Properties

Label 2.3e3_5.6t3.2c1
Dimension 2
Group $D_{6}$
Conductor $ 3^{3} \cdot 5 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:$135= 3^{3} \cdot 5 $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 3 x^{4} - 3 x^{2} + 3 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{6}$
Parity: Odd
Determinant: 1.3_5.2t1.1c1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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