Properties

Label 6.307_853.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 307 \cdot 853 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$261871= 307 \cdot 853 $
Artin number field: Splitting field of $f= x^{7} - x^{6} - x^{5} + 2 x^{4} - x^{3} - x^{2} + x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.307_853.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 11 + 31\cdot 41 + 35\cdot 41^{2} + 10\cdot 41^{3} + 35\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 7 a + 11 + \left(8 a + 21\right)\cdot 41 + \left(12 a + 11\right)\cdot 41^{2} + \left(21 a + 15\right)\cdot 41^{3} + \left(2 a + 8\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 34 a + 32 + \left(32 a + 38\right)\cdot 41 + \left(28 a + 39\right)\cdot 41^{2} + \left(19 a + 25\right)\cdot 41^{3} + \left(38 a + 35\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 22 a + \left(29 a + 24\right)\cdot 41 + \left(8 a + 24\right)\cdot 41^{2} + \left(a + 10\right)\cdot 41^{3} + \left(39 a + 24\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 19 a + 25 + \left(11 a + 8\right)\cdot 41 + \left(32 a + 21\right)\cdot 41^{2} + \left(39 a + 5\right)\cdot 41^{3} + \left(a + 17\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 37 + 19\cdot 41 + 39\cdot 41^{2} + 16\cdot 41^{3} + 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 8 + 20\cdot 41 + 32\cdot 41^{2} + 37\cdot 41^{3} +O\left(41^{ 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.