Properties

Label 1.3e2_13.6t1.4c1
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} - 91 x^{3} + 2197 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{117}(74,\cdot)\)

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:
$r_{ 1 }$ $=$ $ 12 a + 2 + \left(10 a + 17\right)\cdot 23 + \left(15 a + 7\right)\cdot 23^{2} + \left(10 a + 4\right)\cdot 23^{3} + \left(16 a + 1\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 19 a + 12 + \left(20 a + 22\right)\cdot 23 + \left(4 a + 1\right)\cdot 23^{2} + \left(a + 12\right)\cdot 23^{3} + 11\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 4 a + 4 + \left(2 a + 22\right)\cdot 23 + \left(18 a + 13\right)\cdot 23^{2} + \left(21 a + 9\right)\cdot 23^{3} + \left(22 a + 10\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 11 a + 3 + \left(12 a + 3\right)\cdot 23 + \left(7 a + 5\right)\cdot 23^{2} + \left(12 a + 10\right)\cdot 23^{3} + 6 a\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 7 a + 17 + \left(10 a + 6\right)\cdot 23 + \left(12 a + 1\right)\cdot 23^{2} + \left(13 a + 9\right)\cdot 23^{3} + \left(6 a + 11\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 16 a + 8 + \left(12 a + 20\right)\cdot 23 + \left(10 a + 15\right)\cdot 23^{2} + 9 a\cdot 23^{3} + \left(16 a + 11\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$

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

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

Character values on conjugacy classes

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