Properties

Label 1.3e2_13.6t1.8c1
Dimension 1
Group $C_6$
Conductor $ 3^{2} \cdot 13 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

Learn more about

Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$117= 3^{2} \cdot 13 $
Artin number field: Splitting field of $f= x^{6} - 26 x^{3} + 351 x^{2} - 936 x + 793 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{117}(56,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 13 a + 9 + \left(a + 14\right)\cdot 23 + \left(22 a + 4\right)\cdot 23^{2} + \left(a + 4\right)\cdot 23^{3} + \left(3 a + 21\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 20 a + 12 + \left(9 a + 1\right)\cdot 23 + \left(3 a + 18\right)\cdot 23^{2} + \left(10 a + 18\right)\cdot 23^{3} + \left(14 a + 12\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 13 a + 2 + \left(11 a + 7\right)\cdot 23 + 20 a\cdot 23^{2} + 10 a\cdot 23^{3} + \left(5 a + 12\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 10 a + 5 + \left(11 a + 17\right)\cdot 23 + \left(2 a + 6\right)\cdot 23^{2} + \left(12 a + 1\right)\cdot 23^{3} + \left(17 a + 12\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 3 a + 6 + \left(13 a + 1\right)\cdot 23 + \left(19 a + 15\right)\cdot 23^{2} + \left(12 a + 12\right)\cdot 23^{3} + \left(8 a + 8\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 10 a + 12 + \left(21 a + 4\right)\cdot 23 + 23^{2} + \left(21 a + 9\right)\cdot 23^{3} + \left(19 a + 2\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$

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

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

Character values on conjugacy classes

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