Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:C_4$ |
| Conductor: | \(117128000\)\(\medspace = 2^{6} \cdot 5^{3} \cdot 11^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.2.4840000.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:C_4$ |
| Parity: | even |
| Projective image: | $C_3^2:C_4$ |
| Projective field: | Galois closure of 6.2.4840000.2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{2} + 60x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 2 a + 32 + \left(41 a + 43\right)\cdot 61 + \left(2 a + 26\right)\cdot 61^{2} + \left(47 a + 25\right)\cdot 61^{3} + \left(50 a + 25\right)\cdot 61^{4} + \left(8 a + 11\right)\cdot 61^{5} + \left(7 a + 15\right)\cdot 61^{6} + \left(29 a + 13\right)\cdot 61^{7} + \left(28 a + 6\right)\cdot 61^{8} + \left(60 a + 25\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 59 a + 34 + \left(19 a + 21\right)\cdot 61 + \left(58 a + 49\right)\cdot 61^{2} + \left(13 a + 8\right)\cdot 61^{3} + \left(10 a + 29\right)\cdot 61^{4} + \left(52 a + 30\right)\cdot 61^{5} + \left(53 a + 13\right)\cdot 61^{6} + \left(31 a + 35\right)\cdot 61^{7} + \left(32 a + 5\right)\cdot 61^{8} + 57\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 58 + 13\cdot 61 + 14\cdot 61^{2} + 34\cdot 61^{3} + 28\cdot 61^{4} + 6\cdot 61^{5} + 16\cdot 61^{6} + 55\cdot 61^{7} + 2\cdot 61^{8} + 44\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 45 a + 49 + \left(60 a + 28\right)\cdot 61 + 44\cdot 61^{2} + \left(16 a + 14\right)\cdot 61^{3} + \left(54 a + 27\right)\cdot 61^{4} + \left(19 a + 45\right)\cdot 61^{5} + \left(a + 28\right)\cdot 61^{6} + \left(46 a + 33\right)\cdot 61^{7} + 25\cdot 61^{8} + \left(58 a + 7\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 39 + 30\cdot 61 + 2\cdot 61^{2} + 9\cdot 61^{3} + 7\cdot 61^{4} + 17\cdot 61^{5} + 38\cdot 61^{6} + 28\cdot 61^{7} + 40\cdot 61^{8} + 45\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a + 33 + 44\cdot 61 + \left(60 a + 45\right)\cdot 61^{2} + \left(44 a + 29\right)\cdot 61^{3} + \left(6 a + 4\right)\cdot 61^{4} + \left(41 a + 11\right)\cdot 61^{5} + \left(59 a + 10\right)\cdot 61^{6} + \left(14 a + 17\right)\cdot 61^{7} + \left(60 a + 41\right)\cdot 61^{8} + \left(2 a + 3\right)\cdot 61^{9} +O(61^{10})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $4$ |
| $9$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $4$ | $3$ | $(1,2,5)$ | $-2$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $1$ |
| $9$ | $4$ | $(1,4,2,3)(5,6)$ | $0$ |
| $9$ | $4$ | $(1,3,2,4)(5,6)$ | $0$ |