Properties

 Label 6.237751.7t7.a.a Dimension $6$ Group $S_7$ Conductor $237751$ Root number $1$ Indicator $1$

Related objects

Basic invariants

 Dimension: $6$ Group: $S_7$ Conductor: $$237751$$$$\medspace = 23 \cdot 10337$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 7.1.237751.1 Galois orbit size: $1$ Smallest permutation container: $S_7$ Parity: odd Determinant: 1.237751.2t1.a.a Projective image: $S_7$ Projective field: Galois closure of 7.1.237751.1

Defining polynomial

 $f(x)$ $=$ $x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1$.

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $x^{2} + 38 x + 6$

Roots:
 $r_{ 1 }$ $=$ $30 a + 17 + \left(16 a + 20\right)\cdot 41 + \left(17 a + 22\right)\cdot 41^{2} + \left(14 a + 22\right)\cdot 41^{3} + \left(13 a + 8\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 2 }$ $=$ $34 a + 37 + \left(18 a + 38\right)\cdot 41 + \left(8 a + 25\right)\cdot 41^{2} + \left(17 a + 13\right)\cdot 41^{3} + \left(3 a + 29\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 3 }$ $=$ $35 + 31\cdot 41 + 28\cdot 41^{2} + 33\cdot 41^{3} + 6\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 4 }$ $=$ $7 a + 16 + \left(22 a + 20\right)\cdot 41 + \left(32 a + 32\right)\cdot 41^{2} + \left(23 a + 15\right)\cdot 41^{3} + \left(37 a + 22\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 5 }$ $=$ $22 + 36\cdot 41 + 29\cdot 41^{2} + 14\cdot 41^{3} + 24\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 6 }$ $=$ $13 + 16\cdot 41 + 7\cdot 41^{2} + 15\cdot 41^{3} + 38\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 7 }$ $=$ $11 a + 25 + \left(24 a + 40\right)\cdot 41 + \left(23 a + 16\right)\cdot 41^{2} + \left(26 a + 7\right)\cdot 41^{3} + \left(27 a + 34\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$

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

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

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character value $1$ $1$ $()$ $6$ $21$ $2$ $(1,2)$ $4$ $105$ $2$ $(1,2)(3,4)(5,6)$ $0$ $105$ $2$ $(1,2)(3,4)$ $2$ $70$ $3$ $(1,2,3)$ $3$ $280$ $3$ $(1,2,3)(4,5,6)$ $0$ $210$ $4$ $(1,2,3,4)$ $2$ $630$ $4$ $(1,2,3,4)(5,6)$ $0$ $504$ $5$ $(1,2,3,4,5)$ $1$ $210$ $6$ $(1,2,3)(4,5)(6,7)$ $-1$ $420$ $6$ $(1,2,3)(4,5)$ $1$ $840$ $6$ $(1,2,3,4,5,6)$ $0$ $720$ $7$ $(1,2,3,4,5,6,7)$ $-1$ $504$ $10$ $(1,2,3,4,5)(6,7)$ $-1$ $420$ $12$ $(1,2,3,4)(5,6,7)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.