Properties

Label 2.401.5t2.1c2
Dimension 2
Group $D_{5}$
Conductor $ 401 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{5}$
Conductor:$401 $
Artin number field: Splitting field of $f= x^{5} - x^{4} - 5 x^{3} + 4 x^{2} + 3 x - 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $D_{5}$
Parity: Even
Determinant: 1.401.2t1.1c1

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: \[ \begin{aligned} r_{ 1 } &= 5 a + 12 + \left(6 a + 6\right)\cdot 13 + \left(4 a + 11\right)\cdot 13^{2} + \left(11 a + 5\right)\cdot 13^{3} + \left(a + 3\right)\cdot 13^{4} +O\left(13^{ 5 }\right) \\ r_{ 2 } &= 11 a + 4 + \left(3 a + 3\right)\cdot 13 + \left(12 a + 5\right)\cdot 13^{2} + 5\cdot 13^{3} + \left(6 a + 11\right)\cdot 13^{4} +O\left(13^{ 5 }\right) \\ r_{ 3 } &= 2 a + 2 + \left(9 a + 9\right)\cdot 13 + \left(12 a + 7\right)\cdot 13^{3} + \left(6 a + 3\right)\cdot 13^{4} +O\left(13^{ 5 }\right) \\ r_{ 4 } &= 5 + 11\cdot 13 + 11\cdot 13^{2} + 7\cdot 13^{3} +O\left(13^{ 5 }\right) \\ r_{ 5 } &= 8 a + 4 + \left(6 a + 8\right)\cdot 13 + \left(8 a + 9\right)\cdot 13^{2} + \left(a + 12\right)\cdot 13^{3} + \left(11 a + 6\right)\cdot 13^{4} +O\left(13^{ 5 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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