Properties

Label 6.371131.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 371131 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$371131 $
Artin number field: Splitting field of $f= x^{7} - x^{4} - 2 x^{2} - x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.371131.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 223 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 223 }$: $ x^{2} + 221 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 10 a + 57 + \left(101 a + 4\right)\cdot 223 + 98\cdot 223^{2} + \left(91 a + 34\right)\cdot 223^{3} + \left(125 a + 89\right)\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 174 a + 59 + \left(86 a + 158\right)\cdot 223 + \left(116 a + 140\right)\cdot 223^{2} + \left(25 a + 36\right)\cdot 223^{3} + \left(33 a + 162\right)\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 83 + 56\cdot 223 + 204\cdot 223^{2} + 218\cdot 223^{3} + 116\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 49 a + 184 + \left(136 a + 157\right)\cdot 223 + \left(106 a + 63\right)\cdot 223^{2} + \left(197 a + 194\right)\cdot 223^{3} + \left(189 a + 202\right)\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 213 a + 77 + \left(121 a + 196\right)\cdot 223 + \left(222 a + 220\right)\cdot 223^{2} + \left(131 a + 215\right)\cdot 223^{3} + \left(97 a + 25\right)\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 155 + 129\cdot 223 + 145\cdot 223^{2} + 45\cdot 223^{3} + 136\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 54 + 189\cdot 223 + 18\cdot 223^{2} + 146\cdot 223^{3} + 158\cdot 223^{4} +O\left(223^{ 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.