Basic invariants
Dimension: | $3$ |
Group: | $S_4\times C_2$ |
Conductor: | \(31211\)\(\medspace = 23^{2} \cdot 59 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.31211.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_4\times C_2$ |
Parity: | odd |
Determinant: | 1.59.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.80063.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{4} - x^{3} + 2x^{2} + 3x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: \( x^{2} + 78x + 3 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 6 a + 38 + \left(41 a + 19\right)\cdot 79 + \left(74 a + 66\right)\cdot 79^{2} + \left(50 a + 19\right)\cdot 79^{3} + \left(15 a + 70\right)\cdot 79^{4} + \left(76 a + 23\right)\cdot 79^{5} +O(79^{6})\)
$r_{ 2 }$ |
$=$ |
\( 72 a + 12 + \left(26 a + 75\right)\cdot 79 + \left(20 a + 7\right)\cdot 79^{2} + \left(66 a + 45\right)\cdot 79^{3} + \left(69 a + 4\right)\cdot 79^{4} + \left(16 a + 13\right)\cdot 79^{5} +O(79^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 68 + 12\cdot 79 + 16\cdot 79^{2} + 44\cdot 79^{3} + 43\cdot 79^{4} + 6\cdot 79^{5} +O(79^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 73 a + 44 + \left(37 a + 54\right)\cdot 79 + \left(4 a + 20\right)\cdot 79^{2} + \left(28 a + 75\right)\cdot 79^{3} + \left(63 a + 34\right)\cdot 79^{4} + \left(2 a + 5\right)\cdot 79^{5} +O(79^{6})\)
| $r_{ 5 }$ |
$=$ |
\( 70 + 44\cdot 79 + 45\cdot 79^{2} + 40\cdot 79^{3} + 75\cdot 79^{4} + 69\cdot 79^{5} +O(79^{6})\)
| $r_{ 6 }$ |
$=$ |
\( 7 a + 5 + \left(52 a + 30\right)\cdot 79 + \left(58 a + 1\right)\cdot 79^{2} + \left(12 a + 12\right)\cdot 79^{3} + \left(9 a + 8\right)\cdot 79^{4} + \left(62 a + 39\right)\cdot 79^{5} +O(79^{6})\)
| |
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 |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-3$ |
$3$ | $2$ | $(1,2)(4,6)$ | $-1$ |
$3$ | $2$ | $(1,2)$ | $1$ |
$6$ | $2$ | $(1,4)(2,6)$ | $-1$ |
$6$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
$8$ | $3$ | $(1,3,4)(2,5,6)$ | $0$ |
$6$ | $4$ | $(1,6,2,4)$ | $-1$ |
$6$ | $4$ | $(1,6,2,4)(3,5)$ | $1$ |
$8$ | $6$ | $(1,6,5,2,4,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.