Properties

Label 2.980.6t3.b
Dimension $2$
Group $D_{6}$
Conductor $980$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:\(980\)\(\medspace = 2^{2} \cdot 5 \cdot 7^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.26891200.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Projective image: $S_3$
Projective field: 3.1.980.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \(x^{2} + 18 x + 2\)  Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 4 + 19 + 15\cdot 19^{2} + 5\cdot 19^{3} + 10\cdot 19^{4} + 10\cdot 19^{5} + 14\cdot 19^{6} +O(19^{7})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 8 + 10\cdot 19 + 18\cdot 19^{2} + 18\cdot 19^{3} + 5\cdot 19^{4} + 4\cdot 19^{5} + 4\cdot 19^{6} +O(19^{7})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 7 + \left(3 a + 8\right)\cdot 19 + \left(2 a + 2\right)\cdot 19^{2} + 15 a\cdot 19^{3} + 2\cdot 19^{4} + \left(14 a + 7\right)\cdot 19^{5} + \left(18 a + 9\right)\cdot 19^{6} +O(19^{7})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 15 a + 8 + \left(2 a + 10\right)\cdot 19 + 2 a\cdot 19^{2} + \left(7 a + 7\right)\cdot 19^{3} + \left(4 a + 17\right)\cdot 19^{4} + \left(7 a + 5\right)\cdot 19^{5} + 2 a\cdot 19^{6} +O(19^{7})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 4 a + 4 + \left(16 a + 17\right)\cdot 19 + \left(16 a + 18\right)\cdot 19^{2} + \left(11 a + 11\right)\cdot 19^{3} + \left(14 a + 14\right)\cdot 19^{4} + \left(11 a + 8\right)\cdot 19^{5} + \left(16 a + 14\right)\cdot 19^{6} +O(19^{7})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 17 a + 9 + \left(15 a + 9\right)\cdot 19 + \left(16 a + 1\right)\cdot 19^{2} + \left(3 a + 13\right)\cdot 19^{3} + \left(18 a + 6\right)\cdot 19^{4} + \left(4 a + 1\right)\cdot 19^{5} + 14\cdot 19^{6} +O(19^{7})\)  Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2)(3,5)(4,6)$
$(3,6)(4,5)$
$(1,3)(2,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $2$
$1$ $2$ $(1,2)(3,5)(4,6)$ $-2$
$3$ $2$ $(1,3)(2,5)$ $0$
$3$ $2$ $(1,5)(2,3)(4,6)$ $0$
$2$ $3$ $(1,6,3)(2,4,5)$ $-1$
$2$ $6$ $(1,4,3,2,6,5)$ $1$
The blue line marks the conjugacy class containing complex conjugation.