Properties

Label 2.2200.6t3.a.a
Dimension $2$
Group $D_{6}$
Conductor $2200$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(2200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.193600000.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.88.2t1.b.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.2200.1

Defining polynomial

$f(x)$$=$ \( x^{6} + 10x^{4} - 20x^{3} + 15x^{2} - 80x + 90 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{2} + 33x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 31 + 22\cdot 37 + 21\cdot 37^{2} + 25\cdot 37^{3} + 3\cdot 37^{4} + 10\cdot 37^{5} +O(37^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 36 a + 29 + \left(23 a + 32\right)\cdot 37 + \left(21 a + 5\right)\cdot 37^{2} + \left(17 a + 6\right)\cdot 37^{3} + \left(34 a + 7\right)\cdot 37^{4} + \left(14 a + 6\right)\cdot 37^{5} +O(37^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 16 a + 8 + \left(31 a + 26\right)\cdot 37 + \left(24 a + 10\right)\cdot 37^{2} + \left(34 a + 4\right)\cdot 37^{3} + \left(33 a + 3\right)\cdot 37^{4} + \left(a + 8\right)\cdot 37^{5} +O(37^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( a + 25 + \left(13 a + 18\right)\cdot 37 + \left(15 a + 31\right)\cdot 37^{2} + \left(19 a + 17\right)\cdot 37^{3} + \left(2 a + 16\right)\cdot 37^{4} + \left(22 a + 31\right)\cdot 37^{5} +O(37^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 20 + 22\cdot 37 + 36\cdot 37^{2} + 12\cdot 37^{3} + 13\cdot 37^{4} + 36\cdot 37^{5} +O(37^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 21 a + 35 + \left(5 a + 24\right)\cdot 37 + \left(12 a + 4\right)\cdot 37^{2} + \left(2 a + 7\right)\cdot 37^{3} + \left(3 a + 30\right)\cdot 37^{4} + \left(35 a + 18\right)\cdot 37^{5} +O(37^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.