Properties

Label 3.3e4_5e2_11e2.12t33.1c1
Dimension 3
Group $A_5$
Conductor $ 3^{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:$245025= 3^{4} \cdot 5^{2} \cdot 11^{2} $
Artin number field: Splitting field of $f= x^{5} - x^{4} - 3 x^{3} - 2 x^{2} + 5 x - 3 $ 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 $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 4 a + 46 + \left(5 a + 11\right)\cdot 53 + \left(20 a + 5\right)\cdot 53^{2} + 31 a\cdot 53^{3} + \left(30 a + 47\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 22 a + 41 + \left(33 a + 5\right)\cdot 53 + \left(45 a + 47\right)\cdot 53^{2} + \left(38 a + 16\right)\cdot 53^{3} + \left(46 a + 44\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 49 a + 9 + \left(47 a + 28\right)\cdot 53 + \left(32 a + 27\right)\cdot 53^{2} + \left(21 a + 52\right)\cdot 53^{3} + \left(22 a + 31\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 31 a + 23 + \left(19 a + 11\right)\cdot 53 + \left(7 a + 37\right)\cdot 53^{2} + \left(14 a + 20\right)\cdot 53^{3} + \left(6 a + 33\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 41 + 48\cdot 53 + 41\cdot 53^{2} + 15\cdot 53^{3} + 2\cdot 53^{4} +O\left(53^{ 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.