Properties

 Label 3.376712.42t37.a Dimension 3 Group $\GL(3,2)$ Conductor $2^{3} \cdot 7^{2} \cdot 31^{2}$ Frobenius-Schur indicator 0

Related objects

Basic invariants

 Dimension: $3$ Group: $\GL(3,2)$ Conductor: $376712= 2^{3} \cdot 7^{2} \cdot 31^{2}$ Artin number field: Splitting field of $f= x^{7} + 2 x^{5} - 12 x^{3} - 2 x^{2} + 6 x - 2$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $\PSL(2,7)$ Parity: Even Projective image: $\PSL(2,7)$ Projective field: Galois closure of 7.3.147671104.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $x^{3} + 2 x + 9$
Roots:
 $r_{ 1 }$ $=$ $8 a^{2} + 7 a + 8 + \left(8 a^{2} + 7 a + 5\right)\cdot 11 + \left(7 a^{2} + 4 a + 5\right)\cdot 11^{2} + \left(2 a^{2} + 6 a + 8\right)\cdot 11^{3} + \left(a^{2} + 7 a + 9\right)\cdot 11^{4} + \left(2 a + 1\right)\cdot 11^{5} + 4 a\cdot 11^{6} +O\left(11^{ 7 }\right)$ $r_{ 2 }$ $=$ $3 a^{2} + 8 a + 4 + \left(9 a^{2} + 8 a + 3\right)\cdot 11 + \left(a^{2} + 9 a + 8\right)\cdot 11^{2} + \left(9 a^{2} + 6 a\right)\cdot 11^{3} + \left(2 a^{2} + 7 a + 2\right)\cdot 11^{4} + \left(9 a + 5\right)\cdot 11^{5} + \left(7 a^{2} + 3 a + 4\right)\cdot 11^{6} +O\left(11^{ 7 }\right)$ $r_{ 3 }$ $=$ $2 a^{2} + 5 a + \left(3 a^{2} + 3 a + 2\right)\cdot 11 + \left(2 a + 10\right)\cdot 11^{2} + \left(7 a + 4\right)\cdot 11^{3} + \left(4 a^{2} + 3 a + 2\right)\cdot 11^{4} + \left(a^{2} + 10 a + 7\right)\cdot 11^{5} + \left(3 a^{2} + 9 a\right)\cdot 11^{6} +O\left(11^{ 7 }\right)$ $r_{ 4 }$ $=$ $a^{2} + 10 a + 6 + \left(10 a^{2} + 10 a + 7\right)\cdot 11 + \left(2 a^{2} + 3 a + 2\right)\cdot 11^{2} + \left(8 a^{2} + 8 a + 1\right)\cdot 11^{3} + \left(5 a^{2} + 10 a + 1\right)\cdot 11^{4} + \left(9 a^{2} + 8 a + 7\right)\cdot 11^{5} + \left(7 a^{2} + 7 a + 10\right)\cdot 11^{6} +O\left(11^{ 7 }\right)$ $r_{ 5 }$ $=$ $8 + 8\cdot 11^{2} + 8\cdot 11^{3} + 2\cdot 11^{4} + 2\cdot 11^{5} + 3\cdot 11^{6} +O\left(11^{ 7 }\right)$ $r_{ 6 }$ $=$ $2 a^{2} + 10 a + 10 + \left(6 a + 5\right)\cdot 11 + \left(6 a^{2} + 3 a + 6\right)\cdot 11^{2} + \left(2 a^{2} + a + 6\right)\cdot 11^{3} + \left(a^{2} + 10 a + 3\right)\cdot 11^{4} + \left(10 a^{2} + 3 a + 7\right)\cdot 11^{5} + \left(3 a^{2} + 7 a + 7\right)\cdot 11^{6} +O\left(11^{ 7 }\right)$ $r_{ 7 }$ $=$ $6 a^{2} + 4 a + 8 + \left(a^{2} + 6 a + 7\right)\cdot 11 + \left(3 a^{2} + 8 a + 2\right)\cdot 11^{2} + \left(10 a^{2} + 2 a + 2\right)\cdot 11^{3} + \left(6 a^{2} + 4 a\right)\cdot 11^{4} + \left(8 a + 2\right)\cdot 11^{5} + \left(10 a + 6\right)\cdot 11^{6} +O\left(11^{ 7 }\right)$

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

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

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $3$ $3$ $21$ $2$ $(1,7)(5,6)$ $-1$ $-1$ $56$ $3$ $(1,2,5)(3,6,4)$ $0$ $0$ $42$ $4$ $(1,3,2,6)(4,7)$ $1$ $1$ $24$ $7$ $(1,3,2,5,6,7,4)$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ $24$ $7$ $(1,5,4,2,7,3,6)$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
The blue line marks the conjugacy class containing complex conjugation.