Properties

Label 3.2e4_5e2_11e2.12t33.1c1
Dimension 3
Group $A_5$
Conductor $ 2^{4} \cdot 5^{2} \cdot 11^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$48400= 2^{4} \cdot 5^{2} \cdot 11^{2} $
Artin number field: Splitting field of $f= x^{5} - 2 x^{4} + 6 x^{3} - 20 x^{2} + 4 x - 32 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $A_5$
Parity: Even
Determinant: 1.1.1t1.1c1

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:
$r_{ 1 }$ $=$ $ 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\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 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\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 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\left(41^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 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\left(41^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 7 + 14\cdot 41 + 30\cdot 41^{2} + 2\cdot 41^{3} + 9\cdot 41^{4} +O\left(41^{ 5 }\right)$

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}$
$12$$5$$(1,3,4,5,2)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
The blue line marks the conjugacy class containing complex conjugation.