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Properties

Label 20.0.13350881126...0625.5
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 31^{10}$
Root discriminant $57.05$
Ramified primes $3, 5, 7, 31$
Class number $2904$ (GRH)
Class group $[2, 22, 66]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2141055, -409185, -423366, -4728816, 9109972, -8122839, 5617475, -3724529, 2110378, -1054816, 542891, -235835, 99874, -38851, 13097, -4124, 1120, -244, 53, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 53*x^18 - 244*x^17 + 1120*x^16 - 4124*x^15 + 13097*x^14 - 38851*x^13 + 99874*x^12 - 235835*x^11 + 542891*x^10 - 1054816*x^9 + 2110378*x^8 - 3724529*x^7 + 5617475*x^6 - 8122839*x^5 + 9109972*x^4 - 4728816*x^3 - 423366*x^2 - 409185*x + 2141055)
 
gp: K = bnfinit(x^20 - 6*x^19 + 53*x^18 - 244*x^17 + 1120*x^16 - 4124*x^15 + 13097*x^14 - 38851*x^13 + 99874*x^12 - 235835*x^11 + 542891*x^10 - 1054816*x^9 + 2110378*x^8 - 3724529*x^7 + 5617475*x^6 - 8122839*x^5 + 9109972*x^4 - 4728816*x^3 - 423366*x^2 - 409185*x + 2141055, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 53 x^{18} - 244 x^{17} + 1120 x^{16} - 4124 x^{15} + 13097 x^{14} - 38851 x^{13} + 99874 x^{12} - 235835 x^{11} + 542891 x^{10} - 1054816 x^{9} + 2110378 x^{8} - 3724529 x^{7} + 5617475 x^{6} - 8122839 x^{5} + 9109972 x^{4} - 4728816 x^{3} - 423366 x^{2} - 409185 x + 2141055 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(133508811261782548210805849619140625=3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.05$
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:  $3, 5, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{15} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{5} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{15} a^{2}$, $\frac{1}{105} a^{11} + \frac{1}{105} a^{10} + \frac{4}{21} a^{9} + \frac{5}{21} a^{8} + \frac{13}{105} a^{7} - \frac{2}{105} a^{6} - \frac{2}{7} a^{5} + \frac{2}{7} a^{4} + \frac{2}{35} a^{3} - \frac{19}{105} a^{2} - \frac{3}{7} a$, $\frac{1}{105} a^{12} - \frac{2}{105} a^{10} + \frac{1}{21} a^{9} - \frac{4}{35} a^{8} - \frac{1}{7} a^{7} + \frac{2}{15} a^{6} - \frac{3}{7} a^{5} - \frac{8}{35} a^{4} - \frac{5}{21} a^{3} - \frac{47}{105} a^{2} + \frac{3}{7} a$, $\frac{1}{525} a^{13} + \frac{2}{525} a^{12} + \frac{1}{525} a^{11} - \frac{1}{175} a^{10} + \frac{23}{525} a^{9} - \frac{34}{525} a^{8} + \frac{66}{175} a^{7} + \frac{61}{525} a^{6} + \frac{146}{525} a^{5} + \frac{157}{525} a^{4} - \frac{254}{525} a^{3} - \frac{218}{525} a^{2} - \frac{2}{7} a - \frac{2}{5}$, $\frac{1}{525} a^{14} + \frac{2}{525} a^{12} - \frac{11}{525} a^{10} - \frac{26}{105} a^{9} - \frac{19}{525} a^{8} + \frac{1}{105} a^{7} - \frac{1}{25} a^{6} + \frac{38}{105} a^{5} + \frac{54}{175} a^{4} - \frac{31}{105} a^{3} - \frac{79}{525} a^{2} + \frac{6}{35} a - \frac{1}{5}$, $\frac{1}{7875} a^{15} + \frac{1}{1125} a^{14} + \frac{1}{1125} a^{13} - \frac{2}{2625} a^{12} + \frac{1}{875} a^{11} + \frac{29}{1125} a^{10} + \frac{887}{2625} a^{9} + \frac{3412}{7875} a^{8} - \frac{89}{875} a^{7} + \frac{397}{875} a^{6} + \frac{346}{1125} a^{5} + \frac{1709}{7875} a^{4} - \frac{3869}{7875} a^{3} + \frac{1024}{2625} a^{2} + \frac{47}{105} a - \frac{9}{25}$, $\frac{1}{7875} a^{16} + \frac{1}{2625} a^{14} + \frac{1}{1575} a^{13} + \frac{4}{875} a^{12} - \frac{1}{315} a^{11} - \frac{52}{1575} a^{10} + \frac{188}{1575} a^{9} - \frac{326}{1575} a^{8} + \frac{7}{75} a^{7} - \frac{2099}{7875} a^{6} + \frac{113}{1575} a^{5} - \frac{3097}{7875} a^{4} + \frac{49}{225} a^{3} + \frac{937}{2625} a^{2} - \frac{64}{525} a + \frac{8}{25}$, $\frac{1}{7875} a^{17} - \frac{1}{7875} a^{14} - \frac{1}{1125} a^{12} - \frac{2}{7875} a^{11} - \frac{2}{1125} a^{10} - \frac{94}{1125} a^{9} + \frac{283}{2625} a^{8} - \frac{188}{1125} a^{7} + \frac{313}{1125} a^{6} - \frac{328}{7875} a^{5} + \frac{59}{1125} a^{4} - \frac{157}{375} a^{3} + \frac{478}{2625} a^{2} - \frac{7}{25} a + \frac{7}{25}$, $\frac{1}{275625} a^{18} + \frac{2}{39375} a^{17} + \frac{2}{275625} a^{16} - \frac{8}{275625} a^{15} + \frac{76}{91875} a^{14} + \frac{179}{275625} a^{13} - \frac{166}{275625} a^{12} - \frac{166}{55125} a^{11} - \frac{1651}{55125} a^{10} - \frac{5924}{18375} a^{9} + \frac{64636}{275625} a^{8} + \frac{6541}{30625} a^{7} + \frac{42059}{91875} a^{6} + \frac{14122}{275625} a^{5} - \frac{16834}{91875} a^{4} - \frac{12729}{30625} a^{3} - \frac{1664}{30625} a^{2} + \frac{256}{875} a - \frac{49}{125}$, $\frac{1}{30511424632224854367057242378936176875} a^{19} + \frac{34649460723204884130285142560967}{30511424632224854367057242378936176875} a^{18} + \frac{1161947165332036796784649470390374}{30511424632224854367057242378936176875} a^{17} - \frac{197185096088603833534625935842332}{30511424632224854367057242378936176875} a^{16} - \frac{124805260115565815430166622896612}{10170474877408284789019080792978725625} a^{15} + \frac{562693865494541665704563767412938}{30511424632224854367057242378936176875} a^{14} - \frac{11899572304140706296447405045761959}{30511424632224854367057242378936176875} a^{13} + \frac{2602109751637136394902641945295249}{10170474877408284789019080792978725625} a^{12} + \frac{1588052139750557819204019904777238}{678031658493885652601272052865248375} a^{11} + \frac{63311102301415453820725774391880697}{6102284926444970873411448475787235375} a^{10} - \frac{1561998202815228006762576219964149299}{30511424632224854367057242378936176875} a^{9} + \frac{6727376814343535543215270597041361567}{30511424632224854367057242378936176875} a^{8} - \frac{9636461887301446131795179278713498706}{30511424632224854367057242378936176875} a^{7} - \frac{6780612892673414852705490351030431617}{30511424632224854367057242378936176875} a^{6} + \frac{300238208087150348831505193748659641}{3390158292469428263006360264326241875} a^{5} + \frac{11535379255562794712288895805760626273}{30511424632224854367057242378936176875} a^{4} - \frac{10695628134535241813153920652681435884}{30511424632224854367057242378936176875} a^{3} + \frac{1068082675494275011573994779352789283}{3390158292469428263006360264326241875} a^{2} + \frac{14603744057587228948641386863720891}{58116999299475913080109033102735575} a - \frac{2116322014548188416312861988414982}{13837380785589503114311674548270375}$

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

Class group and class number

$C_{2}\times C_{22}\times C_{66}$, which has order $2904$ (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:  $9$
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:  \( 25137930.4797 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-3255}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-31}) \), \(\Q(\sqrt{-31}, \sqrt{105})\), 5.1.10595025.1 x5, 10.0.365388575713284375.1, 10.2.11786728248815625.1 x5, 10.0.3479891197269375.1 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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$