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Properties

Label 20.0.81818105394...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 401^{9}$
Root discriminant $49.62$
Ramified primes $5, 401$
Class number $404$ (GRH)
Class group $[404]$ (GRH)
Galois group $D_{20}$ (as 20T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15161, -52120, 85694, 23262, -87665, 88164, 94740, -203394, 158294, -45010, -13069, 13317, -797, -2988, 1875, -447, 2, 29, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 5*x^18 + 29*x^17 + 2*x^16 - 447*x^15 + 1875*x^14 - 2988*x^13 - 797*x^12 + 13317*x^11 - 13069*x^10 - 45010*x^9 + 158294*x^8 - 203394*x^7 + 94740*x^6 + 88164*x^5 - 87665*x^4 + 23262*x^3 + 85694*x^2 - 52120*x + 15161)
 
gp: K = bnfinit(x^20 - 3*x^19 + 5*x^18 + 29*x^17 + 2*x^16 - 447*x^15 + 1875*x^14 - 2988*x^13 - 797*x^12 + 13317*x^11 - 13069*x^10 - 45010*x^9 + 158294*x^8 - 203394*x^7 + 94740*x^6 + 88164*x^5 - 87665*x^4 + 23262*x^3 + 85694*x^2 - 52120*x + 15161, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 5 x^{18} + 29 x^{17} + 2 x^{16} - 447 x^{15} + 1875 x^{14} - 2988 x^{13} - 797 x^{12} + 13317 x^{11} - 13069 x^{10} - 45010 x^{9} + 158294 x^{8} - 203394 x^{7} + 94740 x^{6} + 88164 x^{5} - 87665 x^{4} + 23262 x^{3} + 85694 x^{2} - 52120 x + 15161 \)

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:  \(8181810539473601738391143798828125=5^{15}\cdot 401^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.62$
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:  $5, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{1989} a^{18} + \frac{5}{1989} a^{17} - \frac{16}{1989} a^{16} - \frac{61}{663} a^{15} + \frac{59}{663} a^{14} + \frac{22}{221} a^{13} - \frac{5}{221} a^{12} - \frac{59}{663} a^{11} - \frac{131}{1989} a^{10} - \frac{139}{1989} a^{9} - \frac{223}{1989} a^{8} - \frac{101}{663} a^{7} + \frac{8}{39} a^{6} - \frac{72}{221} a^{5} - \frac{2}{13} a^{4} + \frac{200}{663} a^{3} + \frac{112}{1989} a^{2} + \frac{818}{1989} a - \frac{121}{1989}$, $\frac{1}{2833377915191259193499690583197615177943482141139} a^{19} + \frac{263664347720278055054755435307884314036624770}{2833377915191259193499690583197615177943482141139} a^{18} + \frac{14394736345690425907900525510671472358339948371}{944459305063753064499896861065871725981160713713} a^{17} - \frac{112779451141376641998211810430473708248690957904}{2833377915191259193499690583197615177943482141139} a^{16} - \frac{35094214895099938105661218769412353948189694739}{314819768354584354833298953688623908660386904571} a^{15} - \frac{11015844201677377225576278868593836337711767264}{944459305063753064499896861065871725981160713713} a^{14} + \frac{6314991616764765390114870009857284816224649736}{104939922784861451611099651229541302886795634857} a^{13} + \frac{30376395179744959438725196785486260126164876456}{314819768354584354833298953688623908660386904571} a^{12} - \frac{422424462108414251667625731556001030310406901222}{2833377915191259193499690583197615177943482141139} a^{11} + \frac{81564111121073115635047059998758148611930852}{2527544973408794998661632991255678124838075059} a^{10} - \frac{12020854916615115223826868398092170325202241650}{104939922784861451611099651229541302886795634857} a^{9} + \frac{244862108464567103129562700781317028312788680752}{2833377915191259193499690583197615177943482141139} a^{8} - \frac{4739072728535594851692231004725309503153764706}{18518809903210844401958761981683759332963935563} a^{7} - \frac{221358375031494665309079800380832586828320159905}{944459305063753064499896861065871725981160713713} a^{6} - \frac{2016876881462778382899476511298434467475848131}{18518809903210844401958761981683759332963935563} a^{5} - \frac{15850376102749145972379334265587035298766162816}{34979974261620483870366550409847100962265211619} a^{4} + \frac{137731132647884093679986976533290682229383194501}{2833377915191259193499690583197615177943482141139} a^{3} - \frac{1065077548304064360800017890139732230699044941006}{2833377915191259193499690583197615177943482141139} a^{2} + \frac{287777742153456195101053498421732620363490862347}{944459305063753064499896861065871725981160713713} a + \frac{629916454594035841096527707607993477494841892}{12820714548376738432125296756550294922821186159}$

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

Class group and class number

$C_{404}$, which has order $404$ (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:  \( 2526424.45141 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{20}$ (as 20T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 13 conjugacy class representatives for $D_{20}$
Character table for $D_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.50125.1, 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

Galois closure: data not computed
Degree 20 sibling: 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 $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
401Data not computed