Properties

Label 6.13e2_17_127.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 13^{2} \cdot 17 \cdot 127 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$364871= 13^{2} \cdot 17 \cdot 127 $
Artin number field: Splitting field of $f= x^{7} - x^{6} + 2 x^{4} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.17_127.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $ x^{2} + 82 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 9 a + 71 + \left(75 a + 82\right)\cdot 83 + \left(23 a + 70\right)\cdot 83^{2} + \left(28 a + 32\right)\cdot 83^{3} + \left(7 a + 37\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 22 + 67\cdot 83 + 82\cdot 83^{2} + 33\cdot 83^{3} + 42\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 82 a + 71 + \left(37 a + 64\right)\cdot 83 + \left(58 a + 26\right)\cdot 83^{2} + \left(67 a + 33\right)\cdot 83^{3} + \left(a + 6\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 48 a + 27 + \left(11 a + 33\right)\cdot 83 + \left(27 a + 34\right)\cdot 83^{2} + \left(33 a + 31\right)\cdot 83^{3} + \left(56 a + 8\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 5 }$ $=$ $ a + 70 + \left(45 a + 20\right)\cdot 83 + \left(24 a + 47\right)\cdot 83^{2} + \left(15 a + 42\right)\cdot 83^{3} + \left(81 a + 23\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 74 a + 80 + \left(7 a + 65\right)\cdot 83 + \left(59 a + 19\right)\cdot 83^{2} + \left(54 a + 37\right)\cdot 83^{3} + \left(75 a + 16\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 35 a + 75 + \left(71 a + 79\right)\cdot 83 + \left(55 a + 49\right)\cdot 83^{2} + \left(49 a + 37\right)\cdot 83^{3} + \left(26 a + 31\right)\cdot 83^{4} +O\left(83^{ 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.