Properties

Label 3.2e2_3e4_5e4.12t33.2c1
Dimension 3
Group $A_5$
Conductor $ 2^{2} \cdot 3^{4} \cdot 5^{4}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$202500= 2^{2} \cdot 3^{4} \cdot 5^{4} $
Artin number field: Splitting field of $f= x^{5} - 10 x^{2} - 24 $ 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 $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 20 + 17\cdot 31 + 3\cdot 31^{2} + 30\cdot 31^{3} + 13\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 25 a + \left(9 a + 17\right)\cdot 31 + \left(4 a + 27\right)\cdot 31^{2} + \left(2 a + 25\right)\cdot 31^{3} + \left(12 a + 5\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 6 a + 19 + \left(21 a + 11\right)\cdot 31 + \left(26 a + 26\right)\cdot 31^{2} + \left(28 a + 25\right)\cdot 31^{3} + \left(18 a + 27\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 11 a + 16 + \left(26 a + 2\right)\cdot 31 + \left(20 a + 10\right)\cdot 31^{2} + \left(28 a + 18\right)\cdot 31^{3} + \left(9 a + 11\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 20 a + 7 + \left(4 a + 13\right)\cdot 31 + \left(10 a + 25\right)\cdot 31^{2} + \left(2 a + 23\right)\cdot 31^{3} + \left(21 a + 2\right)\cdot 31^{4} +O\left(31^{ 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.