Properties

Label 6.29_43_241.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 29 \cdot 43 \cdot 241 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$300527= 29 \cdot 43 \cdot 241 $
Artin number field: Splitting field of $f= x^{7} - 2 x^{6} + 2 x^{5} - 2 x^{3} + 4 x^{2} - 3 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.29_43_241.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: $ x^{2} + 192 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 60 a + 4 + \left(135 a + 54\right)\cdot 193 + \left(147 a + 108\right)\cdot 193^{2} + \left(3 a + 104\right)\cdot 193^{3} + \left(a + 164\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 133 a + 64 + \left(57 a + 129\right)\cdot 193 + \left(45 a + 120\right)\cdot 193^{2} + \left(189 a + 153\right)\cdot 193^{3} + \left(191 a + 161\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 167 a + 190 + \left(35 a + 118\right)\cdot 193 + \left(24 a + 111\right)\cdot 193^{2} + \left(113 a + 131\right)\cdot 193^{3} + \left(87 a + 34\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 78 + 154\cdot 193 + 18\cdot 193^{2} + 148\cdot 193^{3} + 158\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 26 a + 164 + \left(157 a + 180\right)\cdot 193 + \left(168 a + 99\right)\cdot 193^{2} + \left(79 a + 27\right)\cdot 193^{3} + \left(105 a + 9\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 45 a + 18 + \left(175 a + 2\right)\cdot 193 + \left(46 a + 124\right)\cdot 193^{2} + \left(118 a + 67\right)\cdot 193^{3} + \left(169 a + 192\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 148 a + 63 + \left(17 a + 132\right)\cdot 193 + \left(146 a + 188\right)\cdot 193^{2} + \left(74 a + 138\right)\cdot 193^{3} + \left(23 a + 50\right)\cdot 193^{4} +O\left(193^{ 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.