Properties

Label 6.461_587.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 461 \cdot 587 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$270607= 461 \cdot 587 $
Artin number field: Splitting field of $f= x^{7} - x^{6} + x^{5} - 3 x^{4} + 2 x^{3} - 2 x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.461_587.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 2 + 8\cdot 59 + 49\cdot 59^{2} + 43\cdot 59^{3} + 54\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 56 + 58\cdot 59 + 51\cdot 59^{2} + 24\cdot 59^{3} + 21\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 42 a + 14 + \left(29 a + 33\right)\cdot 59 + \left(29 a + 19\right)\cdot 59^{2} + \left(22 a + 6\right)\cdot 59^{3} + \left(14 a + 51\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 54 + 29\cdot 59 + 11\cdot 59^{2} + 28\cdot 59^{3} + 44\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 17 a + 56 + \left(29 a + 20\right)\cdot 59 + \left(29 a + 19\right)\cdot 59^{2} + \left(36 a + 58\right)\cdot 59^{3} + \left(44 a + 42\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 11 a + 22 + \left(8 a + 14\right)\cdot 59 + \left(22 a + 35\right)\cdot 59^{2} + \left(34 a + 1\right)\cdot 59^{3} + \left(57 a + 58\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 48 a + 33 + \left(50 a + 11\right)\cdot 59 + \left(36 a + 49\right)\cdot 59^{2} + \left(24 a + 13\right)\cdot 59^{3} + \left(a + 22\right)\cdot 59^{4} +O\left(59^{ 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

SizeOrderAction 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.