Properties

Label 2.1352.6t5.a.b
Dimension $2$
Group $S_3\times C_3$
Conductor $1352$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(1352\)\(\medspace = 2^{3} \cdot 13^{2}\)
Artin stem field: Galois closure of 6.0.190102016.3
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.104.6t1.b.b
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.104.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} - 7x^{4} + 6x^{3} + 44x^{2} + 60x + 27 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: \( x^{2} + 49x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 52 a + 3 + \left(a + 12\right)\cdot 53 + \left(34 a + 30\right)\cdot 53^{2} + \left(8 a + 50\right)\cdot 53^{3} + \left(39 a + 24\right)\cdot 53^{4} + \left(41 a + 51\right)\cdot 53^{5} +O(53^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 13 a + 24 + \left(12 a + 23\right)\cdot 53 + \left(41 a + 11\right)\cdot 53^{2} + \left(51 a + 31\right)\cdot 53^{3} + \left(48 a + 32\right)\cdot 53^{4} + \left(47 a + 19\right)\cdot 53^{5} +O(53^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 14 a + 28 + \left(10 a + 34\right)\cdot 53 + \left(7 a + 17\right)\cdot 53^{2} + \left(43 a + 17\right)\cdot 53^{3} + \left(9 a + 10\right)\cdot 53^{4} + \left(6 a + 51\right)\cdot 53^{5} +O(53^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 39 a + 31 + \left(42 a + 8\right)\cdot 53 + \left(45 a + 36\right)\cdot 53^{2} + \left(9 a + 23\right)\cdot 53^{3} + \left(43 a + 6\right)\cdot 53^{4} + \left(46 a + 13\right)\cdot 53^{5} +O(53^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( a + 52 + \left(51 a + 20\right)\cdot 53 + \left(18 a + 5\right)\cdot 53^{2} + \left(44 a + 51\right)\cdot 53^{3} + \left(13 a + 13\right)\cdot 53^{4} + \left(11 a + 20\right)\cdot 53^{5} +O(53^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 40 a + 23 + \left(40 a + 6\right)\cdot 53 + \left(11 a + 5\right)\cdot 53^{2} + \left(a + 38\right)\cdot 53^{3} + \left(4 a + 17\right)\cdot 53^{4} + \left(5 a + 3\right)\cdot 53^{5} +O(53^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.