Properties

Label 3.2e6_5e2_13e2.12t33.1c2
Dimension 3
Group $A_5$
Conductor $ 2^{6} \cdot 5^{2} \cdot 13^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$270400= 2^{6} \cdot 5^{2} \cdot 13^{2} $
Artin number field: Splitting field of $f= x^{5} - 2 x^{4} + 5 x^{3} - 2 x^{2} + 7 x + 4 $ 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:
$r_{ 1 }$ $=$ $ 24 + 16\cdot 31 + 8\cdot 31^{2} + 17\cdot 31^{3} + 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 11 a + 25 + \left(17 a + 24\right)\cdot 31 + \left(10 a + 18\right)\cdot 31^{2} + \left(a + 11\right)\cdot 31^{3} + \left(11 a + 6\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 17 a + 29 + \left(18 a + 6\right)\cdot 31 + \left(22 a + 8\right)\cdot 31^{2} + \left(18 a + 7\right)\cdot 31^{3} + \left(13 a + 9\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 20 a + 16 + \left(13 a + 17\right)\cdot 31 + \left(20 a + 22\right)\cdot 31^{2} + \left(29 a + 3\right)\cdot 31^{3} + \left(19 a + 27\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 14 a + 1 + \left(12 a + 27\right)\cdot 31 + \left(8 a + 3\right)\cdot 31^{2} + \left(12 a + 22\right)\cdot 31^{3} + \left(17 a + 17\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} + 1$
$12$$5$$(1,3,4,5,2)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.