Properties

Label 2.5915.6t3.f.a
Dimension $2$
Group $D_{6}$
Conductor $5915$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(5915\)\(\medspace = 5 \cdot 7 \cdot 13^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.2.174936125.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.35.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.5915.1

Defining polynomial

$f(x)$$=$\(x^{6} - 3 x^{5} - 14 x^{4} + 33 x^{3} + 23 x^{2} - 40 x - 64\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \(x^{2} + 18 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 2 a + \left(3 a + 11\right)\cdot 19 + \left(17 a + 13\right)\cdot 19^{2} + 8 a\cdot 19^{3} + \left(15 a + 11\right)\cdot 19^{4} + \left(12 a + 2\right)\cdot 19^{5} + 15\cdot 19^{6} + \left(17 a + 15\right)\cdot 19^{7} +O(19^{8})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 17 a + 2 + \left(15 a + 12\right)\cdot 19 + \left(a + 8\right)\cdot 19^{2} + \left(10 a + 11\right)\cdot 19^{3} + \left(3 a + 17\right)\cdot 19^{4} + \left(6 a + 18\right)\cdot 19^{5} + \left(18 a + 2\right)\cdot 19^{6} + \left(a + 13\right)\cdot 19^{7} +O(19^{8})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 4 + 17\cdot 19 + 6\cdot 19^{2} + 2\cdot 19^{3} + 17\cdot 19^{4} + 10\cdot 19^{5} + 6\cdot 19^{6} + 2\cdot 19^{7} +O(19^{8})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 16 + 19 + 12\cdot 19^{2} + 16\cdot 19^{3} + 19^{4} + 8\cdot 19^{5} + 12\cdot 19^{6} + 16\cdot 19^{7} +O(19^{8})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 17 a + 1 + \left(15 a + 8\right)\cdot 19 + \left(a + 5\right)\cdot 19^{2} + \left(10 a + 18\right)\cdot 19^{3} + \left(3 a + 7\right)\cdot 19^{4} + \left(6 a + 16\right)\cdot 19^{5} + \left(18 a + 3\right)\cdot 19^{6} + \left(a + 3\right)\cdot 19^{7} +O(19^{8})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 2 a + 18 + \left(3 a + 6\right)\cdot 19 + \left(17 a + 10\right)\cdot 19^{2} + \left(8 a + 7\right)\cdot 19^{3} + \left(15 a + 1\right)\cdot 19^{4} + 12 a\cdot 19^{5} + 16\cdot 19^{6} + \left(17 a + 5\right)\cdot 19^{7} +O(19^{8})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.