Properties

 Label 3.3143529.42t37.a Dimension 3 Group $\GL(3,2)$ Conductor $3^{4} \cdot 197^{2}$ Frobenius-Schur indicator 0

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Basic invariants

 Dimension: $3$ Group: $\GL(3,2)$ Conductor: $3143529= 3^{4} \cdot 197^{2}$ Artin number field: Splitting field of $f= x^{7} - 6 x^{5} - 6 x^{4} + 3 x^{3} + 6 x^{2} + 2 x - 3$ 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 7.3.28291761.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $x^{3} + 2 x + 11$
Roots:
 $r_{ 1 }$ $=$ $2 a^{2} + 12 a + 11 + \left(2 a^{2} + 3 a\right)\cdot 13 + \left(10 a^{2} + 4 a + 9\right)\cdot 13^{2} + \left(2 a^{2} + a + 1\right)\cdot 13^{3} + \left(a^{2} + 7 a + 8\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 2 }$ $=$ $5 a^{2} + 8 a + 12 + \left(7 a^{2} + 4 a + 3\right)\cdot 13 + \left(12 a^{2} + 2 a + 7\right)\cdot 13^{2} + \left(2 a^{2} + 9 a + 5\right)\cdot 13^{3} + \left(a^{2} + a\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 3 }$ $=$ $6 a^{2} + 10 a + 12 + \left(10 a^{2} + 2 a + 11\right)\cdot 13 + \left(2 a^{2} + 8 a + 7\right)\cdot 13^{2} + \left(2 a + 2\right)\cdot 13^{3} + \left(6 a^{2} + 2 a + 10\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 4 }$ $=$ $5 a^{2} + 4 a + 2 + \left(6 a + 7\right)\cdot 13 + 8\cdot 13^{2} + \left(10 a^{2} + 9 a + 2\right)\cdot 13^{3} + \left(5 a^{2} + 3 a + 1\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 5 }$ $=$ $11 + 10\cdot 13 + 2\cdot 13^{2} + 13^{3} + 10\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 6 }$ $=$ $12 a + 1 + \left(a^{2} + 5 a + 4\right)\cdot 13 + \left(12 a^{2} + a + 2\right)\cdot 13^{2} + \left(6 a^{2} + 7 a + 2\right)\cdot 13^{3} + \left(3 a^{2} + 9 a + 12\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 7 }$ $=$ $8 a^{2} + 6 a + 3 + \left(4 a^{2} + 2 a\right)\cdot 13 + \left(a^{2} + 9 a + 1\right)\cdot 13^{2} + \left(3 a^{2} + 9 a + 10\right)\cdot 13^{3} + \left(8 a^{2} + a + 9\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$

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

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

Character values on conjugacy classes

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