Properties

Label 1.3e2_13.6t1.7c2
Dimension 1
Group $C_6$
Conductor $ 3^{2} \cdot 13 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$117= 3^{2} \cdot 13 $
Artin number field: Splitting field of $f= x^{6} - 39 x^{4} - 26 x^{3} + 351 x^{2} + 585 x + 117 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Even
Corresponding Dirichlet character: \(\chi_{117}(43,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 38 a + 39 + 8\cdot 41 + \left(37 a + 8\right)\cdot 41^{2} + \left(31 a + 31\right)\cdot 41^{3} + \left(21 a + 16\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 3 a + 30 + \left(40 a + 14\right)\cdot 41 + \left(3 a + 36\right)\cdot 41^{2} + \left(9 a + 7\right)\cdot 41^{3} + \left(19 a + 9\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 9 a + 23 + \left(33 a + 12\right)\cdot 41 + \left(40 a + 14\right)\cdot 41^{2} + \left(21 a + 10\right)\cdot 41^{3} + \left(35 a + 18\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 32 a + 9 + \left(7 a + 21\right)\cdot 41 + 21\cdot 41^{2} + \left(19 a + 35\right)\cdot 41^{3} + \left(5 a + 20\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 35 a + 20 + \left(6 a + 19\right)\cdot 41 + \left(4 a + 18\right)\cdot 41^{2} + \left(28 a + 40\right)\cdot 41^{3} + \left(24 a + 5\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 6 a + 2 + \left(34 a + 5\right)\cdot 41 + \left(36 a + 24\right)\cdot 41^{2} + \left(12 a + 38\right)\cdot 41^{3} + \left(16 a + 10\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$

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

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

Character values on conjugacy classes

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