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Properties

Label 20.10.1051522209...1875.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,5^{10}\cdot 11^{5}\cdot 401^{8}$
Root discriminant $44.78$
Ramified primes $5, 11, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_4\times D_5$ (as 20T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 6, 16, -88, 79, 582, -765, -1116, 566, 1926, -191, -1603, 577, 471, -360, -63, 38, 12, -5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 5*x^18 + 12*x^17 + 38*x^16 - 63*x^15 - 360*x^14 + 471*x^13 + 577*x^12 - 1603*x^11 - 191*x^10 + 1926*x^9 + 566*x^8 - 1116*x^7 - 765*x^6 + 582*x^5 + 79*x^4 - 88*x^3 + 16*x^2 + 6*x - 1)
 
gp: K = bnfinit(x^20 - x^19 - 5*x^18 + 12*x^17 + 38*x^16 - 63*x^15 - 360*x^14 + 471*x^13 + 577*x^12 - 1603*x^11 - 191*x^10 + 1926*x^9 + 566*x^8 - 1116*x^7 - 765*x^6 + 582*x^5 + 79*x^4 - 88*x^3 + 16*x^2 + 6*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 5 x^{18} + 12 x^{17} + 38 x^{16} - 63 x^{15} - 360 x^{14} + 471 x^{13} + 577 x^{12} - 1603 x^{11} - 191 x^{10} + 1926 x^{9} + 566 x^{8} - 1116 x^{7} - 765 x^{6} + 582 x^{5} + 79 x^{4} - 88 x^{3} + 16 x^{2} + 6 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1051522209829636336015666513671875=-\,5^{10}\cdot 11^{5}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.78$
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:  $5, 11, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{4}{9} a^{7} + \frac{4}{9} a^{6} - \frac{4}{9} a^{5} + \frac{4}{9} a^{4} - \frac{4}{9} a^{3} + \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{27} a^{18} + \frac{1}{27} a^{17} - \frac{1}{27} a^{16} + \frac{1}{9} a^{14} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{27} a^{10} + \frac{1}{27} a^{9} - \frac{1}{27} a^{8} - \frac{1}{3} a^{7} - \frac{4}{9} a^{6} - \frac{1}{3} a^{5} + \frac{2}{9} a^{4} - \frac{4}{9} a^{3} - \frac{11}{27} a^{2} - \frac{11}{27} a + \frac{11}{27}$, $\frac{1}{572267200295315044629} a^{19} - \frac{894387249127916980}{190755733431771681543} a^{18} + \frac{832886453257379983}{572267200295315044629} a^{17} - \frac{3414899406621246164}{572267200295315044629} a^{16} - \frac{188540051339220599}{7065027164139691909} a^{15} + \frac{2540492520737121142}{63585244477257227181} a^{14} - \frac{6724324718475272528}{63585244477257227181} a^{13} - \frac{27004163137010470643}{190755733431771681543} a^{12} + \frac{59075112157559350033}{572267200295315044629} a^{11} - \frac{3319179452745167311}{21195081492419075727} a^{10} - \frac{33693977656411349195}{572267200295315044629} a^{9} + \frac{12966774017517876343}{572267200295315044629} a^{8} + \frac{14671505254030603597}{63585244477257227181} a^{7} - \frac{16532103391218339833}{63585244477257227181} a^{6} - \frac{3895567192545950941}{63585244477257227181} a^{5} - \frac{13022750059706045893}{190755733431771681543} a^{4} + \frac{238967184838387726993}{572267200295315044629} a^{3} + \frac{82294846590474208903}{190755733431771681543} a^{2} + \frac{274870324360223702605}{572267200295315044629} a - \frac{155566402391904215819}{572267200295315044629}$

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:  $14$
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:  \( 3242123709.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times D_5$ (as 20T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $D_4\times D_5$
Character table for $D_4\times D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 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 $20$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ R $20$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$11$11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
401Data not computed