\cdot 97^{3} + \left(25 a + 77\right)\cdot 97^{4} +O(97^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 31 a + 69 + \left(68 a + 5\right)\cdot 97 + \left(80 a + 63\right)\cdot 97^{2} + \left(26 a + 23\right)\cdot 97^{3} + \left(70 a + 88\right)\cdot 97^{4} +O(97^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $5$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
$15$ | $2$ | $(1,2)$ | $3$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$40$ | $3$ | $(1,2,3)$ | $2$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$90$ | $4$ | $(1,2,3,4)$ | $1$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.