Properties

Label 1.3_5_7.6t1.1c1
Dimension 1
Group $C_6$
Conductor $ 3 \cdot 5 \cdot 7 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$105= 3 \cdot 5 \cdot 7 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 27 x^{4} + 27 x^{3} + 197 x^{2} - 197 x - 251 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Even
Corresponding Dirichlet character: \(\chi_{105}(89,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 17 a + 14 + \left(25 a + 24\right)\cdot 29 + \left(6 a + 17\right)\cdot 29^{2} + \left(8 a + 18\right)\cdot 29^{3} + \left(28 a + 19\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 9 a + 12 + \left(20 a + 17\right)\cdot 29 + \left(10 a + 12\right)\cdot 29^{2} + \left(26 a + 19\right)\cdot 29^{3} + \left(27 a + 17\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 12 a + 12 + \left(3 a + 19\right)\cdot 29 + \left(22 a + 26\right)\cdot 29^{2} + \left(20 a + 23\right)\cdot 29^{3} + 7\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 7 a + 8 + \left(4 a + 23\right)\cdot 29 + \left(9 a + 14\right)\cdot 29^{2} + \left(16 a + 7\right)\cdot 29^{3} + \left(5 a + 22\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 20 a + 28 + \left(8 a + 22\right)\cdot 29 + \left(18 a + 16\right)\cdot 29^{2} + \left(2 a + 24\right)\cdot 29^{3} + \left(a + 14\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 22 a + 14 + \left(24 a + 8\right)\cdot 29 + \left(19 a + 27\right)\cdot 29^{2} + \left(12 a + 21\right)\cdot 29^{3} + \left(23 a + 4\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

Character values on conjugacy classes

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