Properties

Label 2.980.6t5.a
Dimension $2$
Group $S_3\times C_3$
Conductor $980$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$S_3\times C_3$
Conductor:\(980\)\(\medspace = 2^{2} \cdot 5 \cdot 7^{2} \)
Artin number field: Galois closure of 6.0.33614000.1
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Projective image: $S_3$
Projective field: Galois closure of 3.1.140.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 14 a + 28 + \left(5 a + 20\right)\cdot 41 + \left(22 a + 37\right)\cdot 41^{2} + \left(36 a + 27\right)\cdot 41^{3} + \left(2 a + 13\right)\cdot 41^{4} + \left(6 a + 31\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 13 a + 15 + \left(6 a + 20\right)\cdot 41 + \left(33 a + 19\right)\cdot 41^{2} + \left(20 a + 16\right)\cdot 41^{3} + \left(34 a + 6\right)\cdot 41^{4} + \left(39 a + 23\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 28 a + 13 + \left(34 a + 26\right)\cdot 41 + \left(7 a + 30\right)\cdot 41^{2} + \left(20 a + 4\right)\cdot 41^{3} + \left(6 a + 7\right)\cdot 41^{4} + \left(a + 26\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 27 a + 29 + \left(35 a + 22\right)\cdot 41 + \left(18 a + 16\right)\cdot 41^{2} + \left(4 a + 33\right)\cdot 41^{3} + \left(38 a + 26\right)\cdot 41^{4} + \left(34 a + 5\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( a + 39 + \left(40 a + 38\right)\cdot 41 + \left(29 a + 4\right)\cdot 41^{2} + \left(15 a + 32\right)\cdot 41^{3} + \left(9 a + 7\right)\cdot 41^{4} + \left(7 a + 12\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 40 a + 1 + 35\cdot 41 + \left(11 a + 13\right)\cdot 41^{2} + \left(25 a + 8\right)\cdot 41^{3} + \left(31 a + 20\right)\cdot 41^{4} + \left(33 a + 24\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $2$ $2$
$3$ $2$ $(1,4)(2,3)(5,6)$ $0$ $0$
$1$ $3$ $(1,3,6)(2,5,4)$ $2 \zeta_{3}$ $-2 \zeta_{3} - 2$
$1$ $3$ $(1,6,3)(2,4,5)$ $-2 \zeta_{3} - 2$ $2 \zeta_{3}$
$2$ $3$ $(1,3,6)$ $\zeta_{3} + 1$ $-\zeta_{3}$
$2$ $3$ $(1,6,3)$ $-\zeta_{3}$ $\zeta_{3} + 1$
$2$ $3$ $(1,6,3)(2,5,4)$ $-1$ $-1$
$3$ $6$ $(1,5,3,4,6,2)$ $0$ $0$
$3$ $6$ $(1,2,6,4,3,5)$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.