Properties

Label 2.135.6t3.a.a
Dimension $2$
Group $D_{6}$
Conductor $135$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(135\)\(\medspace = 3^{3} \cdot 5 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.2.91125.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.15.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.135.1

Defining polynomial

$f(x)$$=$\(x^{6} - x^{3} - 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \(x^{2} + 7 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 5 a + 4 + 10\cdot 11 + \left(5 a + 10\right)\cdot 11^{2} + \left(5 a + 2\right)\cdot 11^{3} + \left(8 a + 2\right)\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 2 + 7\cdot 11 + 6\cdot 11^{2} + 7\cdot 11^{3} + 3\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 5 + 4\cdot 11 + 2\cdot 11^{2} + 10\cdot 11^{3} + 10\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 9 a + 3 + \left(9 a + 3\right)\cdot 11 + \left(a + 3\right)\cdot 11^{2} + \left(8 a + 8\right)\cdot 11^{3} + \left(4 a + 3\right)\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 6 a + 2 + \left(10 a + 7\right)\cdot 11 + \left(5 a + 8\right)\cdot 11^{2} + \left(5 a + 8\right)\cdot 11^{3} + \left(2 a + 8\right)\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 2 a + 6 + a\cdot 11 + \left(9 a + 1\right)\cdot 11^{2} + \left(2 a + 6\right)\cdot 11^{3} + \left(6 a + 3\right)\cdot 11^{4} +O(11^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,4)(2,3)(5,6)$$-2$
$3$$2$$(1,2)(3,4)(5,6)$$0$
$3$$2$$(1,5)(4,6)$$0$
$2$$3$$(1,3,5)(2,6,4)$$-1$
$2$$6$$(1,6,3,4,5,2)$$1$

The blue line marks the conjugacy class containing complex conjugation.