Properties

Label 2.2548.6t5.d.a
Dimension $2$
Group $S_3\times C_3$
Conductor $2548$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(2548\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 13 \)
Artin stem field: Galois closure of 6.0.25969216.3
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.364.6t1.g.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.676.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} - 12x^{4} + 46x^{3} + 23x^{2} - 240x + 225 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{2} + 18x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 7 a + 8 + \left(12 a + 12\right)\cdot 19 + \left(15 a + 18\right)\cdot 19^{2} + \left(10 a + 18\right)\cdot 19^{3} + \left(12 a + 17\right)\cdot 19^{4} + 4 a\cdot 19^{5} + 11\cdot 19^{6} +O(19^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 12 a + 15 + \left(6 a + 17\right)\cdot 19 + \left(3 a + 2\right)\cdot 19^{2} + \left(8 a + 14\right)\cdot 19^{3} + 6 a\cdot 19^{4} + \left(14 a + 12\right)\cdot 19^{5} + \left(18 a + 6\right)\cdot 19^{6} +O(19^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 4 + \left(5 a + 17\right)\cdot 19 + \left(a + 7\right)\cdot 19^{2} + 19^{3} + \left(16 a + 17\right)\cdot 19^{4} + \left(13 a + 17\right)\cdot 19^{5} + \left(2 a + 11\right)\cdot 19^{6} +O(19^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 5 a + 1 + \left(7 a + 3\right)\cdot 19 + 15\cdot 19^{2} + \left(18 a + 11\right)\cdot 19^{3} + \left(16 a + 13\right)\cdot 19^{4} + \left(11 a + 7\right)\cdot 19^{5} + \left(a + 18\right)\cdot 19^{6} +O(19^{7})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 17 a + 6 + \left(13 a + 1\right)\cdot 19 + \left(17 a + 4\right)\cdot 19^{2} + 18 a\cdot 19^{3} + \left(2 a + 14\right)\cdot 19^{4} + \left(5 a + 15\right)\cdot 19^{5} + 16 a\cdot 19^{6} +O(19^{7})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 14 a + 6 + \left(11 a + 5\right)\cdot 19 + \left(18 a + 8\right)\cdot 19^{2} + 10\cdot 19^{3} + \left(2 a + 12\right)\cdot 19^{4} + \left(7 a + 2\right)\cdot 19^{5} + \left(17 a + 8\right)\cdot 19^{6} +O(19^{7})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.