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Properties

Label 20.0.49685095734...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 13^{10}$
Root discriminant $192.67$
Ramified primes $2, 3, 5, 13$
Class number $18622568$ (GRH)
Class group $[2, 2, 4655642]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1584355549, 119407110, 772864335, 13888310, 209760805, -1871502, 35937265, 1269200, 5611600, -37920, 203160, 74300, 91710, 2360, -7840, 636, 1035, -20, -45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 45*x^18 - 20*x^17 + 1035*x^16 + 636*x^15 - 7840*x^14 + 2360*x^13 + 91710*x^12 + 74300*x^11 + 203160*x^10 - 37920*x^9 + 5611600*x^8 + 1269200*x^7 + 35937265*x^6 - 1871502*x^5 + 209760805*x^4 + 13888310*x^3 + 772864335*x^2 + 119407110*x + 1584355549)
 
gp: K = bnfinit(x^20 - 45*x^18 - 20*x^17 + 1035*x^16 + 636*x^15 - 7840*x^14 + 2360*x^13 + 91710*x^12 + 74300*x^11 + 203160*x^10 - 37920*x^9 + 5611600*x^8 + 1269200*x^7 + 35937265*x^6 - 1871502*x^5 + 209760805*x^4 + 13888310*x^3 + 772864335*x^2 + 119407110*x + 1584355549, 1)
 

Normalized defining polynomial

\( x^{20} - 45 x^{18} - 20 x^{17} + 1035 x^{16} + 636 x^{15} - 7840 x^{14} + 2360 x^{13} + 91710 x^{12} + 74300 x^{11} + 203160 x^{10} - 37920 x^{9} + 5611600 x^{8} + 1269200 x^{7} + 35937265 x^{6} - 1871502 x^{5} + 209760805 x^{4} + 13888310 x^{3} + 772864335 x^{2} + 119407110 x + 1584355549 \)

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:  \(4968509573481201782226562500000000000000000000=2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $192.67$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3900=2^{2}\cdot 3\cdot 5^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{3900}(1,·)$, $\chi_{3900}(389,·)$, $\chi_{3900}(571,·)$, $\chi_{3900}(1351,·)$, $\chi_{3900}(3719,·)$, $\chi_{3900}(781,·)$, $\chi_{3900}(1169,·)$, $\chi_{3900}(2131,·)$, $\chi_{3900}(599,·)$, $\chi_{3900}(1561,·)$, $\chi_{3900}(1949,·)$, $\chi_{3900}(2911,·)$, $\chi_{3900}(1379,·)$, $\chi_{3900}(2341,·)$, $\chi_{3900}(2729,·)$, $\chi_{3900}(3691,·)$, $\chi_{3900}(2159,·)$, $\chi_{3900}(3121,·)$, $\chi_{3900}(3509,·)$, $\chi_{3900}(2939,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{175} a^{10} - \frac{2}{35} a^{8} - \frac{2}{35} a^{7} - \frac{2}{5} a^{6} - \frac{1}{25} a^{5} + \frac{11}{35} a^{4} + \frac{1}{5} a^{3} + \frac{2}{35} a^{2} + \frac{2}{35} a - \frac{1}{25}$, $\frac{1}{175} a^{11} - \frac{2}{35} a^{9} - \frac{2}{35} a^{8} + \frac{1}{35} a^{7} - \frac{1}{25} a^{6} + \frac{11}{35} a^{5} + \frac{1}{5} a^{4} + \frac{2}{35} a^{3} + \frac{2}{35} a^{2} - \frac{82}{175} a$, $\frac{1}{1225} a^{12} + \frac{1}{1225} a^{11} + \frac{1}{1225} a^{10} - \frac{4}{245} a^{9} + \frac{17}{245} a^{8} - \frac{62}{1225} a^{7} - \frac{372}{1225} a^{6} + \frac{538}{1225} a^{5} - \frac{2}{49} a^{4} - \frac{94}{245} a^{3} + \frac{13}{1225} a^{2} + \frac{503}{1225} a + \frac{39}{175}$, $\frac{1}{1225} a^{13} - \frac{2}{35} a^{9} - \frac{1}{175} a^{8} + \frac{1}{245} a^{7} - \frac{16}{35} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{24}{175} a^{3} + \frac{2}{7} a^{2} - \frac{109}{245} a - \frac{12}{35}$, $\frac{1}{8575} a^{14} - \frac{1}{8575} a^{13} - \frac{2}{8575} a^{12} + \frac{12}{8575} a^{11} - \frac{16}{8575} a^{10} - \frac{387}{8575} a^{9} + \frac{542}{8575} a^{8} - \frac{83}{1225} a^{7} - \frac{3449}{8575} a^{6} + \frac{3222}{8575} a^{5} - \frac{572}{8575} a^{4} - \frac{3778}{8575} a^{3} + \frac{3279}{8575} a^{2} - \frac{594}{8575} a + \frac{276}{1225}$, $\frac{1}{8575} a^{15} - \frac{3}{8575} a^{13} + \frac{3}{8575} a^{12} - \frac{11}{8575} a^{11} - \frac{18}{8575} a^{10} + \frac{59}{1715} a^{9} + \frac{346}{8575} a^{8} - \frac{166}{8575} a^{7} + \frac{662}{8575} a^{6} - \frac{772}{1715} a^{5} + \frac{82}{1715} a^{4} - \frac{639}{8575} a^{3} + \frac{1614}{8575} a^{2} + \frac{2962}{8575} a - \frac{389}{1225}$, $\frac{1}{8575} a^{16} - \frac{3}{8575} a^{12} - \frac{17}{8575} a^{11} + \frac{16}{8575} a^{10} - \frac{121}{1715} a^{9} - \frac{107}{1715} a^{8} + \frac{256}{8575} a^{7} - \frac{1922}{8575} a^{6} - \frac{522}{8575} a^{5} - \frac{219}{1715} a^{4} + \frac{72}{1715} a^{3} - \frac{984}{8575} a^{2} + \frac{821}{1715} a + \frac{394}{1225}$, $\frac{1}{8575} a^{17} - \frac{3}{8575} a^{13} - \frac{3}{8575} a^{12} - \frac{19}{8575} a^{11} - \frac{3}{8575} a^{10} - \frac{13}{343} a^{9} - \frac{269}{8575} a^{8} - \frac{68}{1715} a^{7} - \frac{3672}{8575} a^{6} - \frac{374}{8575} a^{5} - \frac{803}{1715} a^{4} + \frac{3951}{8575} a^{3} - \frac{2573}{8575} a^{2} + \frac{52}{175} a - \frac{6}{175}$, $\frac{1}{20402308787822675} a^{18} - \frac{347916039343}{20402308787822675} a^{17} + \frac{823483440516}{20402308787822675} a^{16} - \frac{102215307981}{2914615541117525} a^{15} + \frac{148661584597}{20402308787822675} a^{14} + \frac{6727995921543}{20402308787822675} a^{13} + \frac{477964817927}{4080461757564535} a^{12} - \frac{33800038161672}{20402308787822675} a^{11} + \frac{39683221728739}{20402308787822675} a^{10} - \frac{186469479958939}{20402308787822675} a^{9} + \frac{457804812328923}{20402308787822675} a^{8} + \frac{855188327173743}{20402308787822675} a^{7} + \frac{7084172097360298}{20402308787822675} a^{6} + \frac{46064818040044}{202003057305175} a^{5} - \frac{4498052800503839}{20402308787822675} a^{4} + \frac{776661118538101}{4080461757564535} a^{3} - \frac{2794617918073468}{20402308787822675} a^{2} + \frac{1997373147195967}{4080461757564535} a + \frac{46282465170911}{116584621644701}$, $\frac{1}{66123682595790746221250796258813017148575} a^{19} + \frac{168039457126470156059159}{66123682595790746221250796258813017148575} a^{18} + \frac{473118850421992455073427996555703707}{13224736519158149244250159251762603429715} a^{17} + \frac{1647591453080124863195033193901565064}{66123682595790746221250796258813017148575} a^{16} - \frac{629111386201339487632726269976136281}{66123682595790746221250796258813017148575} a^{15} + \frac{593193464824905146859829987485343901}{66123682595790746221250796258813017148575} a^{14} - \frac{18985952760047198282684626363928264066}{66123682595790746221250796258813017148575} a^{13} - \frac{14977050387818327852594488739026175409}{66123682595790746221250796258813017148575} a^{12} - \frac{119995620429155029732570161142227646646}{66123682595790746221250796258813017148575} a^{11} + \frac{9213147378619488789408148328002601061}{13224736519158149244250159251762603429715} a^{10} - \frac{1102809314465697764832745333694995056873}{66123682595790746221250796258813017148575} a^{9} - \frac{1810037948948948726774579534968438917517}{66123682595790746221250796258813017148575} a^{8} - \frac{291209897449762873771615442612486927997}{66123682595790746221250796258813017148575} a^{7} - \frac{30222599892518648151316644177207830027562}{66123682595790746221250796258813017148575} a^{6} - \frac{31893409124420895140247949264850627786379}{66123682595790746221250796258813017148575} a^{5} + \frac{2247200864826474155875370447648980741942}{9446240370827249460178685179830431021225} a^{4} - \frac{22204699826143152648704194758266881943494}{66123682595790746221250796258813017148575} a^{3} - \frac{25931642812244163225985129140219898659057}{66123682595790746221250796258813017148575} a^{2} + \frac{12074096767812434017662357672669426773781}{66123682595790746221250796258813017148575} a - \frac{1313181643321762350755842194990948974703}{9446240370827249460178685179830431021225}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4655642}$, which has order $18622568$ (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:  \( 180801817.57689384 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-195}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-13}, \sqrt{15})\), 5.5.390625.1, 10.0.68835601043701171875.3, 10.0.58014531250000000000.3, 10.10.189843750000000000.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
3Data not computed
5Data not computed
13Data not computed