Properties

 Label 2.3143.7t2.a.b Dimension $2$ Group $D_{7}$ Conductor $3143$ Root number $1$ Indicator $1$

Related objects

Basic invariants

 Dimension: $2$ Group: $D_{7}$ Conductor: $$3143$$$$\medspace = 7 \cdot 449$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 7.1.31047965207.1 Galois orbit size: $3$ Smallest permutation container: $D_{7}$ Parity: odd Determinant: 1.3143.2t1.a.a Projective image: $D_7$ Projective stem field: 7.1.31047965207.1

Defining polynomial

 $f(x)$ $=$ $$x^{7} - 3 x^{6} - 4 x^{5} + 30 x^{4} - 24 x^{3} - 70 x^{2} + 112 x - 47$$  .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $$x^{2} + 24 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$11 a + 6 + 10\cdot 29 + 9\cdot 29^{2} + \left(23 a + 7\right)\cdot 29^{3} + \left(13 a + 13\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 2 }$ $=$ $$19 a + 9 + \left(6 a + 14\right)\cdot 29 + \left(19 a + 22\right)\cdot 29^{2} + \left(28 a + 22\right)\cdot 29^{3} + \left(7 a + 18\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 3 }$ $=$ $$a + 8 + \left(27 a + 10\right)\cdot 29 + \left(19 a + 20\right)\cdot 29^{2} + \left(27 a + 10\right)\cdot 29^{3} + \left(13 a + 16\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 4 }$ $=$ $$18 a + 3 + \left(28 a + 1\right)\cdot 29 + \left(28 a + 9\right)\cdot 29^{2} + \left(5 a + 6\right)\cdot 29^{3} + \left(15 a + 1\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 5 }$ $=$ $$10 a + 17 + \left(22 a + 28\right)\cdot 29 + \left(9 a + 24\right)\cdot 29^{2} + 29^{3} + \left(21 a + 1\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 6 }$ $=$ $$5 + 23\cdot 29 + 23\cdot 29^{2} + 24\cdot 29^{3} + 6\cdot 29^{4} +O(29^{5})$$ $r_{ 7 }$ $=$ $$28 a + 13 + \left(a + 28\right)\cdot 29 + \left(9 a + 5\right)\cdot 29^{2} + \left(a + 13\right)\cdot 29^{3} + 15 a\cdot 29^{4} +O(29^{5})$$

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

 Cycle notation $(1,5)(2,7)(3,6)$ $(1,7)(2,6)(4,5)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.