Properties

Label 5.5.104060401.1
Degree $5$
Signature $[5, 0]$
Discriminant $101^{4}$
Root discriminant $40.13$
Ramified prime $101$
Class number $1$
Class group Trivial
Galois Group $C_5$ (as 5T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17, -21, -93, -40, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 - 40*x^3 - 93*x^2 - 21*x + 17)
gp: K = bnfinit(x^5 - x^4 - 40*x^3 - 93*x^2 - 21*x + 17, 1)

Normalized defining polynomial

\(x^{5} \) \(\mathstrut -\mathstrut x^{4} \) \(\mathstrut -\mathstrut 40 x^{3} \) \(\mathstrut -\mathstrut 93 x^{2} \) \(\mathstrut -\mathstrut 21 x \) \(\mathstrut +\mathstrut 17 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $5$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[5, 0]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(104060401=101^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $40.13$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $101$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois and abelian over $\Q$.
Conductor:  \(101\)
Dirichlet character group:    $\lbrace$$\chi_{101}(1,·)$, $\chi_{101}(87,·)$, $\chi_{101}(84,·)$, $\chi_{101}(36,·)$, $\chi_{101}(95,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{17} a^{4} + \frac{4}{17} a^{3} - \frac{3}{17} a^{2} - \frac{6}{17} a$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $4$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{5}{17} a^{4} - \frac{14}{17} a^{3} - \frac{168}{17} a^{2} - \frac{200}{17} a + 7 \),  \( \frac{4}{17} a^{4} - \frac{18}{17} a^{3} - \frac{97}{17} a^{2} - \frac{24}{17} a + 1 \),  \( a + 3 \),  \( \frac{6}{17} a^{4} - \frac{27}{17} a^{3} - \frac{154}{17} a^{2} + \frac{15}{17} a + 1 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 119.525694644 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_5$ (as 5T1):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A cyclic group of order 5
The 5 conjugacy class representatives for $C_5$
Character table for $C_5$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }$ ${\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$101$101.5.4.1$x^{5} - 101$$5$$1$$4$$C_5$$[\ ]_{5}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.101.5t1.1c1$1$ $ 101 $ $x^{5} - x^{4} - 40 x^{3} - 93 x^{2} - 21 x + 17$ $C_5$ (as 5T1) $0$ $1$
* 1.101.5t1.1c2$1$ $ 101 $ $x^{5} - x^{4} - 40 x^{3} - 93 x^{2} - 21 x + 17$ $C_5$ (as 5T1) $0$ $1$
* 1.101.5t1.1c3$1$ $ 101 $ $x^{5} - x^{4} - 40 x^{3} - 93 x^{2} - 21 x + 17$ $C_5$ (as 5T1) $0$ $1$
* 1.101.5t1.1c4$1$ $ 101 $ $x^{5} - x^{4} - 40 x^{3} - 93 x^{2} - 21 x + 17$ $C_5$ (as 5T1) $0$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.