Properties

Label 1.104.6t1.c.a
Dimension $1$
Group $C_6$
Conductor $104$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(104\)\(\medspace = 2^{3} \cdot 13 \)
Artin field: Galois closure of 6.6.190102016.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{104}(69,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - 26x^{4} + 104x^{2} - 104 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 22 a + 9 + \left(18 a + 23\right)\cdot 31 + \left(28 a + 11\right)\cdot 31^{2} + 13 a\cdot 31^{3} + \left(8 a + 14\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 12 a + 19 + \left(7 a + 29\right)\cdot 31 + \left(6 a + 12\right)\cdot 31^{2} + \left(6 a + 12\right)\cdot 31^{3} + \left(24 a + 25\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 14 a + 17 + \left(22 a + 15\right)\cdot 31 + \left(23 a + 18\right)\cdot 31^{2} + \left(3 a + 23\right)\cdot 31^{3} + \left(11 a + 21\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 9 a + 22 + \left(12 a + 7\right)\cdot 31 + \left(2 a + 19\right)\cdot 31^{2} + \left(17 a + 30\right)\cdot 31^{3} + \left(22 a + 16\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 19 a + 12 + \left(23 a + 1\right)\cdot 31 + \left(24 a + 18\right)\cdot 31^{2} + \left(24 a + 18\right)\cdot 31^{3} + \left(6 a + 5\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 17 a + 14 + \left(8 a + 15\right)\cdot 31 + \left(7 a + 12\right)\cdot 31^{2} + \left(27 a + 7\right)\cdot 31^{3} + \left(19 a + 9\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display

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)$$\zeta_{3}$
$1$$3$$(1,5,3)(2,6,4)$$-\zeta_{3} - 1$
$1$$6$$(1,2,3,4,5,6)$$\zeta_{3} + 1$
$1$$6$$(1,6,5,4,3,2)$$-\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.