Properties

Label 2.10575.6t3.b.a
Dimension $2$
Group $D_{6}$
Conductor $10575$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(10575\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 47 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.2.8760065625.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.47.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.1175.1

Defining polynomial

$f(x)$$=$\(x^{6} - 2 x^{5} + 5 x^{4} - 60 x^{3} + 60 x^{2} - 112 x + 79\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 9.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.