Properties

 Label 2.3743.7t2.a.b Dimension $2$ Group $D_{7}$ Conductor $3743$ Root number $1$ Indicator $1$

Related objects

Basic invariants

 Dimension: $2$ Group: $D_{7}$ Conductor: $$3743$$$$\medspace = 19 \cdot 197$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 7.1.52439613407.1 Galois orbit size: $3$ Smallest permutation container: $D_{7}$ Parity: odd Determinant: 1.3743.2t1.a.a Projective image: $D_7$ Projective stem field: 7.1.52439613407.1

Defining polynomial

 $f(x)$ $=$ $$x^{7} - 2 x^{6} + 11 x^{5} - 26 x^{4} + 45 x^{3} - 60 x^{2} + 39 x + 11$$  .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{2} + 16 x + 3$$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.