Properties

Label 1.5_19.6t1.1c2
Dimension 1
Group $C_6$
Conductor $ 5 \cdot 19 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$95= 5 \cdot 19 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 16 x^{4} + x^{3} + 47 x^{2} + 10 x - 11 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Even
Corresponding Dirichlet character: \(\chi_{95}(49,\cdot)\)

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:
$r_{ 1 }$ $=$ $ 25 a + 35 + \left(8 a + 13\right)\cdot 37 + \left(16 a + 36\right)\cdot 37^{2} + \left(29 a + 13\right)\cdot 37^{3} + \left(13 a + 7\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 12 a + 25 + \left(28 a + 16\right)\cdot 37 + \left(20 a + 32\right)\cdot 37^{2} + \left(7 a + 22\right)\cdot 37^{3} + \left(23 a + 8\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 12 a + 24 + \left(28 a + 23\right)\cdot 37 + \left(20 a + 18\right)\cdot 37^{2} + \left(7 a + 4\right)\cdot 37^{3} + \left(23 a + 33\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 25 a + 36 + \left(8 a + 6\right)\cdot 37 + \left(16 a + 13\right)\cdot 37^{2} + \left(29 a + 32\right)\cdot 37^{3} + \left(13 a + 19\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 12 a + 9 + \left(28 a + 11\right)\cdot 37 + \left(20 a + 33\right)\cdot 37^{2} + \left(7 a + 13\right)\cdot 37^{3} + \left(23 a + 15\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 25 a + 20 + \left(8 a + 1\right)\cdot 37 + \left(16 a + 14\right)\cdot 37^{2} + \left(29 a + 23\right)\cdot 37^{3} + \left(13 a + 26\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$

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

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

Character values on conjugacy classes

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