Properties

Label 3.3e4_23e2.12t33.1c1
Dimension 3
Group $A_5$
Conductor $ 3^{4} \cdot 23^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$42849= 3^{4} \cdot 23^{2} $
Artin number field: Splitting field of $f= x^{5} - x^{4} + x^{3} + 2 x^{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 $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{2} + 12 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 2 a + 9 + \left(a + 5\right)\cdot 13 + \left(3 a + 7\right)\cdot 13^{2} + \left(a + 7\right)\cdot 13^{3} + \left(4 a + 4\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 11 a + 11 + \left(11 a + 4\right)\cdot 13 + \left(9 a + 9\right)\cdot 13^{2} + \left(11 a + 5\right)\cdot 13^{3} + \left(8 a + 7\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 6 + 4\cdot 13 + 7\cdot 13^{2} + 8\cdot 13^{3} + 9\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 7 a + 10 + \left(2 a + 7\right)\cdot 13 + \left(6 a + 5\right)\cdot 13^{2} + \left(10 a + 6\right)\cdot 13^{3} + \left(3 a + 5\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 6 a + 4 + \left(10 a + 3\right)\cdot 13 + \left(6 a + 9\right)\cdot 13^{2} + \left(2 a + 10\right)\cdot 13^{3} + \left(9 a + 11\right)\cdot 13^{4} +O\left(13^{ 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.