Properties

Label 6.277_1123.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 277 \cdot 1123 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$311071= 277 \cdot 1123 $
Artin number field: Splitting field of $f= x^{7} - x^{6} + 3 x^{5} - 3 x^{4} + 4 x^{3} - 4 x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.277_1123.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{2} + 96 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 8 + 75\cdot 97 + 45\cdot 97^{2} + 83\cdot 97^{3} + 21\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 54 a + 47 + \left(16 a + 94\right)\cdot 97 + \left(9 a + 77\right)\cdot 97^{2} + \left(54 a + 62\right)\cdot 97^{3} + \left(20 a + 93\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 43 a + 4 + \left(80 a + 57\right)\cdot 97 + \left(87 a + 70\right)\cdot 97^{2} + \left(42 a + 10\right)\cdot 97^{3} + \left(76 a + 60\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 87 a + 25 + \left(64 a + 27\right)\cdot 97 + \left(4 a + 58\right)\cdot 97^{2} + \left(29 a + 58\right)\cdot 97^{3} + \left(53 a + 63\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 10 a + 15 + \left(32 a + 5\right)\cdot 97 + \left(92 a + 95\right)\cdot 97^{2} + \left(67 a + 82\right)\cdot 97^{3} + \left(43 a + 87\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 41 + 97 + 15\cdot 97^{2} + 97^{3} + 41\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 55 + 30\cdot 97 + 25\cdot 97^{2} + 88\cdot 97^{3} + 19\cdot 97^{4} +O\left(97^{ 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.