Properties

 Label 2.72.6t5.b.a Dimension $2$ Group $S_3\times C_3$ Conductor $72$ Root number not computed Indicator $0$

Related objects

Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$72$$$$\medspace = 2^{3} \cdot 3^{2}$$ Artin stem field: 6.0.41472.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.72.6t1.a.b Projective image: $S_3$ Projective stem field: 3.1.648.1

Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2 x^{5} + 3 x^{4} - 2 x^{3} + 2 x^{2} + 1$$  .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $$x^{2} + 33 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$24 a + 9 + \left(21 a + 32\right)\cdot 37 + \left(4 a + 18\right)\cdot 37^{2} + \left(31 a + 8\right)\cdot 37^{3} + \left(35 a + 15\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 2 }$ $=$ $$13 a + 31 + \left(15 a + 20\right)\cdot 37 + \left(32 a + 15\right)\cdot 37^{2} + \left(5 a + 17\right)\cdot 37^{3} + \left(a + 16\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 3 }$ $=$ $$29 a + 18 + \left(35 a + 13\right)\cdot 37 + \left(27 a + 30\right)\cdot 37^{2} + \left(12 a + 29\right)\cdot 37^{3} + 35\cdot 37^{4} +O(37^{5})$$ $r_{ 4 }$ $=$ $$26 a + 1 + \left(16 a + 12\right)\cdot 37 + \left(12 a + 27\right)\cdot 37^{2} + \left(8 a + 27\right)\cdot 37^{3} + \left(24 a + 20\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 5 }$ $=$ $$8 a + 23 + \left(a + 16\right)\cdot 37 + \left(9 a + 32\right)\cdot 37^{2} + \left(24 a + 15\right)\cdot 37^{3} + \left(36 a + 24\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 6 }$ $=$ $$11 a + 31 + \left(20 a + 15\right)\cdot 37 + \left(24 a + 23\right)\cdot 37^{2} + \left(28 a + 11\right)\cdot 37^{3} + \left(12 a + 35\right)\cdot 37^{4} +O(37^{5})$$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.