Properties

Label 1.2e2_13.6t1.1c1
Dimension 1
Group $C_6$
Conductor $ 2^{2} \cdot 13 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$52= 2^{2} \cdot 13 $
Artin number field: Splitting field of $f= x^{6} + 9 x^{4} + 14 x^{2} + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{52}(3,\cdot)\)

Galois action

Roots of defining polynomial

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} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 24 a + 7 + \left(2 a + 9\right)\cdot 31 + \left(17 a + 15\right)\cdot 31^{2} + \left(30 a + 24\right)\cdot 31^{3} + \left(3 a + 26\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 5 a + 26 + \left(18 a + 30\right)\cdot 31 + \left(3 a + 20\right)\cdot 31^{2} + \left(18 a + 14\right)\cdot 31^{3} + 8\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 15 a + 16 + \left(18 a + 4\right)\cdot 31 + \left(22 a + 2\right)\cdot 31^{2} + \left(11 a + 15\right)\cdot 31^{3} + \left(13 a + 23\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 7 a + 24 + \left(28 a + 21\right)\cdot 31 + \left(13 a + 15\right)\cdot 31^{2} + 6\cdot 31^{3} + \left(27 a + 4\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 26 a + 5 + 12 a\cdot 31 + \left(27 a + 10\right)\cdot 31^{2} + \left(12 a + 16\right)\cdot 31^{3} + \left(30 a + 22\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 16 a + 15 + \left(12 a + 26\right)\cdot 31 + \left(8 a + 28\right)\cdot 31^{2} + \left(19 a + 15\right)\cdot 31^{3} + \left(17 a + 7\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

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$$3$$(1,5,6)(2,3,4)$$\zeta_{3}$
$1$$3$$(1,6,5)(2,4,3)$$-\zeta_{3} - 1$
$1$$6$$(1,3,5,4,6,2)$$\zeta_{3} + 1$
$1$$6$$(1,2,6,4,5,3)$$-\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.