Properties

Label 1.13_31.6t1.2c2
Dimension 1
Group $C_6$
Conductor $ 13 \cdot 31 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$403= 13 \cdot 31 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 135 x^{4} - 74 x^{3} + 3672 x^{2} + 985 x - 10687 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Even
Corresponding Dirichlet character: \(\chi_{403}(160,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $ x^{2} + 6 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ a + 4\cdot 7 + \left(a + 1\right)\cdot 7^{2} + 4\cdot 7^{3} + \left(3 a + 6\right)\cdot 7^{4} + \left(5 a + 5\right)\cdot 7^{5} +O\left(7^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 5 a + 4 + \left(5 a + 3\right)\cdot 7 + \left(2 a + 4\right)\cdot 7^{2} + \left(4 a + 6\right)\cdot 7^{3} + 3\cdot 7^{4} + \left(4 a + 3\right)\cdot 7^{5} +O\left(7^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 6 a + 1 + \left(6 a + 3\right)\cdot 7 + \left(5 a + 2\right)\cdot 7^{2} + \left(6 a + 3\right)\cdot 7^{3} + \left(3 a + 2\right)\cdot 7^{4} + \left(a + 1\right)\cdot 7^{5} +O\left(7^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 2 a + 2 + \left(a + 4\right)\cdot 7 + \left(4 a + 1\right)\cdot 7^{2} + \left(2 a + 1\right)\cdot 7^{3} + 6 a\cdot 7^{4} + 2 a\cdot 7^{5} +O\left(7^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 4 a + 2 + \left(6 a + 5\right)\cdot 7 + \left(3 a + 6\right)\cdot 7^{2} + \left(3 a + 2\right)\cdot 7^{3} + \left(6 a + 2\right)\cdot 7^{4} + 4\cdot 7^{5} +O\left(7^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 3 a + 6 + \left(3 a + 4\right)\cdot 7^{2} + \left(3 a + 2\right)\cdot 7^{3} + 5\cdot 7^{4} + \left(6 a + 5\right)\cdot 7^{5} +O\left(7^{ 6 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$1$
$1$$2$$(1,3)(2,4)(5,6)$$-1$
$1$$3$$(1,2,5)(3,4,6)$$-\zeta_{3} - 1$
$1$$3$$(1,5,2)(3,6,4)$$\zeta_{3}$
$1$$6$$(1,4,5,3,2,6)$$\zeta_{3} + 1$
$1$$6$$(1,6,2,3,5,4)$$-\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.