Properties

Label 6.73_5507.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 73 \cdot 5507 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$402011= 73 \cdot 5507 $
Artin number field: Splitting field of $f= x^{7} - x^{6} + 2 x^{5} - 4 x^{4} + 3 x^{3} - 3 x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.73_5507.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 503 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 503 }$: $ x^{2} + 498 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 35 + 315\cdot 503 + 173\cdot 503^{2} + 434\cdot 503^{3} + 350\cdot 503^{4} +O\left(503^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 191 a + 312 + \left(355 a + 370\right)\cdot 503 + \left(160 a + 383\right)\cdot 503^{2} + \left(127 a + 261\right)\cdot 503^{3} + \left(410 a + 84\right)\cdot 503^{4} +O\left(503^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 312 a + 261 + \left(147 a + 447\right)\cdot 503 + \left(342 a + 328\right)\cdot 503^{2} + \left(375 a + 234\right)\cdot 503^{3} + \left(92 a + 499\right)\cdot 503^{4} +O\left(503^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 461 + 246\cdot 503 + 411\cdot 503^{2} + 410\cdot 503^{3} + 20\cdot 503^{4} +O\left(503^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 210 a + 130 + \left(452 a + 464\right)\cdot 503 + \left(81 a + 428\right)\cdot 503^{2} + \left(68 a + 299\right)\cdot 503^{3} + \left(81 a + 307\right)\cdot 503^{4} +O\left(503^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 293 a + 174 + \left(50 a + 1\right)\cdot 503 + \left(421 a + 386\right)\cdot 503^{2} + \left(434 a + 55\right)\cdot 503^{3} + \left(421 a + 142\right)\cdot 503^{4} +O\left(503^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 137 + 166\cdot 503 + 402\cdot 503^{2} + 314\cdot 503^{3} + 103\cdot 503^{4} +O\left(503^{ 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.