Properties

Label 1.7_13.6t1.9c2
Dimension 1
Group $C_6$
Conductor $ 7 \cdot 13 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$91= 7 \cdot 13 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{4} + 13 x^{3} + 71 x^{2} + 419 x + 827 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{91}(3,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 10 a + 39 + 24 a\cdot 61 + \left(53 a + 29\right)\cdot 61^{2} + \left(59 a + 26\right)\cdot 61^{3} + \left(12 a + 32\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 49 a + 15 + \left(4 a + 57\right)\cdot 61 + \left(45 a + 42\right)\cdot 61^{2} + \left(47 a + 24\right)\cdot 61^{3} + \left(26 a + 41\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 12 a + 3 + \left(56 a + 13\right)\cdot 61 + \left(15 a + 22\right)\cdot 61^{2} + \left(13 a + 27\right)\cdot 61^{3} + \left(34 a + 20\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 58 a + 10 + \left(2 a + 15\right)\cdot 61 + \left(56 a + 19\right)\cdot 61^{2} + \left(50 a + 38\right)\cdot 61^{3} + \left(43 a + 24\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 51 a + 49 + \left(36 a + 14\right)\cdot 61 + \left(7 a + 58\right)\cdot 61^{2} + \left(a + 32\right)\cdot 61^{3} + \left(48 a + 46\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 3 a + 7 + \left(58 a + 21\right)\cdot 61 + \left(4 a + 11\right)\cdot 61^{2} + \left(10 a + 33\right)\cdot 61^{3} + \left(17 a + 17\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$

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

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

Character values on conjugacy classes

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