Properties

Label 2.3_37.6t5.1c1
Dimension 2
Group $S_3\times C_3$
Conductor $ 3 \cdot 37 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$S_3\times C_3$
Conductor:$111= 3 \cdot 37 $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 4 x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $S_3\times C_3$
Parity: Odd
Determinant: 1.3_37.6t1.1c2

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$2$
$3$$2$$(1,5)(2,4)(3,6)$$0$
$1$$3$$(1,6,2)(3,4,5)$$2 \zeta_{3}$
$1$$3$$(1,2,6)(3,5,4)$$-2 \zeta_{3} - 2$
$2$$3$$(1,2,6)$$-\zeta_{3}$
$2$$3$$(1,6,2)$$\zeta_{3} + 1$
$2$$3$$(1,6,2)(3,5,4)$$-1$
$3$$6$$(1,4,6,5,2,3)$$0$
$3$$6$$(1,3,2,5,6,4)$$0$
The blue line marks the conjugacy class containing complex conjugation.