Properties

Label 1.3e2_13.6t1.1c1
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} - 39 x^{4} - 26 x^{3} + 351 x^{2} + 234 x - 468 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Even
Corresponding Dirichlet character: \(\chi_{117}(88,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 3 a + 40 + \left(24 a + 2\right)\cdot 59 + \left(22 a + 30\right)\cdot 59^{2} + \left(18 a + 54\right)\cdot 59^{3} + \left(17 a + 18\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 45 a + 9 + \left(26 a + 55\right)\cdot 59 + \left(58 a + 57\right)\cdot 59^{2} + \left(21 a + 43\right)\cdot 59^{3} + \left(33 a + 15\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 11 a + 10 + \left(8 a + 1\right)\cdot 59 + \left(37 a + 30\right)\cdot 59^{2} + \left(18 a + 19\right)\cdot 59^{3} + \left(8 a + 24\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 48 a + 21 + \left(50 a + 57\right)\cdot 59 + \left(21 a + 58\right)\cdot 59^{2} + 40 a\cdot 59^{3} + \left(50 a + 14\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 56 a + 43 + \left(34 a + 23\right)\cdot 59 + \left(36 a + 28\right)\cdot 59^{2} + \left(40 a + 50\right)\cdot 59^{3} + \left(41 a + 17\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 14 a + 54 + \left(32 a + 36\right)\cdot 59 + 30\cdot 59^{2} + \left(37 a + 7\right)\cdot 59^{3} + \left(25 a + 27\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$

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

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

Character values on conjugacy classes

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