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: SageMath / Pari/GP / Magma

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)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17, -21, -93, -40, -1, 1]);
 

Normalized defining polynomial

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

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

Invariants

Degree:  $5$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[5, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(104060401=101^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $40.13$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $101$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $5$
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 \(a\))

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

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

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

Galois group

$C_5$ (as 5T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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.

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]])
 
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];
 

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.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.101.5t1.a.a$1$ $ 101 $ $x^{5} - x^{4} - 40 x^{3} - 93 x^{2} - 21 x + 17$ $C_5$ (as 5T1) $0$ $1$
* 1.101.5t1.a.b$1$ $ 101 $ $x^{5} - x^{4} - 40 x^{3} - 93 x^{2} - 21 x + 17$ $C_5$ (as 5T1) $0$ $1$
* 1.101.5t1.a.c$1$ $ 101 $ $x^{5} - x^{4} - 40 x^{3} - 93 x^{2} - 21 x + 17$ $C_5$ (as 5T1) $0$ $1$
* 1.101.5t1.a.d$1$ $ 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 *.