Properties

Label 6.31_12377.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 31 \cdot 12377 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$383687= 31 \cdot 12377 $
Artin number field: Splitting field of $f= x^{7} - x^{6} + 2 x^{5} - 2 x^{4} + x^{3} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.31_12377.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 10 + 60\cdot 73 + 52\cdot 73^{2} + 34\cdot 73^{3} + 70\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 34 + 33\cdot 73 + 71\cdot 73^{2} + 7\cdot 73^{3} + 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 38 a + 66 + \left(63 a + 43\right)\cdot 73 + \left(49 a + 24\right)\cdot 73^{2} + \left(45 a + 60\right)\cdot 73^{3} + \left(54 a + 49\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 53 + 64\cdot 73 + 4\cdot 73^{2} + 73^{3} + 56\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 35 a + 34 + \left(9 a + 50\right)\cdot 73 + \left(23 a + 37\right)\cdot 73^{2} + \left(27 a + 1\right)\cdot 73^{3} + \left(18 a + 22\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 33 a + 35 + \left(33 a + 22\right)\cdot 73 + \left(59 a + 14\right)\cdot 73^{2} + \left(53 a + 42\right)\cdot 73^{3} + \left(56 a + 24\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 40 a + 61 + \left(39 a + 16\right)\cdot 73 + \left(13 a + 13\right)\cdot 73^{2} + \left(19 a + 71\right)\cdot 73^{3} + \left(16 a + 67\right)\cdot 73^{4} +O\left(73^{ 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.