Properties

Label 2.3447.6t5.b.b
Dimension $2$
Group $S_3\times C_3$
Conductor $3447$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(3447\)\(\medspace = 3^{2} \cdot 383 \)
Artin stem field: Galois closure of 6.0.4550732847.1
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.3447.6t1.b.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.31023.1

Defining polynomial

$f(x)$$=$ \( x^{6} + 23x^{4} - 19x^{3} + 228x^{2} + 356x + 952 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 7.

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

Roots:
$r_{ 1 }$ $=$ \( 10 a + 1 + \left(26 a + 1\right)\cdot 37 + \left(2 a + 17\right)\cdot 37^{2} + \left(12 a + 20\right)\cdot 37^{3} + \left(23 a + 11\right)\cdot 37^{4} + \left(22 a + 20\right)\cdot 37^{5} + \left(a + 5\right)\cdot 37^{6} +O(37^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 13 a + 23 + \left(3 a + 30\right)\cdot 37 + \left(5 a + 8\right)\cdot 37^{2} + \left(2 a + 28\right)\cdot 37^{3} + \left(8 a + 4\right)\cdot 37^{4} + \left(32 a + 32\right)\cdot 37^{5} + \left(26 a + 28\right)\cdot 37^{6} +O(37^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 23 a + 32 + \left(29 a + 20\right)\cdot 37 + \left(7 a + 9\right)\cdot 37^{2} + \left(14 a + 13\right)\cdot 37^{3} + \left(31 a + 20\right)\cdot 37^{4} + \left(17 a + 18\right)\cdot 37^{5} + \left(28 a + 17\right)\cdot 37^{6} +O(37^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 14 a + 13 + \left(7 a + 5\right)\cdot 37 + \left(29 a + 11\right)\cdot 37^{2} + \left(22 a + 25\right)\cdot 37^{3} + \left(5 a + 20\right)\cdot 37^{4} + \left(19 a + 21\right)\cdot 37^{5} + \left(8 a + 2\right)\cdot 37^{6} +O(37^{7})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 24 a + 1 + \left(33 a + 31\right)\cdot 37 + \left(31 a + 25\right)\cdot 37^{2} + \left(34 a + 31\right)\cdot 37^{3} + \left(28 a + 34\right)\cdot 37^{4} + \left(4 a + 4\right)\cdot 37^{5} + \left(10 a + 30\right)\cdot 37^{6} +O(37^{7})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 27 a + 4 + \left(10 a + 22\right)\cdot 37 + \left(34 a + 1\right)\cdot 37^{2} + \left(24 a + 29\right)\cdot 37^{3} + \left(13 a + 18\right)\cdot 37^{4} + \left(14 a + 13\right)\cdot 37^{5} + \left(35 a + 26\right)\cdot 37^{6} +O(37^{7})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.