Properties

Label 1.7_11_13.6t1.1c2
Dimension 1
Group $C_6$
Conductor $ 7 \cdot 11 \cdot 13 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$1001= 7 \cdot 11 \cdot 13 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 103 x^{4} - 69 x^{3} + 3857 x^{2} - 1225 x + 51947 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{1001}(571,\cdot)\)

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:
$r_{ 1 }$ $=$ $ 16 a + 22 + 21 a\cdot 29 + \left(18 a + 7\right)\cdot 29^{2} + \left(12 a + 6\right)\cdot 29^{3} + \left(23 a + 2\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 16 a + 11 + \left(21 a + 9\right)\cdot 29 + \left(18 a + 23\right)\cdot 29^{2} + \left(12 a + 20\right)\cdot 29^{3} + \left(23 a + 7\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 13 a + 4 + \left(7 a + 14\right)\cdot 29 + \left(10 a + 8\right)\cdot 29^{2} + \left(16 a + 7\right)\cdot 29^{3} + \left(5 a + 25\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 13 a + \left(7 a + 2\right)\cdot 29 + \left(10 a + 6\right)\cdot 29^{2} + \left(16 a + 23\right)\cdot 29^{3} + \left(5 a + 24\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 16 a + 7 + \left(21 a + 26\right)\cdot 29 + \left(18 a + 20\right)\cdot 29^{2} + \left(12 a + 7\right)\cdot 29^{3} + \left(23 a + 7\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 13 a + 15 + \left(7 a + 5\right)\cdot 29 + \left(10 a + 21\right)\cdot 29^{2} + \left(16 a + 21\right)\cdot 29^{3} + \left(5 a + 19\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

Character values on conjugacy classes

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