Properties

Label 1.273.6t1.g.a
Dimension $1$
Group $C_6$
Conductor $273$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \)
Artin field: Galois closure of 6.0.24069811311.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{273}(23,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} + 34x^{4} + 82x^{3} - 215x^{2} - 81x + 4276 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{2} + 16x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 15 a + 14 + \left(9 a + 11\right)\cdot 17 + \left(10 a + 4\right)\cdot 17^{2} + \left(13 a + 16\right)\cdot 17^{3} + \left(10 a + 3\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 a + 10 + \left(6 a + 5\right)\cdot 17 + \left(a + 8\right)\cdot 17^{2} + \left(12 a + 11\right)\cdot 17^{3} + \left(9 a + 8\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 12 + \left(7 a + 6\right)\cdot 17 + \left(6 a + 5\right)\cdot 17^{2} + \left(3 a + 2\right)\cdot 17^{3} + \left(6 a + 1\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 a + 1 + \left(5 a + 15\right)\cdot 17 + \left(7 a + 13\right)\cdot 17^{2} + \left(7 a + 7\right)\cdot 17^{3} + \left(a + 1\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 2 a + 8 + \left(10 a + 14\right)\cdot 17 + \left(15 a + 2\right)\cdot 17^{2} + \left(4 a + 5\right)\cdot 17^{3} + \left(7 a + 6\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 11 a + 7 + \left(11 a + 14\right)\cdot 17 + \left(9 a + 15\right)\cdot 17^{2} + \left(9 a + 7\right)\cdot 17^{3} + \left(15 a + 12\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$1$
$1$$2$$(1,3)(2,5)(4,6)$$-1$
$1$$3$$(1,5,6)(2,4,3)$$\zeta_{3}$
$1$$3$$(1,6,5)(2,3,4)$$-\zeta_{3} - 1$
$1$$6$$(1,4,5,3,6,2)$$\zeta_{3} + 1$
$1$$6$$(1,2,6,3,5,4)$$-\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.