Basic invariants
Dimension: | $2$ |
Group: | $S_3\times C_3$ |
Conductor: | \(1225\)\(\medspace = 5^{2} \cdot 7^{2} \) |
Artin stem field: | Galois closure of 6.0.10504375.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $S_3\times C_3$ |
Parity: | odd |
Determinant: | 1.7.6t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.175.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + x^{4} + 6x^{3} + 8x^{2} - 29x + 22 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 31 a + 31 + \left(25 a + 6\right)\cdot 41 + \left(15 a + 25\right)\cdot 41^{2} + \left(28 a + 38\right)\cdot 41^{3} + \left(7 a + 13\right)\cdot 41^{4} +O(41^{5})\)
$r_{ 2 }$ |
$=$ |
\( 5 a + 33 + \left(11 a + 10\right)\cdot 41 + \left(27 a + 29\right)\cdot 41^{2} + \left(23 a + 9\right)\cdot 41^{3} + \left(35 a + 23\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 30 a + 22 + \left(17 a + 15\right)\cdot 41 + \left(8 a + 39\right)\cdot 41^{2} + \left(34 a + 11\right)\cdot 41^{3} + \left(23 a + 28\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 11 a + 30 + \left(23 a + 38\right)\cdot 41 + \left(32 a + 5\right)\cdot 41^{2} + \left(6 a + 24\right)\cdot 41^{3} + \left(17 a + 24\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 10 a + 1 + \left(15 a + 12\right)\cdot 41 + \left(25 a + 5\right)\cdot 41^{2} + \left(12 a + 26\right)\cdot 41^{3} + \left(33 a + 8\right)\cdot 41^{4} +O(41^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 36 a + 7 + \left(29 a + 39\right)\cdot 41 + \left(13 a + 17\right)\cdot 41^{2} + \left(17 a + 12\right)\cdot 41^{3} + \left(5 a + 24\right)\cdot 41^{4} +O(41^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$3$ | $2$ | $(1,5)(2,6)(3,4)$ | $0$ |
$1$ | $3$ | $(1,4,2)(3,6,5)$ | $2 \zeta_{3}$ |
$1$ | $3$ | $(1,2,4)(3,5,6)$ | $-2 \zeta_{3} - 2$ |
$2$ | $3$ | $(3,6,5)$ | $\zeta_{3} + 1$ |
$2$ | $3$ | $(3,5,6)$ | $-\zeta_{3}$ |
$2$ | $3$ | $(1,4,2)(3,5,6)$ | $-1$ |
$3$ | $6$ | $(1,5,4,3,2,6)$ | $0$ |
$3$ | $6$ | $(1,6,2,3,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.