# 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)

# Related objects

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]);

## Normalizeddefining 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

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

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 *.