Properties

Label 1.3e2_1117.6t1.1c2
Dimension 1
Group $C_6$
Conductor $ 3^{2} \cdot 1117 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$10053= 3^{2} \cdot 1117 $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} - 840 x^{4} + 1683 x^{3} + 232689 x^{2} - 235200 x - 21251431 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Even
Corresponding Dirichlet character: \(\chi_{10053}(5584,\cdot)\)

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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