Properties

Label 1.2e3_13.6t1.2c2
Dimension 1
Group $C_6$
Conductor $ 2^{3} \cdot 13 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$104= 2^{3} \cdot 13 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} - 13 x^{4} + 14 x^{3} + 26 x^{2} - 28 x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Even
Corresponding Dirichlet character: \(\chi_{104}(61,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 52 a + 16 + \left(25 a + 15\right)\cdot 53 + \left(29 a + 37\right)\cdot 53^{2} + \left(34 a + 49\right)\cdot 53^{3} + \left(38 a + 19\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 52 a + 35 + \left(25 a + 29\right)\cdot 53 + \left(29 a + 52\right)\cdot 53^{2} + \left(34 a + 37\right)\cdot 53^{3} + \left(38 a + 50\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$ $=$ $ a + 5 + \left(27 a + 35\right)\cdot 53 + \left(23 a + 49\right)\cdot 53^{2} + \left(18 a + 16\right)\cdot 53^{3} + \left(14 a + 28\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 52 a + 9 + \left(25 a + 36\right)\cdot 53 + \left(29 a + 10\right)\cdot 53^{2} + \left(34 a + 14\right)\cdot 53^{3} + \left(38 a + 14\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 5 }$ $=$ $ a + 31 + \left(27 a + 28\right)\cdot 53 + \left(23 a + 38\right)\cdot 53^{2} + \left(18 a + 40\right)\cdot 53^{3} + \left(14 a + 11\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 6 }$ $=$ $ a + 12 + \left(27 a + 14\right)\cdot 53 + \left(23 a + 23\right)\cdot 53^{2} + \left(18 a + 52\right)\cdot 53^{3} + \left(14 a + 33\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$

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

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

Character values on conjugacy classes

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