Properties

Label 1.7_11_13.6t1.2c2
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} - 52 x^{4} + 109 x^{3} + 897 x^{2} - 3168 x + 4887 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{1001}(263,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 8 a + 5 + \left(9 a + 5\right)\cdot 19 + 8 a\cdot 19^{2} + \left(3 a + 5\right)\cdot 19^{3} + \left(4 a + 5\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 11 a + 2 + \left(9 a + 16\right)\cdot 19 + \left(10 a + 9\right)\cdot 19^{2} + \left(15 a + 3\right)\cdot 19^{3} + \left(14 a + 7\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 11 a + 7 + \left(9 a + 17\right)\cdot 19 + \left(10 a + 17\right)\cdot 19^{2} + \left(15 a + 7\right)\cdot 19^{3} + \left(14 a + 16\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 11 a + 13 + \left(9 a + 6\right)\cdot 19 + \left(10 a + 18\right)\cdot 19^{2} + \left(15 a + 18\right)\cdot 19^{3} + \left(14 a + 5\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 8 a + 13 + \left(9 a + 14\right)\cdot 19 + \left(8 a + 10\right)\cdot 19^{2} + \left(3 a + 8\right)\cdot 19^{3} + \left(4 a + 6\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 8 a + 18 + \left(9 a + 15\right)\cdot 19 + \left(8 a + 18\right)\cdot 19^{2} + \left(3 a + 12\right)\cdot 19^{3} + \left(4 a + 15\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$

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

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

Character values on conjugacy classes

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