Properties

Label 6.499_797.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 499 \cdot 797 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$397703= 499 \cdot 797 $
Artin number field: Splitting field of $f= x^{7} + x^{5} + x^{3} - 2 x^{2} + x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.499_797.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 35 a + 50 + \left(55 a + 29\right)\cdot 59 + \left(30 a + 14\right)\cdot 59^{2} + \left(23 a + 29\right)\cdot 59^{3} + \left(40 a + 30\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 24 a + 26 + \left(3 a + 50\right)\cdot 59 + \left(28 a + 48\right)\cdot 59^{2} + \left(35 a + 21\right)\cdot 59^{3} + \left(18 a + 47\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 16 a + 15 + \left(7 a + 38\right)\cdot 59 + \left(38 a + 6\right)\cdot 59^{2} + \left(54 a + 31\right)\cdot 59^{3} + \left(12 a + 11\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 36 + 37\cdot 59 + 25\cdot 59^{3} + 47\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 13 + 29\cdot 59 + 52\cdot 59^{2} + 40\cdot 59^{3} + 16\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 43 a + 31 + \left(51 a + 29\right)\cdot 59 + \left(20 a + 37\right)\cdot 59^{2} + \left(4 a + 47\right)\cdot 59^{3} + \left(46 a + 28\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 6 + 21\cdot 59 + 16\cdot 59^{2} + 40\cdot 59^{3} + 53\cdot 59^{4} +O\left(59^{ 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.