Properties

Label 1.19_53.6t1.1c2
Dimension 1
Group $C_6$
Conductor $ 19 \cdot 53 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$1007= 19 \cdot 53 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 249 x^{4} + 255 x^{3} + 6421 x^{2} + 6417 x + 51877 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{1007}(582,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 6 a + 10 + \left(35 a + 19\right)\cdot 37 + \left(23 a + 4\right)\cdot 37^{2} + \left(12 a + 19\right)\cdot 37^{3} + \left(25 a + 22\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 30 a + 18 + \left(29 a + 1\right)\cdot 37 + \left(6 a + 6\right)\cdot 37^{2} + \left(25 a + 13\right)\cdot 37^{3} + \left(33 a + 18\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 7 a + 27 + \left(7 a + 16\right)\cdot 37 + \left(30 a + 3\right)\cdot 37^{2} + \left(11 a + 33\right)\cdot 37^{3} + \left(3 a + 16\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 31 a + 34 + \left(a + 5\right)\cdot 37 + \left(13 a + 28\right)\cdot 37^{2} + \left(24 a + 8\right)\cdot 37^{3} + 11 a\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 5 a + 20 + \left(21 a + 12\right)\cdot 37 + \left(3 a + 19\right)\cdot 37^{2} + \left(2 a + 34\right)\cdot 37^{3} + 8\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 32 a + 3 + \left(15 a + 18\right)\cdot 37 + \left(33 a + 12\right)\cdot 37^{2} + \left(34 a + 2\right)\cdot 37^{3} + \left(36 a + 7\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$

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

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

Character values on conjugacy classes

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