Properties

Label 2.103.5t2.1c2
Dimension 2
Group $D_{5}$
Conductor $ 103 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{5}$
Conductor:$103 $
Artin number field: Splitting field of $f= x^{5} - 2 x^{4} + 3 x^{3} - 3 x^{2} + x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $D_{5}$
Parity: Odd
Determinant: 1.103.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 7 a + 1 + \left(5 a + 2\right)\cdot 11 + \left(7 a + 1\right)\cdot 11^{2} + \left(4 a + 3\right)\cdot 11^{3} + \left(8 a + 2\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 8 a + \left(5 a + 7\right)\cdot 11 + \left(10 a + 5\right)\cdot 11^{2} + \left(3 a + 2\right)\cdot 11^{3} + \left(4 a + 2\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 3 a + 10 + \left(5 a + 10\right)\cdot 11 + 8\cdot 11^{2} + \left(7 a + 7\right)\cdot 11^{3} + \left(6 a + 4\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 4 a + 7 + \left(5 a + 6\right)\cdot 11 + \left(3 a + 3\right)\cdot 11^{2} + \left(6 a + 3\right)\cdot 11^{3} + \left(2 a + 9\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 6 + 6\cdot 11 + 2\cdot 11^{2} + 5\cdot 11^{3} + 3\cdot 11^{4} +O\left(11^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

Cycle notation
$(1,2)(3,5)$
$(1,3)(4,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$2$
$5$$2$$(1,2)(3,5)$$0$
$2$$5$$(1,5,4,3,2)$$-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$
$2$$5$$(1,4,2,5,3)$$\zeta_{5}^{3} + \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.