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Properties

Label 20.10.3933111925...9375.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,5^{10}\cdot 3319^{5}$
Root discriminant $16.97$
Ramified primes $5, 3319$
Class number $1$
Class group Trivial
Galois group $D_5\wr C_2$ (as 20T48)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -30, 43, 17, -155, 305, -42, -215, -9, -167, -79, 266, -3, -54, 61, -53, 11, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^18 + 11*x^17 - 53*x^16 + 61*x^15 - 54*x^14 - 3*x^13 + 266*x^12 - 79*x^11 - 167*x^10 - 9*x^9 - 215*x^8 - 42*x^7 + 305*x^6 - 155*x^5 + 17*x^4 + 43*x^3 - 30*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^20 - x^19 - x^18 + 11*x^17 - 53*x^16 + 61*x^15 - 54*x^14 - 3*x^13 + 266*x^12 - 79*x^11 - 167*x^10 - 9*x^9 - 215*x^8 - 42*x^7 + 305*x^6 - 155*x^5 + 17*x^4 + 43*x^3 - 30*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - x^{18} + 11 x^{17} - 53 x^{16} + 61 x^{15} - 54 x^{14} - 3 x^{13} + 266 x^{12} - 79 x^{11} - 167 x^{10} - 9 x^{9} - 215 x^{8} - 42 x^{7} + 305 x^{6} - 155 x^{5} + 17 x^{4} + 43 x^{3} - 30 x^{2} + 3 x + 1 \)

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:  \(-3933111925467505849609375=-\,5^{10}\cdot 3319^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.97$
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:  $5, 3319$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{17} + \frac{1}{3} a^{16} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{1155299300826237665267667} a^{19} - \frac{145267702955855027283125}{1155299300826237665267667} a^{18} - \frac{218707461314085751274830}{1155299300826237665267667} a^{17} - \frac{537433754013259616458840}{1155299300826237665267667} a^{16} + \frac{131345783128784218001062}{385099766942079221755889} a^{15} + \frac{376595731875375459005080}{1155299300826237665267667} a^{14} + \frac{426239324821869386129393}{1155299300826237665267667} a^{13} + \frac{394305135790251340352882}{1155299300826237665267667} a^{12} - \frac{179399813929925802774344}{385099766942079221755889} a^{11} + \frac{532137477656134428751402}{1155299300826237665267667} a^{10} + \frac{546059677154062561231445}{1155299300826237665267667} a^{9} + \frac{545362479292713856823614}{1155299300826237665267667} a^{8} + \frac{540965448336581727038471}{1155299300826237665267667} a^{7} + \frac{218838586107479040667096}{1155299300826237665267667} a^{6} - \frac{100007959764612732933800}{385099766942079221755889} a^{5} - \frac{75761802561332109809978}{1155299300826237665267667} a^{4} - \frac{78293238673742600523008}{1155299300826237665267667} a^{3} + \frac{507861157526331911105659}{1155299300826237665267667} a^{2} + \frac{351978446112085768355017}{1155299300826237665267667} a - \frac{211466389013873540978065}{1155299300826237665267667}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 53457.8405658 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2$ (as 20T48):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.82975.1, 10.6.34424253125.1 x5

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3319Data not computed