Properties

Label 3.2e2_7e2_31e2.12t33.1c2
Dimension 3
Group $A_5$
Conductor $ 2^{2} \cdot 7^{2} \cdot 31^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$188356= 2^{2} \cdot 7^{2} \cdot 31^{2} $
Artin number field: Splitting field of $f= x^{5} - x^{4} - 12 x^{3} - 10 x^{2} + 63 x + 119 $ 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 $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{2} + 16 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 15\cdot 17 + 10\cdot 17^{2} + 6\cdot 17^{3} + 14\cdot 17^{4} + 14\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 4 a + 10 + \left(5 a + 4\right)\cdot 17 + \left(9 a + 7\right)\cdot 17^{2} + \left(12 a + 3\right)\cdot 17^{3} + \left(10 a + 2\right)\cdot 17^{4} + 10 a\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 7 a + 2 + 16\cdot 17 + \left(8 a + 6\right)\cdot 17^{2} + \left(5 a + 1\right)\cdot 17^{3} + \left(13 a + 13\right)\cdot 17^{4} + \left(15 a + 16\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 10 a + 9 + \left(16 a + 9\right)\cdot 17 + \left(8 a + 14\right)\cdot 17^{2} + \left(11 a + 15\right)\cdot 17^{3} + \left(3 a + 3\right)\cdot 17^{4} + \left(a + 2\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 13 a + 14 + \left(11 a + 5\right)\cdot 17 + \left(7 a + 11\right)\cdot 17^{2} + \left(4 a + 6\right)\cdot 17^{3} + 6 a\cdot 17^{4} + 6 a\cdot 17^{5} +O\left(17^{ 6 }\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.