Properties

Label 2.2240.6t3.f.a
Dimension $2$
Group $D_{6}$
Conductor $2240$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(2240\)\(\medspace = 2^{6} \cdot 5 \cdot 7 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.2.12544000.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.35.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.140.1

Defining polynomial

$f(x)$$=$\(x^{6} - 8 x^{4} + 16 x^{2} - 10\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \(x^{2} + 29 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 27 + 16\cdot 31 + 29\cdot 31^{2} + 22\cdot 31^{3} + 19\cdot 31^{4} + 15\cdot 31^{5} + 11\cdot 31^{6} +O(31^{7})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 29 a + \left(20 a + 2\right)\cdot 31 + \left(2 a + 7\right)\cdot 31^{2} + \left(29 a + 30\right)\cdot 31^{3} + \left(12 a + 26\right)\cdot 31^{4} + \left(25 a + 19\right)\cdot 31^{5} + \left(2 a + 15\right)\cdot 31^{6} +O(31^{7})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 27 + \left(10 a + 14\right)\cdot 31 + \left(28 a + 22\right)\cdot 31^{2} + \left(a + 23\right)\cdot 31^{3} + \left(18 a + 23\right)\cdot 31^{4} + \left(5 a + 26\right)\cdot 31^{5} + \left(28 a + 26\right)\cdot 31^{6} +O(31^{7})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 4 + 14\cdot 31 + 31^{2} + 8\cdot 31^{3} + 11\cdot 31^{4} + 15\cdot 31^{5} + 19\cdot 31^{6} +O(31^{7})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 2 a + \left(10 a + 29\right)\cdot 31 + \left(28 a + 23\right)\cdot 31^{2} + a\cdot 31^{3} + \left(18 a + 4\right)\cdot 31^{4} + \left(5 a + 11\right)\cdot 31^{5} + \left(28 a + 15\right)\cdot 31^{6} +O(31^{7})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 29 a + 4 + \left(20 a + 16\right)\cdot 31 + \left(2 a + 8\right)\cdot 31^{2} + \left(29 a + 7\right)\cdot 31^{3} + \left(12 a + 7\right)\cdot 31^{4} + \left(25 a + 4\right)\cdot 31^{5} + \left(2 a + 4\right)\cdot 31^{6} +O(31^{7})\)  Toggle raw display

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

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

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$$2$$(2,3)(5,6)$$0$
$3$$2$$(1,2)(3,6)(4,5)$$0$
$2$$3$$(1,6,5)(2,4,3)$$-1$
$2$$6$$(1,2,6,4,5,3)$$1$

The blue line marks the conjugacy class containing complex conjugation.