Properties

Label 3.48400.12t33.a.b
Dimension $3$
Group $A_5$
Conductor $48400$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_5$
Conductor: \(48400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.1.5856400.1
Galois orbit size: $2$
Smallest permutation container: $A_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_5$
Projective stem field: Galois closure of 5.1.5856400.1

Defining polynomial

$f(x)$$=$ \( x^{5} - 2x^{4} + 6x^{3} - 20x^{2} + 4x - 32 \) 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:
$r_{ 1 }$ $=$ \( 7 a + 13 + \left(30 a + 15\right)\cdot 41 + \left(7 a + 30\right)\cdot 41^{2} + \left(a + 27\right)\cdot 41^{3} + \left(14 a + 11\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 29 a + 33 + \left(27 a + 31\right)\cdot 41 + \left(36 a + 39\right)\cdot 41^{2} + \left(9 a + 37\right)\cdot 41^{3} + \left(3 a + 24\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 12 a + 38 + \left(13 a + 3\right)\cdot 41 + \left(4 a + 40\right)\cdot 41^{2} + \left(31 a + 30\right)\cdot 41^{3} + \left(37 a + 24\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 34 a + 34 + \left(10 a + 16\right)\cdot 41 + \left(33 a + 23\right)\cdot 41^{2} + \left(39 a + 23\right)\cdot 41^{3} + \left(26 a + 11\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 7 + 14\cdot 41 + 30\cdot 41^{2} + 2\cdot 41^{3} + 9\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$3$
$15$$2$$(1,2)(3,4)$$-1$
$20$$3$$(1,2,3)$$0$
$12$$5$$(1,2,3,4,5)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
$12$$5$$(1,3,4,5,2)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$

The blue line marks the conjugacy class containing complex conjugation.