Basic invariants
Dimension: | $3$ |
Group: | $S_4\times C_2$ |
Conductor: | \(17712697\)\(\medspace = 41^{3} \cdot 257 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.4552163129.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4\times C_2$ |
Parity: | even |
Determinant: | 1.10537.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.0.257.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} + 18x^{4} + 56x^{3} - 257x^{2} - 794x - 1671 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: \( x^{2} + 60x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 4 + 37\cdot 61 + 43\cdot 61^{2} + 14\cdot 61^{3} + 56\cdot 61^{4} + 52\cdot 61^{5} + 34\cdot 61^{6} + 35\cdot 61^{7} + 55\cdot 61^{8} +O(61^{10})\)
$r_{ 2 }$ |
$=$ |
\( 49 a + 51 + \left(34 a + 50\right)\cdot 61 + \left(52 a + 59\right)\cdot 61^{2} + \left(45 a + 60\right)\cdot 61^{3} + \left(54 a + 20\right)\cdot 61^{4} + \left(6 a + 56\right)\cdot 61^{5} + \left(5 a + 36\right)\cdot 61^{6} + \left(17 a + 30\right)\cdot 61^{7} + \left(44 a + 17\right)\cdot 61^{8} + \left(41 a + 24\right)\cdot 61^{9} +O(61^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 59 a + 9 + \left(57 a + 17\right)\cdot 61 + \left(43 a + 3\right)\cdot 61^{2} + \left(36 a + 41\right)\cdot 61^{3} + \left(5 a + 39\right)\cdot 61^{4} + \left(57 a + 8\right)\cdot 61^{5} + \left(25 a + 4\right)\cdot 61^{6} + \left(4 a + 22\right)\cdot 61^{7} + \left(34 a + 53\right)\cdot 61^{8} + \left(3 a + 49\right)\cdot 61^{9} +O(61^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 14 + 25\cdot 61 + 9\cdot 61^{2} + 39\cdot 61^{3} + 27\cdot 61^{4} + 57\cdot 61^{5} + 37\cdot 61^{6} + 51\cdot 61^{7} + 50\cdot 61^{8} + 5\cdot 61^{9} +O(61^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 2 a + 7 + \left(3 a + 16\right)\cdot 61 + \left(17 a + 50\right)\cdot 61^{2} + \left(24 a + 33\right)\cdot 61^{3} + \left(55 a + 8\right)\cdot 61^{4} + \left(3 a + 60\right)\cdot 61^{5} + \left(35 a + 33\right)\cdot 61^{6} + 56 a\cdot 61^{7} + \left(26 a + 22\right)\cdot 61^{8} + \left(57 a + 19\right)\cdot 61^{9} +O(61^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 12 a + 39 + \left(26 a + 36\right)\cdot 61 + \left(8 a + 16\right)\cdot 61^{2} + \left(15 a + 54\right)\cdot 61^{3} + \left(6 a + 29\right)\cdot 61^{4} + \left(54 a + 8\right)\cdot 61^{5} + \left(55 a + 35\right)\cdot 61^{6} + \left(43 a + 42\right)\cdot 61^{7} + \left(16 a + 44\right)\cdot 61^{8} + \left(19 a + 21\right)\cdot 61^{9} +O(61^{10})\)
| |
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 |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,4)(2,6)(3,5)$ | $-3$ |
$3$ | $2$ | $(3,5)$ | $1$ |
$3$ | $2$ | $(2,6)(3,5)$ | $-1$ |
$6$ | $2$ | $(1,2)(4,6)$ | $1$ |
$6$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ |
$8$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
$6$ | $4$ | $(2,3,6,5)$ | $1$ |
$6$ | $4$ | $(1,4)(2,3,6,5)$ | $-1$ |
$8$ | $6$ | $(1,3,6,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.