Properties

Label 2.127.5t2.1c1
Dimension 2
Group $D_{5}$
Conductor $ 127 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{5}$
Conductor:$127 $
Artin number field: Splitting field of $f= x^{5} - x^{4} - 2 x^{3} + x^{2} + 3 x - 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $D_{5}$
Parity: Odd
Determinant: 1.127.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $ x^{2} + 6 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 6 a + 3 + \left(5 a + 6\right)\cdot 7 + \left(2 a + 1\right)\cdot 7^{2} + \left(6 a + 4\right)\cdot 7^{3} + \left(a + 6\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 2 }$ $=$ $ a + 2 + \left(a + 6\right)\cdot 7 + \left(4 a + 5\right)\cdot 7^{2} + \left(5 a + 2\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 5 + 3\cdot 7 + 5\cdot 7^{3} + 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 5 a + \left(4 a + 6\right)\cdot 7 + 4\cdot 7^{2} + 5\cdot 7^{3} + 2 a\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 2 a + 5 + \left(2 a + 5\right)\cdot 7 + 6 a\cdot 7^{2} + \left(6 a + 5\right)\cdot 7^{3} + \left(4 a + 2\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$

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

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

Character values on conjugacy classes

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