Properties

Label 4.4417.5t5.b.a
Dimension $4$
Group $S_5$
Conductor $4417$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(4417\)\(\medspace = 7 \cdot 631 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.1.4417.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.4417.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.1.4417.1

Defining polynomial

$f(x)$$=$ \( x^{5} - 2x^{4} + 2x^{3} - x^{2} - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 11 a + 18 + \left(31 a + 26\right)\cdot 41 + \left(4 a + 3\right)\cdot 41^{2} + \left(13 a + 17\right)\cdot 41^{3} + \left(7 a + 22\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 30 a + 10 + \left(9 a + 27\right)\cdot 41 + \left(36 a + 27\right)\cdot 41^{2} + \left(27 a + 10\right)\cdot 41^{3} + \left(33 a + 31\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 26 + 35\cdot 41 + 40\cdot 41^{2} + 4\cdot 41^{3} + 24\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 34 + 33\cdot 41 + 13\cdot 41^{2} + 10\cdot 41^{3} + 13\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 37 + 40\cdot 41 + 36\cdot 41^{2} + 38\cdot 41^{3} + 31\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display

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.