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Properties

Label 20.8.11392857233...0000.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 5^{17}\cdot 11^{4}\cdot 9931^{4}$
Root discriminant $79.95$
Ramified primes $2, 5, 11, 9931$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1010

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![120125, 0, 377250, 0, 56325, 0, -477300, 0, -48270, 0, 73620, 0, -690, 0, -2465, 0, 0, 0, 25, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 25*x^18 - 2465*x^14 - 690*x^12 + 73620*x^10 - 48270*x^8 - 477300*x^6 + 56325*x^4 + 377250*x^2 + 120125)
 
gp: K = bnfinit(x^20 + 25*x^18 - 2465*x^14 - 690*x^12 + 73620*x^10 - 48270*x^8 - 477300*x^6 + 56325*x^4 + 377250*x^2 + 120125, 1)
 

Normalized defining polynomial

\( x^{20} + 25 x^{18} - 2465 x^{14} - 690 x^{12} + 73620 x^{10} - 48270 x^{8} - 477300 x^{6} + 56325 x^{4} + 377250 x^{2} + 120125 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(113928572339566846848800000000000000000=2^{20}\cdot 5^{17}\cdot 11^{4}\cdot 9931^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.95$
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:  $2, 5, 11, 9931$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12}$, $\frac{1}{5} a^{13}$, $\frac{1}{5} a^{14}$, $\frac{1}{5} a^{15}$, $\frac{1}{125} a^{16} + \frac{2}{25} a^{14} + \frac{1}{25} a^{12} + \frac{2}{25} a^{10} + \frac{12}{25} a^{8} + \frac{4}{25} a^{6} - \frac{4}{25} a^{4} - \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{125} a^{17} + \frac{2}{25} a^{15} + \frac{1}{25} a^{13} + \frac{2}{25} a^{11} + \frac{12}{25} a^{9} + \frac{4}{25} a^{7} - \frac{4}{25} a^{5} - \frac{1}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{103303072195362268372375} a^{18} + \frac{19852024262637554756}{20660614439072453674475} a^{16} - \frac{1302817778441542984559}{20660614439072453674475} a^{14} - \frac{1527450642592600978568}{20660614439072453674475} a^{12} + \frac{590962048441307465706}{1215330261121909039675} a^{10} + \frac{369056248061830967994}{20660614439072453674475} a^{8} + \frac{7804063415865193214976}{20660614439072453674475} a^{6} + \frac{1270771411257877465518}{4132122887814490734895} a^{4} - \frac{643837112893630370671}{4132122887814490734895} a^{2} - \frac{64832185056205175795}{826424577562898146979}$, $\frac{1}{3202395238056230319543625} a^{19} + \frac{19852024262637554756}{640479047611246063908725} a^{17} - \frac{708497813468193440614}{20660614439072453674475} a^{15} + \frac{6736795133036380491222}{640479047611246063908725} a^{13} - \frac{7916349779412055812019}{37675238094779180229925} a^{11} - \frac{20291558191010622706481}{640479047611246063908725} a^{9} - \frac{95499008779497075157399}{640479047611246063908725} a^{7} - \frac{23521965915629066943852}{128095809522249212781745} a^{5} + \frac{3488285774920860364224}{128095809522249212781745} a^{3} - \frac{8329077960685186645585}{25619161904449842556349} a$

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:  $13$
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:  \( 445021430517 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1010:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 180 conjugacy class representatives for t20n1010 are not computed
Character table for t20n1010 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.932312193828125.1

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

Sibling fields

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 R $20$ R $20$ R $20$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ $20$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
9931Data not computed