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Properties

Label 20.16.1919766966...8125.1
Degree $20$
Signature $[16, 2]$
Discriminant $5^{15}\cdot 97^{2}\cdot 401^{8}$
Root discriminant $58.10$
Ramified primes $5, 97, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![171, 1665, 1645, -13680, -18255, 46314, 51670, -83990, -53965, 78985, 14616, -30280, 3710, 1610, -890, 974, -285, -30, 20, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 20*x^18 - 30*x^17 - 285*x^16 + 974*x^15 - 890*x^14 + 1610*x^13 + 3710*x^12 - 30280*x^11 + 14616*x^10 + 78985*x^9 - 53965*x^8 - 83990*x^7 + 51670*x^6 + 46314*x^5 - 18255*x^4 - 13680*x^3 + 1645*x^2 + 1665*x + 171)
 
gp: K = bnfinit(x^20 - 5*x^19 + 20*x^18 - 30*x^17 - 285*x^16 + 974*x^15 - 890*x^14 + 1610*x^13 + 3710*x^12 - 30280*x^11 + 14616*x^10 + 78985*x^9 - 53965*x^8 - 83990*x^7 + 51670*x^6 + 46314*x^5 - 18255*x^4 - 13680*x^3 + 1645*x^2 + 1665*x + 171, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 20 x^{18} - 30 x^{17} - 285 x^{16} + 974 x^{15} - 890 x^{14} + 1610 x^{13} + 3710 x^{12} - 30280 x^{11} + 14616 x^{10} + 78985 x^{9} - 53965 x^{8} - 83990 x^{7} + 51670 x^{6} + 46314 x^{5} - 18255 x^{4} - 13680 x^{3} + 1645 x^{2} + 1665 x + 171 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(191976696673085084180853546142578125=5^{15}\cdot 97^{2}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.10$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 97, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{5} a^{7} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{8} - \frac{2}{25} a^{7} - \frac{2}{25} a^{6} + \frac{1}{25} a^{5} + \frac{1}{25} a^{3} - \frac{2}{25} a^{2} - \frac{2}{25} a + \frac{1}{25}$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{7} + \frac{2}{25} a^{6} + \frac{2}{25} a^{5} + \frac{1}{25} a^{4} - \frac{1}{25} a^{2} + \frac{2}{25} a + \frac{2}{25}$, $\frac{1}{25} a^{10} + \frac{2}{25} a^{5} + \frac{1}{25}$, $\frac{1}{25} a^{11} + \frac{2}{25} a^{6} + \frac{1}{25} a$, $\frac{1}{125} a^{12} + \frac{2}{125} a^{11} + \frac{1}{125} a^{10} + \frac{2}{125} a^{7} + \frac{4}{125} a^{6} + \frac{2}{125} a^{5} + \frac{1}{125} a^{2} + \frac{2}{125} a + \frac{1}{125}$, $\frac{1}{125} a^{13} + \frac{2}{125} a^{11} - \frac{2}{125} a^{10} + \frac{2}{125} a^{8} + \frac{4}{125} a^{6} - \frac{4}{125} a^{5} + \frac{1}{125} a^{3} + \frac{2}{125} a - \frac{2}{125}$, $\frac{1}{125} a^{14} - \frac{1}{125} a^{11} - \frac{2}{125} a^{10} + \frac{2}{125} a^{9} - \frac{2}{125} a^{6} - \frac{4}{125} a^{5} + \frac{1}{125} a^{4} - \frac{1}{125} a - \frac{2}{125}$, $\frac{1}{125} a^{15} - \frac{2}{125} a^{10} - \frac{7}{125} a^{5} - \frac{4}{125}$, $\frac{1}{625} a^{16} + \frac{1}{625} a^{15} + \frac{3}{625} a^{11} + \frac{3}{625} a^{10} + \frac{3}{625} a^{6} + \frac{3}{625} a^{5} + \frac{1}{625} a + \frac{1}{625}$, $\frac{1}{1875} a^{17} - \frac{1}{1875} a^{16} - \frac{7}{1875} a^{15} - \frac{1}{375} a^{14} + \frac{1}{625} a^{12} - \frac{23}{1875} a^{11} - \frac{12}{625} a^{10} + \frac{1}{125} a^{9} - \frac{1}{75} a^{8} - \frac{97}{1875} a^{7} + \frac{182}{1875} a^{6} + \frac{33}{625} a^{5} - \frac{7}{125} a^{4} + \frac{4}{75} a^{3} - \frac{224}{1875} a^{2} + \frac{329}{1875} a + \frac{1}{625}$, $\frac{1}{1875} a^{18} + \frac{1}{1875} a^{16} - \frac{1}{625} a^{15} - \frac{1}{375} a^{14} + \frac{1}{625} a^{13} - \frac{1}{375} a^{12} - \frac{2}{1875} a^{11} + \frac{7}{625} a^{10} - \frac{2}{375} a^{9} + \frac{28}{1875} a^{8} - \frac{37}{375} a^{7} + \frac{68}{1875} a^{6} - \frac{58}{625} a^{5} - \frac{1}{375} a^{4} + \frac{26}{1875} a^{3} - \frac{12}{125} a^{2} + \frac{71}{1875} a - \frac{66}{625}$, $\frac{1}{8610578649375} a^{19} + \frac{1739845057}{8610578649375} a^{18} - \frac{65163747}{574038576625} a^{17} + \frac{875374973}{8610578649375} a^{16} - \frac{4116849166}{8610578649375} a^{15} - \frac{1754764584}{2870192883125} a^{14} + \frac{5502122821}{8610578649375} a^{13} - \frac{98084378}{68884629195} a^{12} + \frac{25596914393}{2870192883125} a^{11} + \frac{81376782677}{8610578649375} a^{10} - \frac{5070765419}{2870192883125} a^{9} - \frac{40976127643}{2870192883125} a^{8} - \frac{36122680042}{1722115729875} a^{7} + \frac{129587569288}{2870192883125} a^{6} - \frac{57897843173}{8610578649375} a^{5} + \frac{23791295582}{2870192883125} a^{4} - \frac{2051673443693}{8610578649375} a^{3} + \frac{705076182392}{1722115729875} a^{2} + \frac{162902893886}{2870192883125} a + \frac{662388145578}{2870192883125}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $17$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 97728351245.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.160801.1, 10.10.80803005003125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

In the table, R denotes a ramified prime. 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
5Data not computed
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed