Properties

Label 1.344.6t1.d.a
Dimension $1$
Group $C_6$
Conductor $344$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(344\)\(\medspace = 2^{3} \cdot 43 \)
Artin number field: Galois closure of 6.6.75268322816.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{344}(179,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{6} - 86 x^{4} + 1376 x^{2} - 5504 $.

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $

Roots:
$r_{ 1 }$ $=$ $ 32 a + 15 + \left(a + 14\right)\cdot 47 + \left(22 a + 2\right)\cdot 47^{2} + \left(28 a + 6\right)\cdot 47^{3} + \left(29 a + 8\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 19 a + 28 + \left(24 a + 8\right)\cdot 47 + \left(28 a + 7\right)\cdot 47^{2} + \left(18 a + 19\right)\cdot 47^{3} + \left(40 a + 39\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 11 a + 36 + \left(46 a + 29\right)\cdot 47 + \left(29 a + 16\right)\cdot 47^{2} + \left(5 a + 9\right)\cdot 47^{3} + \left(a + 25\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 15 a + 32 + \left(45 a + 32\right)\cdot 47 + \left(24 a + 44\right)\cdot 47^{2} + \left(18 a + 40\right)\cdot 47^{3} + \left(17 a + 38\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 28 a + 19 + \left(22 a + 38\right)\cdot 47 + \left(18 a + 39\right)\cdot 47^{2} + \left(28 a + 27\right)\cdot 47^{3} + \left(6 a + 7\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 36 a + 11 + 17\cdot 47 + \left(17 a + 30\right)\cdot 47^{2} + \left(41 a + 37\right)\cdot 47^{3} + \left(45 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$

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

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

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

The blue line marks the conjugacy class containing complex conjugation.