Properties

Label 2.1148.6t5.b.b
Dimension $2$
Group $S_3\times C_3$
Conductor $1148$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(1148\)\(\medspace = 2^{2} \cdot 7 \cdot 41 \)
Artin stem field: 6.0.216136256.1
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.1148.6t1.a.b
Projective image: $S_3$
Projective stem field: 3.1.8036.1

Defining polynomial

$f(x)$$=$\(x^{6} - 2 x^{5} - 3 x^{4} + 8 x^{3} - 8 x + 45\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \(x^{2} + 12 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( a + \left(9 a + 6\right)\cdot 13 + \left(a + 11\right)\cdot 13^{2} + \left(10 a + 6\right)\cdot 13^{3} + \left(7 a + 12\right)\cdot 13^{4} + \left(4 a + 8\right)\cdot 13^{5} + \left(10 a + 12\right)\cdot 13^{6} +O(13^{7})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 12 a + 1 + \left(3 a + 1\right)\cdot 13 + \left(11 a + 4\right)\cdot 13^{2} + \left(2 a + 2\right)\cdot 13^{3} + \left(5 a + 10\right)\cdot 13^{4} + \left(8 a + 5\right)\cdot 13^{5} + \left(2 a + 5\right)\cdot 13^{6} +O(13^{7})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 11 a + 9 + \left(5 a + 2\right)\cdot 13 + \left(12 a + 11\right)\cdot 13^{2} + \left(3 a + 10\right)\cdot 13^{3} + \left(3 a + 1\right)\cdot 13^{4} + \left(2 a + 6\right)\cdot 13^{5} + \left(12 a + 6\right)\cdot 13^{6} +O(13^{7})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 2 a + 7 + \left(7 a + 10\right)\cdot 13 + 4\cdot 13^{2} + \left(9 a + 2\right)\cdot 13^{3} + \left(9 a + 1\right)\cdot 13^{4} + \left(10 a + 5\right)\cdot 13^{5} + 3\cdot 13^{6} +O(13^{7})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 10 a + 7 + \left(9 a + 9\right)\cdot 13 + \left(10 a + 9\right)\cdot 13^{2} + \left(6 a + 3\right)\cdot 13^{3} + \left(8 a + 12\right)\cdot 13^{4} + \left(10 a + 11\right)\cdot 13^{5} + \left(a + 9\right)\cdot 13^{6} +O(13^{7})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 3 a + 4 + \left(3 a + 9\right)\cdot 13 + \left(2 a + 10\right)\cdot 13^{2} + \left(6 a + 12\right)\cdot 13^{3} + 4 a\cdot 13^{4} + \left(2 a + 1\right)\cdot 13^{5} + \left(11 a + 1\right)\cdot 13^{6} +O(13^{7})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.