Properties

Label 2.870124.6t3.a.a
Dimension $2$
Group $D_{6}$
Conductor $870124$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(870124\)\(\medspace = 2^{2} \cdot 19 \cdot 107^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.134441118992.2
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.19.2t1.a.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.76.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 3x^{5} - 131x^{4} + 213x^{3} + 7564x^{2} - 17742x - 355994 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: \( x^{2} + 58x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 25 + 16\cdot 59 + 54\cdot 59^{2} + 14\cdot 59^{3} + 3\cdot 59^{4} + 38\cdot 59^{5} + 39\cdot 59^{6} + 2\cdot 59^{7} +O(59^{8})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 10 a + 21 + \left(10 a + 30\right)\cdot 59 + \left(18 a + 43\right)\cdot 59^{2} + \left(56 a + 21\right)\cdot 59^{3} + \left(37 a + 24\right)\cdot 59^{4} + \left(51 a + 33\right)\cdot 59^{5} + \left(22 a + 37\right)\cdot 59^{6} + \left(47 a + 44\right)\cdot 59^{7} +O(59^{8})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 49 + 24\cdot 59 + 28\cdot 59^{2} + 44\cdot 59^{3} + 47\cdot 59^{4} + 19\cdot 59^{5} + 39\cdot 59^{6} + 13\cdot 59^{7} +O(59^{8})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 49 a + 31 + \left(48 a + 30\right)\cdot 59 + \left(40 a + 51\right)\cdot 59^{2} + 2 a\cdot 59^{3} + \left(21 a + 6\right)\cdot 59^{4} + \left(7 a + 47\right)\cdot 59^{5} + \left(36 a + 8\right)\cdot 59^{6} + \left(11 a + 10\right)\cdot 59^{7} +O(59^{8})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 30 a + 12 + \left(39 a + 3\right)\cdot 59 + \left(14 a + 12\right)\cdot 59^{2} + \left(38 a + 6\right)\cdot 59^{3} + \left(53 a + 40\right)\cdot 59^{4} + \left(42 a + 24\right)\cdot 59^{5} + \left(45 a + 24\right)\cdot 59^{6} + \left(39 a + 26\right)\cdot 59^{7} +O(59^{8})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 29 a + 42 + \left(19 a + 12\right)\cdot 59 + \left(44 a + 46\right)\cdot 59^{2} + \left(20 a + 29\right)\cdot 59^{3} + \left(5 a + 55\right)\cdot 59^{4} + \left(16 a + 13\right)\cdot 59^{5} + \left(13 a + 27\right)\cdot 59^{6} + \left(19 a + 20\right)\cdot 59^{7} +O(59^{8})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.