Properties

Label 4.4477.5t5.a
Dimension $4$
Group $S_5$
Conductor $4477$
Indicator $1$

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:\(4477\)\(\medspace = 11^{2} \cdot 37 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 5.1.4477.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Projective image: $S_5$
Projective field: Galois closure of 5.1.4477.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: \( x^{2} + 97x + 2 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 90 a + 50 + \left(29 a + 88\right)\cdot 101 + \left(89 a + 37\right)\cdot 101^{2} + \left(83 a + 87\right)\cdot 101^{3} + \left(36 a + 82\right)\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 11 a + 6 + \left(71 a + 17\right)\cdot 101 + \left(11 a + 62\right)\cdot 101^{2} + \left(17 a + 30\right)\cdot 101^{3} + \left(64 a + 45\right)\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 48 + 31\cdot 101 + 100\cdot 101^{2} + 32\cdot 101^{3} + 5\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 72 a + 6 + \left(42 a + 84\right)\cdot 101 + \left(46 a + 80\right)\cdot 101^{2} + \left(83 a + 83\right)\cdot 101^{3} + \left(76 a + 73\right)\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 29 a + 92 + \left(58 a + 81\right)\cdot 101 + \left(54 a + 21\right)\cdot 101^{2} + \left(17 a + 68\right)\cdot 101^{3} + \left(24 a + 95\right)\cdot 101^{4} +O(101^{5})\) 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 values
$c1$
$1$ $1$ $()$ $4$
$10$ $2$ $(1,2)$ $2$
$15$ $2$ $(1,2)(3,4)$ $0$
$20$ $3$ $(1,2,3)$ $1$
$30$ $4$ $(1,2,3,4)$ $0$
$24$ $5$ $(1,2,3,4,5)$ $-1$
$20$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.