Properties

Label 3.13e2_31e2.12t33.1c2
Dimension 3
Group $A_5$
Conductor $ 13^{2} \cdot 31^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$162409= 13^{2} \cdot 31^{2} $
Artin number field: Splitting field of $f= x^{5} + 2 x^{3} - 5 x^{2} + 2 x - 1 $ 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 $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $ x^{2} + 7 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 5 a + 3 + \left(6 a + 7\right)\cdot 11 + \left(9 a + 2\right)\cdot 11^{2} + 4 a\cdot 11^{3} + \left(3 a + 1\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 7 a + 1 + \left(7 a + 3\right)\cdot 11 + \left(4 a + 10\right)\cdot 11^{2} + \left(4 a + 3\right)\cdot 11^{3} + \left(5 a + 4\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 6 a + 1 + \left(4 a + 6\right)\cdot 11 + \left(a + 1\right)\cdot 11^{2} + \left(6 a + 10\right)\cdot 11^{3} + \left(7 a + 9\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 10 + 8\cdot 11^{2} + 11^{3} + 7\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 4 a + 7 + \left(3 a + 4\right)\cdot 11 + \left(6 a + 10\right)\cdot 11^{2} + \left(6 a + 5\right)\cdot 11^{3} + \left(5 a + 10\right)\cdot 11^{4} +O\left(11^{ 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.