Properties

Label 5.37_857.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 37 \cdot 857 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$31709= 37 \cdot 857 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 2 x^{3} - 2 x^{2} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Even
Determinant: 1.37_857.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 39 a + 28 + \left(49 a + 25\right)\cdot 73 + \left(2 a + 71\right)\cdot 73^{2} + \left(48 a + 20\right)\cdot 73^{3} + \left(55 a + 50\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 40 a + 28 + \left(67 a + 43\right)\cdot 73 + 37\cdot 73^{2} + \left(38 a + 30\right)\cdot 73^{3} + \left(49 a + 25\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 34 a + 72 + \left(23 a + 61\right)\cdot 73 + \left(70 a + 29\right)\cdot 73^{2} + \left(24 a + 16\right)\cdot 73^{3} + \left(17 a + 23\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 71 + 6\cdot 73 + 42\cdot 73^{2} + 36\cdot 73^{3} + 34\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 33 a + 2 + \left(5 a + 60\right)\cdot 73 + \left(72 a + 45\right)\cdot 73^{2} + \left(34 a + 70\right)\cdot 73^{3} + \left(23 a + 62\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 19 + 21\cdot 73 + 65\cdot 73^{2} + 43\cdot 73^{3} + 22\cdot 73^{4} +O\left(73^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$5$
$15$$2$$(1,2)(3,4)(5,6)$$-1$
$15$$2$$(1,2)$$3$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$-1$
$40$$3$$(1,2,3)$$2$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$90$$4$$(1,2,3,4)$$1$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$-1$
$120$$6$$(1,2,3)(4,5)$$0$
The blue line marks the conjugacy class containing complex conjugation.