Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(24208\)\(\medspace = 2^{4} \cdot 17 \cdot 89 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.96832.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Projective image: | $\SOPlus(4,2)$ |
| Projective field: | Galois closure of 6.0.96832.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 26 + 21\cdot 37 + 19\cdot 37^{2} + 16\cdot 37^{3} + 25\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 8\cdot 37 + 26\cdot 37^{2} + 6\cdot 37^{3} + 24\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 a + \left(25 a + 2\right)\cdot 37 + \left(34 a + 11\right)\cdot 37^{2} + \left(17 a + 2\right)\cdot 37^{3} + \left(5 a + 2\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 a + 6 + \left(11 a + 10\right)\cdot 37 + \left(2 a + 13\right)\cdot 37^{2} + \left(19 a + 2\right)\cdot 37^{3} + \left(31 a + 6\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 28 a + 17 + \left(23 a + 19\right)\cdot 37 + \left(15 a + 19\right)\cdot 37^{2} + \left(26 a + 33\right)\cdot 37^{3} + \left(20 a + 16\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 a + 18 + \left(13 a + 12\right)\cdot 37 + \left(21 a + 21\right)\cdot 37^{2} + \left(10 a + 12\right)\cdot 37^{3} + \left(16 a + 36\right)\cdot 37^{4} +O(37^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(3,4)$ | $2$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,4)$ | $1$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
| $12$ | $6$ | $(2,5,6)(3,4)$ | $-1$ |