Properties

Label 2.2200.6t3.d.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 6.2.24200000.1
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)$$=$ \( x^{6} - 3x^{5} + 5x^{3} - 20x^{2} + 17x - 61 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 10 + 36\cdot 41 + 29\cdot 41^{2} + 10\cdot 41^{3} + 5\cdot 41^{4} + 33\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 32 + 4\cdot 41 + 11\cdot 41^{2} + 30\cdot 41^{3} + 35\cdot 41^{4} + 7\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 10 + \left(20 a + 3\right)\cdot 41 + \left(4 a + 40\right)\cdot 41^{2} + \left(35 a + 20\right)\cdot 41^{3} + \left(32 a + 33\right)\cdot 41^{4} + 7\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 39 a + 32 + \left(20 a + 37\right)\cdot 41 + 36 a\cdot 41^{2} + \left(5 a + 20\right)\cdot 41^{3} + \left(8 a + 7\right)\cdot 41^{4} + \left(40 a + 33\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 39 a + 16 + \left(20 a + 20\right)\cdot 41 + \left(36 a + 33\right)\cdot 41^{2} + \left(5 a + 39\right)\cdot 41^{3} + \left(8 a + 14\right)\cdot 41^{4} + \left(40 a + 18\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 2 a + 26 + \left(20 a + 20\right)\cdot 41 + \left(4 a + 7\right)\cdot 41^{2} + \left(35 a + 1\right)\cdot 41^{3} + \left(32 a + 26\right)\cdot 41^{4} + 22\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.