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Properties

Label 20.4.97007082560...0625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{10}\cdot 199^{5}\cdot 1471^{3}$
Root discriminant $25.08$
Ramified primes $5, 199, 1471$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3725, 11975, -210, -18410, 14016, 16553, -27900, 201, 9295, 3434, -5079, -333, 1009, 349, -357, -22, 43, 12, -10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 10*x^18 + 12*x^17 + 43*x^16 - 22*x^15 - 357*x^14 + 349*x^13 + 1009*x^12 - 333*x^11 - 5079*x^10 + 3434*x^9 + 9295*x^8 + 201*x^7 - 27900*x^6 + 16553*x^5 + 14016*x^4 - 18410*x^3 - 210*x^2 + 11975*x + 3725)
 
gp: K = bnfinit(x^20 - x^19 - 10*x^18 + 12*x^17 + 43*x^16 - 22*x^15 - 357*x^14 + 349*x^13 + 1009*x^12 - 333*x^11 - 5079*x^10 + 3434*x^9 + 9295*x^8 + 201*x^7 - 27900*x^6 + 16553*x^5 + 14016*x^4 - 18410*x^3 - 210*x^2 + 11975*x + 3725, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 10 x^{18} + 12 x^{17} + 43 x^{16} - 22 x^{15} - 357 x^{14} + 349 x^{13} + 1009 x^{12} - 333 x^{11} - 5079 x^{10} + 3434 x^{9} + 9295 x^{8} + 201 x^{7} - 27900 x^{6} + 16553 x^{5} + 14016 x^{4} - 18410 x^{3} - 210 x^{2} + 11975 x + 3725 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9700708256022096688369140625=5^{10}\cdot 199^{5}\cdot 1471^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.08$
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:  $5, 199, 1471$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{15} - \frac{2}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{15} - \frac{2}{5} a^{13} + \frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{18} - \frac{2}{5} a^{15} - \frac{2}{5} a^{14} + \frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{4771901716743075993308637663691351168757575} a^{19} - \frac{69161595572949407553688939747440124878267}{4771901716743075993308637663691351168757575} a^{18} - \frac{56806402916332288542023286003193109267683}{4771901716743075993308637663691351168757575} a^{17} - \frac{68832747071567198327439600671575279720167}{954380343348615198661727532738270233751515} a^{16} + \frac{263616812080381382841878449764516865799664}{681700245249010856186948237670193024108225} a^{15} + \frac{64811247455985585229557763646817786356703}{954380343348615198661727532738270233751515} a^{14} + \frac{1683582331857874157480660477063955014557943}{4771901716743075993308637663691351168757575} a^{13} - \frac{631942824187342003089544671545788008458109}{4771901716743075993308637663691351168757575} a^{12} + \frac{535309274723180738481720998968305502438558}{4771901716743075993308637663691351168757575} a^{11} + \frac{14641395777733288387851781370335897977609}{4771901716743075993308637663691351168757575} a^{10} - \frac{834731501704685959721424833749875467463093}{4771901716743075993308637663691351168757575} a^{9} + \frac{2269460307376676004966948800552970255852247}{4771901716743075993308637663691351168757575} a^{8} - \frac{1281915370272561110219448251216017760272267}{4771901716743075993308637663691351168757575} a^{7} + \frac{571220516864636612970665178030559188491158}{4771901716743075993308637663691351168757575} a^{6} - \frac{318909001222485909323312819611815863627709}{681700245249010856186948237670193024108225} a^{5} + \frac{1885556432540013502873589705821889749823636}{4771901716743075993308637663691351168757575} a^{4} + \frac{276961806677611653904157225705257030272638}{954380343348615198661727532738270233751515} a^{3} + \frac{426543975259924486746126816048299151937726}{954380343348615198661727532738270233751515} a^{2} - \frac{19095650184112409085694317483867359787622}{190876068669723039732345506547654046750303} a + \frac{15651720413525659876090510109608909717327}{190876068669723039732345506547654046750303}$

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:  $11$
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:  \( 918016.132042 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1022 are not computed
Character table for t20n1022 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.2.914778125.1

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 32 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R $16{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
199Data not computed
1471Data not computed