Properties

Label 3.3627.6t11.b.a
Dimension $3$
Group $S_4\times C_2$
Conductor $3627$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4\times C_2$
Conductor: \(3627\)\(\medspace = 3^{2} \cdot 13 \cdot 31 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.3485547.1
Galois orbit size: $1$
Smallest permutation container: $S_4\times C_2$
Parity: odd
Determinant: 1.403.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.47151.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} - 5x^{4} - 2x^{3} + 17x^{2} + 55x + 47 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.

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

Roots:
$r_{ 1 }$ $=$ \( 1 + 15\cdot 37 + 19\cdot 37^{2} + 37^{3} + 6\cdot 37^{4} + 3\cdot 37^{5} + 6\cdot 37^{6} + 10\cdot 37^{7} + 20\cdot 37^{8} + 20\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 26 a + 32 + \left(36 a + 3\right)\cdot 37 + \left(8 a + 16\right)\cdot 37^{2} + \left(a + 28\right)\cdot 37^{3} + \left(20 a + 33\right)\cdot 37^{4} + \left(7 a + 5\right)\cdot 37^{5} + \left(6 a + 25\right)\cdot 37^{6} + \left(36 a + 2\right)\cdot 37^{7} + \left(33 a + 20\right)\cdot 37^{8} + \left(19 a + 12\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 15 a + 2 + \left(8 a + 29\right)\cdot 37 + \left(35 a + 22\right)\cdot 37^{2} + \left(22 a + 30\right)\cdot 37^{3} + \left(25 a + 7\right)\cdot 37^{4} + \left(16 a + 20\right)\cdot 37^{5} + \left(14 a + 28\right)\cdot 37^{6} + 5 a\cdot 37^{7} + \left(6 a + 25\right)\cdot 37^{8} + \left(26 a + 21\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 22 a + 25 + \left(28 a + 10\right)\cdot 37 + \left(a + 7\right)\cdot 37^{2} + \left(14 a + 13\right)\cdot 37^{3} + \left(11 a + 13\right)\cdot 37^{4} + \left(20 a + 24\right)\cdot 37^{5} + \left(22 a + 32\right)\cdot 37^{6} + \left(31 a + 7\right)\cdot 37^{7} + \left(30 a + 7\right)\cdot 37^{8} + \left(10 a + 9\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 11 a + 25 + 13\cdot 37 + \left(28 a + 15\right)\cdot 37^{2} + \left(35 a + 24\right)\cdot 37^{3} + \left(16 a + 1\right)\cdot 37^{4} + \left(29 a + 16\right)\cdot 37^{5} + \left(30 a + 5\right)\cdot 37^{6} + 30\cdot 37^{7} + \left(3 a + 8\right)\cdot 37^{8} + \left(17 a + 21\right)\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 28 + 37 + 30\cdot 37^{2} + 12\cdot 37^{3} + 11\cdot 37^{4} + 4\cdot 37^{5} + 13\cdot 37^{6} + 22\cdot 37^{7} + 29\cdot 37^{8} + 25\cdot 37^{9} +O(37^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$3$
$1$$2$$(1,6)(2,4)(3,5)$$-3$
$3$$2$$(2,4)(3,5)$$-1$
$3$$2$$(3,5)$$1$
$6$$2$$(2,3)(4,5)$$-1$
$6$$2$$(1,2)(3,5)(4,6)$$1$
$8$$3$$(1,2,3)(4,5,6)$$0$
$6$$4$$(2,3,4,5)$$-1$
$6$$4$$(1,6)(2,3,4,5)$$1$
$8$$6$$(1,3,4,6,5,2)$$0$

The blue line marks the conjugacy class containing complex conjugation.