Properties

Label 6.7_38953.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 7 \cdot 38953 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$272671= 7 \cdot 38953 $
Artin number field: Splitting field of $f= x^{7} - 2 x^{6} + 3 x^{5} - 4 x^{4} + 4 x^{3} - 3 x^{2} + x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.7_38953.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
$r_{ 1 }$ $=$ $ 23 a + 54 + \left(64 a + 36\right)\cdot 71 + \left(6 a + 68\right)\cdot 71^{2} + \left(5 a + 16\right)\cdot 71^{3} + \left(41 a + 27\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 27 a + 69 + \left(55 a + 31\right)\cdot 71 + \left(19 a + 34\right)\cdot 71^{2} + \left(15 a + 10\right)\cdot 71^{3} + \left(52 a + 26\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 58 + 9\cdot 71 + 60\cdot 71^{2} + 7\cdot 71^{3} + 39\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 56 + 25\cdot 71 + 32\cdot 71^{2} + 12\cdot 71^{3} + 58\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 48 a + 29 + 6 a\cdot 71 + \left(64 a + 18\right)\cdot 71^{2} + \left(65 a + 20\right)\cdot 71^{3} + \left(29 a + 33\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 44 a + 52 + \left(15 a + 44\right)\cdot 71 + \left(51 a + 18\right)\cdot 71^{2} + \left(55 a + 21\right)\cdot 71^{3} + \left(18 a + 44\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 39 + 63\cdot 71 + 51\cdot 71^{2} + 52\cdot 71^{3} + 55\cdot 71^{4} +O\left(71^{ 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.