Properties

Label 4.7_631.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 7 \cdot 631 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$4417= 7 \cdot 631 $
Artin number field: Splitting field of $f= x^{5} - 2 x^{4} + 2 x^{3} - x^{2} - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.7_631.2t1.1c1

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 a + 18 + \left(31 a + 26\right)\cdot 41 + \left(4 a + 3\right)\cdot 41^{2} + \left(13 a + 17\right)\cdot 41^{3} + \left(7 a + 22\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 30 a + 10 + \left(9 a + 27\right)\cdot 41 + \left(36 a + 27\right)\cdot 41^{2} + \left(27 a + 10\right)\cdot 41^{3} + \left(33 a + 31\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 26 + 35\cdot 41 + 40\cdot 41^{2} + 4\cdot 41^{3} + 24\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 34 + 33\cdot 41 + 13\cdot 41^{2} + 10\cdot 41^{3} + 13\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 37 + 40\cdot 41 + 36\cdot 41^{2} + 38\cdot 41^{3} + 31\cdot 41^{4} +O\left(41^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$4$
$10$$2$$(1,2)$$2$
$15$$2$$(1,2)(3,4)$$0$
$20$$3$$(1,2,3)$$1$
$30$$4$$(1,2,3,4)$$0$
$24$$5$$(1,2,3,4,5)$$-1$
$20$$6$$(1,2,3)(4,5)$$-1$
The blue line marks the conjugacy class containing complex conjugation.