Properties

Label 1.104.6t1.d.b
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 6.0.14623232.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{104}(35,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} - x^{4} - 2x^{3} + 34x^{2} + 28x + 73 \) 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:
$r_{ 1 }$ $=$ \( a + 3 + \left(23 a + 30\right)\cdot 31 + \left(12 a + 29\right)\cdot 31^{2} + \left(23 a + 23\right)\cdot 31^{3} + \left(6 a + 25\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 30 a + 23 + \left(7 a + 26\right)\cdot 31 + \left(18 a + 25\right)\cdot 31^{2} + \left(7 a + 7\right)\cdot 31^{3} + \left(24 a + 8\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 30 a + 7 + \left(7 a + 12\right)\cdot 31 + \left(18 a + 7\right)\cdot 31^{2} + \left(7 a + 16\right)\cdot 31^{3} + \left(24 a + 7\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( a + 5 + \left(23 a + 29\right)\cdot 31 + \left(12 a + 4\right)\cdot 31^{2} + \left(23 a + 13\right)\cdot 31^{3} + \left(6 a + 17\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( a + 21 + \left(23 a + 12\right)\cdot 31 + \left(12 a + 23\right)\cdot 31^{2} + \left(23 a + 4\right)\cdot 31^{3} + \left(6 a + 18\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 30 a + 5 + \left(7 a + 13\right)\cdot 31 + \left(18 a + 1\right)\cdot 31^{2} + \left(7 a + 27\right)\cdot 31^{3} + \left(24 a + 15\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.