Properties

Label 1.2e3_13.6t1.4c2
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} + 26 x^{4} + 104 x^{2} + 104 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{104}(43,\cdot)\)

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 + 2 + 12 a\cdot 53 + \left(22 a + 15\right)\cdot 53^{2} + \left(46 a + 24\right)\cdot 53^{3} + \left(13 a + 48\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 19 a + 15 + \left(15 a + 5\right)\cdot 53 + \left(21 a + 18\right)\cdot 53^{2} + \left(28 a + 33\right)\cdot 53^{3} + \left(5 a + 29\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 46 a + 14 + \left(28 a + 18\right)\cdot 53 + \left(30 a + 6\right)\cdot 53^{2} + \left(51 a + 18\right)\cdot 53^{3} + \left(30 a + 43\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$ $=$ $ a + 51 + \left(40 a + 52\right)\cdot 53 + \left(30 a + 37\right)\cdot 53^{2} + \left(6 a + 28\right)\cdot 53^{3} + \left(39 a + 4\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 34 a + 38 + \left(37 a + 47\right)\cdot 53 + \left(31 a + 34\right)\cdot 53^{2} + \left(24 a + 19\right)\cdot 53^{3} + \left(47 a + 23\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 7 a + 39 + \left(24 a + 34\right)\cdot 53 + \left(22 a + 46\right)\cdot 53^{2} + \left(a + 34\right)\cdot 53^{3} + \left(22 a + 9\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$

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

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

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$$3$$(1,3,5)(2,4,6)$$-\zeta_{3} - 1$
$1$$3$$(1,5,3)(2,6,4)$$\zeta_{3}$
$1$$6$$(1,2,3,4,5,6)$$-\zeta_{3}$
$1$$6$$(1,6,5,4,3,2)$$\zeta_{3} + 1$
The blue line marks the conjugacy class containing complex conjugation.