Properties

 Label 2.1260.12t18.b.b Dimension $2$ Group $C_6\times S_3$ Conductor $1260$ Root number not computed Indicator $0$

Related objects

Basic invariants

 Dimension: $2$ Group: $C_6\times S_3$ Conductor: $$1260$$$$\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Artin stem field: 12.0.112021056000000.2 Galois orbit size: $2$ Smallest permutation container: $C_6\times S_3$ Parity: odd Determinant: 1.140.6t1.b.a Projective image: $S_3$ Projective stem field: 3.1.980.1

Defining polynomial

 $f(x)$ $=$ $$x^{12} - 2 x^{11} + x^{10} - 6 x^{9} + 5 x^{8} + 18 x^{7} - 6 x^{6} - 14 x^{5} + 13 x^{4} + 8 x^{3} + 2 x + 1$$  .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$14 a^{5} + 3 a^{4} + 8 a^{3} + 2 a^{2} + a + 10 + \left(12 a^{4} + 13 a^{3} + 11 a^{2} + a + 3\right)\cdot 17 + \left(16 a^{5} + 11 a^{4} + 7 a^{3} + a^{2} + 10 a + 10\right)\cdot 17^{2} + \left(a^{5} + 14 a^{4} + 2 a^{3} + 4 a^{2} + 10 a + 6\right)\cdot 17^{3} + \left(14 a^{5} + 4 a^{4} + 15 a^{3} + 3 a^{2} + 15 a + 12\right)\cdot 17^{4} + \left(9 a^{5} + 4 a^{4} + a^{3} + 9 a^{2} + 4 a + 16\right)\cdot 17^{5} + \left(10 a^{5} + 12 a^{4} + 5 a^{3} + a^{2} + a + 4\right)\cdot 17^{6} +O(17^{7})$$ $r_{ 2 }$ $=$ $$10 a^{5} + 14 a^{3} + 4 a^{2} + 12 a + 8 + \left(15 a^{5} + 8 a^{4} + a^{3} + 16 a^{2} + 8 a + 7\right)\cdot 17 + \left(10 a^{5} + 4 a^{4} + 10 a^{3} + 7 a^{2} + 2 a + 5\right)\cdot 17^{2} + \left(11 a^{5} + 13 a^{4} + 14 a^{3} + 8 a^{2} + 6 a + 6\right)\cdot 17^{3} + \left(16 a^{5} + 15 a^{3} + a^{2} + 8 a + 4\right)\cdot 17^{4} + \left(4 a^{5} + 16 a^{4} + 2 a^{3} + 13 a^{2} + 4 a + 13\right)\cdot 17^{5} + \left(a^{5} + 16 a^{4} + 4 a^{3} + 5 a^{2} + 15 a + 6\right)\cdot 17^{6} +O(17^{7})$$ $r_{ 3 }$ $=$ $$12 a^{5} + 14 a^{4} + 5 a^{3} + 12 a^{2} + 4 a + 8 + \left(11 a^{5} + 16 a^{4} + 16 a^{3} + a^{2} + 11 a + 12\right)\cdot 17 + \left(10 a^{5} + 8 a^{4} + 16 a^{3} + 14 a + 6\right)\cdot 17^{2} + \left(14 a^{5} + 14 a^{4} + 2 a^{3} + 12 a^{2} + 13 a + 14\right)\cdot 17^{3} + \left(6 a^{5} + 14 a^{4} + a^{3} + 9 a^{2} + 9 a\right)\cdot 17^{4} + \left(6 a^{5} + 5 a^{4} + 2 a^{3} + a + 8\right)\cdot 17^{5} + \left(3 a^{5} + 15 a^{4} + 10 a^{3} + 11 a^{2} + a + 15\right)\cdot 17^{6} +O(17^{7})$$ $r_{ 4 }$ $=$ $$8 a^{5} + a^{4} + 12 a^{3} + 2 a + \left(15 a^{5} + 12 a^{4} + 14 a^{3} + 13 a^{2} + a + 4\right)\cdot 17 + \left(7 a^{5} + 8 a^{4} + 8 a^{3} + 9 a^{2} + 13 a + 13\right)\cdot 17^{2} + \left(6 a^{5} + 9 a^{4} + 15 a^{3} + 15 a^{2} + 14 a\right)\cdot 17^{3} + \left(9 a^{5} + 10 a^{4} + 1\right)\cdot 17^{4} + \left(4 a^{5} + 12 a^{4} + 8 a^{3} + 10 a^{2} + 2 a + 6\right)\cdot 17^{5} + \left(10 a^{5} + 15 a^{4} + 12 a^{3} + a^{2} + 12 a + 14\right)\cdot 17^{6} +O(17^{7})$$ $r_{ 5 }$ $=$ $$4 a^{5} + a^{4} + 5 a^{3} + 7 a^{2} + 7 a + 6 + \left(11 a^{5} + 8 a^{4} + 15 a^{3} + 6 a^{2} + 3 a + 7\right)\cdot 17 + \left(4 a^{5} + 6 a^{4} + 14 a^{3} + 15 a + 12\right)\cdot 17^{2} + \left(16 a^{5} + 13 a^{4} + 2 a^{3} + 14 a^{2} + 10 a + 13\right)\cdot 17^{3} + \left(8 a^{5} + 11 a^{3} + 9 a + 15\right)\cdot 17^{4} + \left(8 a^{5} + 6 a^{4} + 10 a^{3} + 16 a^{2} + 5 a + 1\right)\cdot 17^{5} + \left(4 a^{5} + 8 a^{3} + 9 a + 8\right)\cdot 17^{6} +O(17^{7})$$ $r_{ 6 }$ $=$ $$11 a^{4} + 3 a^{3} + 14 a^{2} + 5 a + 3 + \left(3 a^{5} + 4 a^{4} + 12 a^{3} + 9 a^{2} + 16\right)\cdot 17 + \left(10 a^{5} + 10 a^{3} + 11 a^{2} + 5 a + 11\right)\cdot 17^{2} + \left(15 a^{5} + 14 a^{4} + 4 a^{2}\right)\cdot 17^{3} + \left(11 a^{5} + 16 a^{4} + 5 a^{2} + a + 4\right)\cdot 17^{4} + \left(5 a^{5} + 15 a^{3} + 14 a^{2} + 15 a + 9\right)\cdot 17^{5} + \left(10 a^{5} + 12 a^{4} + 2 a^{2} + 16 a + 15\right)\cdot 17^{6} +O(17^{7})$$ $r_{ 7 }$ $=$ $$11 a^{5} + 15 a^{4} + 7 a^{3} + 2 a^{2} + 4 a + 7 + \left(10 a^{5} + 10 a^{4} + 3 a^{3} + 10 a^{2} + 14 a + 6\right)\cdot 17 + \left(14 a^{5} + 4 a^{4} + 11 a^{3} + 7 a^{2} + 8 a + 15\right)\cdot 17^{2} + \left(9 a^{5} + 16 a^{4} + 8 a^{3} + 8 a^{2} + 12 a\right)\cdot 17^{3} + \left(4 a^{5} + 5 a^{4} + 14 a^{3} + 6 a^{2} + 3 a + 5\right)\cdot 17^{4} + \left(13 a^{5} + a^{4} + 8 a^{3} + 11 a^{2} + 6 a + 8\right)\cdot 17^{5} + \left(4 a^{5} + 15 a^{4} + 6 a^{3} + 7 a^{2} + 9 a + 12\right)\cdot 17^{6} +O(17^{7})$$ $r_{ 8 }$ $=$ $$12 a^{4} + 7 a^{3} + 14 a^{2} + 16 a + 14 + \left(5 a^{5} + 7 a^{4} + 5 a^{3} + 7 a^{2} + 16 a + 7\right)\cdot 17 + \left(a^{5} + 14 a^{4} + 13 a^{3} + 15 a^{2} + 2 a + 8\right)\cdot 17^{2} + \left(2 a^{5} + 6 a^{4} + 14 a^{3} + 12 a^{2} + 5 a + 13\right)\cdot 17^{3} + \left(7 a^{5} + 2 a^{4} + 9 a^{3} + 16 a^{2} + 2 a + 7\right)\cdot 17^{4} + \left(8 a^{5} + 8 a^{4} + 4 a^{3} + 16 a^{2} + 4 a + 13\right)\cdot 17^{5} + \left(15 a^{5} + 6 a^{4} + 6 a^{2} + 10 a + 12\right)\cdot 17^{6} +O(17^{7})$$ $r_{ 9 }$ $=$ $$4 a^{5} + 7 a^{4} + 10 a^{3} + 14 a + 5 + \left(7 a^{5} + 12 a^{4} + 10 a^{3} + 4 a^{2} + 15 a + 13\right)\cdot 17 + \left(14 a^{5} + 6 a^{4} + 8 a^{3} + a^{2} + 11 a + 4\right)\cdot 17^{2} + \left(10 a^{5} + 2 a^{4} + 7 a^{3} + 5 a^{2} + 2 a + 9\right)\cdot 17^{3} + \left(a^{5} + 5 a^{4} + 11 a^{3} + 13 a^{2} + 11 a + 3\right)\cdot 17^{4} + \left(8 a^{5} + 15 a^{4} + 13 a^{3} + 16 a^{2} + 12 a + 8\right)\cdot 17^{5} + \left(15 a^{5} + a^{4} + 7 a^{3} + 13 a + 15\right)\cdot 17^{6} +O(17^{7})$$ $r_{ 10 }$ $=$ $$12 a^{5} + a^{4} + 8 a^{3} + 9 a^{2} + 16 + \left(12 a^{5} + 2 a^{4} + 6 a^{3} + 6 a^{2} + 3 a + 15\right)\cdot 17 + \left(14 a^{5} + 15 a^{4} + 3 a^{3} + 5 a^{2} + 12 a + 4\right)\cdot 17^{2} + \left(3 a^{5} + 5 a^{4} + 12 a^{3} + 13 a^{2} + 16 a + 10\right)\cdot 17^{3} + \left(10 a^{5} + a^{4} + 12 a^{3} + 12 a^{2} + a + 1\right)\cdot 17^{4} + \left(14 a^{5} + 8 a^{4} + 12 a^{3} + 10 a^{2} + 13 a + 13\right)\cdot 17^{5} + \left(8 a^{5} + 11 a^{4} + 9 a^{3} + 2 a^{2} + 13 a + 5\right)\cdot 17^{6} +O(17^{7})$$ $r_{ 11 }$ $=$ $$7 a^{5} + 5 a^{4} + 16 a^{3} + 12 a^{2} + 12 a + 7 + \left(12 a^{5} + 13 a^{4} + 12 a^{3} + 12 a^{2} + 8\right)\cdot 17 + \left(12 a^{5} + 9 a^{4} + 3 a^{3} + 9 a^{2} + 10 a + 5\right)\cdot 17^{2} + \left(8 a^{5} + 5 a^{4} + 7 a^{3} + 6 a^{2} + 13 a + 16\right)\cdot 17^{3} + \left(15 a^{5} + 2 a^{4} + 2 a^{3} + 13 a^{2} + 13 a + 11\right)\cdot 17^{4} + \left(13 a^{3} + 14 a^{2} + 16 a + 15\right)\cdot 17^{5} + \left(13 a^{5} + 4 a^{4} + 8 a^{3} + 12 a^{2} + 3 a + 5\right)\cdot 17^{6} +O(17^{7})$$ $r_{ 12 }$ $=$ $$3 a^{5} + 15 a^{4} + 7 a^{3} + 9 a^{2} + 8 a + 3 + \left(13 a^{5} + 10 a^{4} + 6 a^{3} + 2 a^{2} + 8 a + 16\right)\cdot 17 + \left(10 a^{4} + 9 a^{3} + 14 a^{2} + 12 a + 2\right)\cdot 17^{2} + \left(2 a^{4} + 12 a^{3} + 13 a^{2} + 11 a + 9\right)\cdot 17^{3} + \left(12 a^{5} + 2 a^{4} + 6 a^{3} + 6 a + 16\right)\cdot 17^{4} + \left(16 a^{5} + 6 a^{4} + 8 a^{3} + 2 a^{2} + 15 a + 4\right)\cdot 17^{5} + \left(3 a^{5} + 7 a^{4} + 10 a^{3} + 13 a^{2} + 11 a + 1\right)\cdot 17^{6} +O(17^{7})$$

Generators of the action on the roots $r_1, \ldots, r_{ 12 }$

 Cycle notation $(1,4)(2,5)(3,12)(6,9)(7,10)(8,11)$ $(1,10)(2,11)(3,6)(4,7)(5,8)(9,12)$ $(1,2,3)(4,5,12)$ $(1,4)(2,5)(3,12)(6,7,11,9,10,8)$

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 12 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,4)(2,5)(3,12)(6,9)(7,10)(8,11)$ $-2$ $3$ $2$ $(1,10)(2,11)(3,6)(4,7)(5,8)(9,12)$ $0$ $3$ $2$ $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$ $0$ $1$ $3$ $(1,2,3)(4,5,12)(6,10,11)(7,8,9)$ $-2 \zeta_{3} - 2$ $1$ $3$ $(1,3,2)(4,12,5)(6,11,10)(7,9,8)$ $2 \zeta_{3}$ $2$ $3$ $(1,2,3)(4,5,12)$ $-\zeta_{3}$ $2$ $3$ $(1,3,2)(4,12,5)$ $\zeta_{3} + 1$ $2$ $3$ $(1,3,2)(4,12,5)(6,10,11)(7,8,9)$ $-1$ $1$ $6$ $(1,5,3,4,2,12)(6,7,11,9,10,8)$ $2 \zeta_{3} + 2$ $1$ $6$ $(1,12,2,4,3,5)(6,8,10,9,11,7)$ $-2 \zeta_{3}$ $2$ $6$ $(1,5,3,4,2,12)(6,9)(7,10)(8,11)$ $\zeta_{3}$ $2$ $6$ $(1,12,2,4,3,5)(6,9)(7,10)(8,11)$ $-\zeta_{3} - 1$ $2$ $6$ $(1,12,2,4,3,5)(6,7,11,9,10,8)$ $1$ $3$ $6$ $(1,8,2,9,3,7)(4,11,5,6,12,10)$ $0$ $3$ $6$ $(1,7,3,9,2,8)(4,10,12,6,5,11)$ $0$ $3$ $6$ $(1,11,2,6,3,10)(4,8,5,9,12,7)$ $0$ $3$ $6$ $(1,10,3,6,2,11)(4,7,12,9,5,8)$ $0$

The blue line marks the conjugacy class containing complex conjugation.