Properties

Label 1.13_79.6t1.2c2
Dimension 1
Group $C_6$
Conductor $ 13 \cdot 79 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

Learn more about

Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$1027= 13 \cdot 79 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 51 x^{4} - 29 x^{3} + 1195 x^{2} - 703 x + 11545 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{1027}(315,\cdot)\)

Galois action

Roots of defining polynomial

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:
$r_{ 1 }$ $=$ $ 40 a + 6 + \left(14 a + 29\right)\cdot 47 + \left(16 a + 34\right)\cdot 47^{2} + \left(2 a + 31\right)\cdot 47^{3} + \left(24 a + 19\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 7 a + 8 + \left(32 a + 7\right)\cdot 47 + \left(30 a + 5\right)\cdot 47^{2} + 44 a\cdot 47^{3} + \left(22 a + 39\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 40 a + 17 + \left(14 a + 39\right)\cdot 47 + \left(16 a + 21\right)\cdot 47^{2} + \left(2 a + 44\right)\cdot 47^{3} + \left(24 a + 35\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 40 a + 22 + \left(14 a + 17\right)\cdot 47 + \left(16 a + 34\right)\cdot 47^{2} + \left(2 a + 11\right)\cdot 47^{3} + \left(24 a + 40\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 7 a + 39 + \left(32 a + 18\right)\cdot 47 + \left(30 a + 5\right)\cdot 47^{2} + \left(44 a + 20\right)\cdot 47^{3} + \left(22 a + 18\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 7 a + 3 + \left(32 a + 29\right)\cdot 47 + \left(30 a + 39\right)\cdot 47^{2} + \left(44 a + 32\right)\cdot 47^{3} + \left(22 a + 34\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$

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

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

Character values on conjugacy classes

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