Properties

Label 6.61_4643.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 61 \cdot 4643 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$283223= 61 \cdot 4643 $
Artin number field: Splitting field of $f= x^{7} + x^{3} - 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.61_4643.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 251 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 251 }$: $ x^{2} + 242 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 151 a + 29 + \left(204 a + 186\right)\cdot 251 + \left(33 a + 43\right)\cdot 251^{2} + \left(43 a + 66\right)\cdot 251^{3} + \left(127 a + 169\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 239 + 165\cdot 251 + 223\cdot 251^{2} + 181\cdot 251^{3} + 187\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 247 a + 57 + \left(15 a + 75\right)\cdot 251 + \left(104 a + 24\right)\cdot 251^{2} + \left(81 a + 75\right)\cdot 251^{3} + \left(110 a + 186\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 132 + 145\cdot 251 + 242\cdot 251^{2} + 242\cdot 251^{3} + 26\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 100 a + 133 + \left(46 a + 119\right)\cdot 251 + \left(217 a + 143\right)\cdot 251^{2} + \left(207 a + 169\right)\cdot 251^{3} + \left(123 a + 15\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 4 a + 21 + \left(235 a + 223\right)\cdot 251 + \left(146 a + 191\right)\cdot 251^{2} + \left(169 a + 201\right)\cdot 251^{3} + \left(140 a + 93\right)\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 142 + 88\cdot 251 + 134\cdot 251^{2} + 66\cdot 251^{3} + 73\cdot 251^{4} +O\left(251^{ 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.