Basic invariants
Dimension: | $4$ |
Group: | $C_3^2:D_4$ |
Conductor: | \(13941\)\(\medspace = 3^{2} \cdot 1549 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.41823.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_3^2:D_4$ |
Parity: | even |
Determinant: | 1.1549.2t1.a.a |
Projective image: | $\SOPlus(4,2)$ |
Projective stem field: | Galois closure of 6.0.41823.1 |
Defining polynomial
$f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 2x^{4} + 5x^{3} + x^{2} - 3x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$:
\( x^{2} + 6x + 3 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 3 + 6\cdot 7 + 3\cdot 7^{3} + 2\cdot 7^{4} +O(7^{5})\)
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$r_{ 2 }$ | $=$ |
\( 2 + 6\cdot 7 + 3\cdot 7^{2} + 2\cdot 7^{3} + 2\cdot 7^{4} +O(7^{5})\)
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$r_{ 3 }$ | $=$ |
\( 6 a + \left(2 a + 2\right)\cdot 7 + \left(2 a + 5\right)\cdot 7^{2} + \left(2 a + 5\right)\cdot 7^{3} + \left(4 a + 4\right)\cdot 7^{4} +O(7^{5})\)
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$r_{ 4 }$ | $=$ |
\( 6 a + 3 + \left(6 a + 3\right)\cdot 7 + \left(a + 5\right)\cdot 7^{2} + \left(6 a + 6\right)\cdot 7^{3} + \left(3 a + 6\right)\cdot 7^{4} +O(7^{5})\)
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$r_{ 5 }$ | $=$ |
\( a + 6 + \left(4 a + 5\right)\cdot 7 + \left(4 a + 4\right)\cdot 7^{2} + \left(4 a + 5\right)\cdot 7^{3} + \left(2 a + 6\right)\cdot 7^{4} +O(7^{5})\)
|
$r_{ 6 }$ | $=$ |
\( a + 2 + 4\cdot 7 + 5 a\cdot 7^{2} + 4\cdot 7^{3} + \left(3 a + 4\right)\cdot 7^{4} +O(7^{5})\)
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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 | Complex conjugation |
$1$ | $1$ | $()$ | $4$ | |
$6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ | ✓ |
$6$ | $2$ | $(3,5)$ | $2$ | |
$9$ | $2$ | $(3,5)(4,6)$ | $0$ | |
$4$ | $3$ | $(1,4,6)$ | $1$ | |
$4$ | $3$ | $(1,4,6)(2,3,5)$ | $-2$ | |
$18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ | |
$12$ | $6$ | $(1,3,4,5,6,2)$ | $0$ | |
$12$ | $6$ | $(1,4,6)(3,5)$ | $-1$ |