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Properties

Label 20.0.11508250083...0000.7
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{10}\cdot 41^{18}$
Root discriminant $252.96$
Ramified primes $2, 5, 41$
Class number $1529936320$ (GRH)
Class group $[2, 2, 382484080]$ (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![1072100788929, 0, 224935514622, 0, 10988455869, 0, 733055832, 0, 301446370, 0, -30018588, 0, 3152130, 0, -156872, 0, 6045, 0, -114, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 114*x^18 + 6045*x^16 - 156872*x^14 + 3152130*x^12 - 30018588*x^10 + 301446370*x^8 + 733055832*x^6 + 10988455869*x^4 + 224935514622*x^2 + 1072100788929)
 
gp: K = bnfinit(x^20 - 114*x^18 + 6045*x^16 - 156872*x^14 + 3152130*x^12 - 30018588*x^10 + 301446370*x^8 + 733055832*x^6 + 10988455869*x^4 + 224935514622*x^2 + 1072100788929, 1)
 

Normalized defining polynomial

\( x^{20} - 114 x^{18} + 6045 x^{16} - 156872 x^{14} + 3152130 x^{12} - 30018588 x^{10} + 301446370 x^{8} + 733055832 x^{6} + 10988455869 x^{4} + 224935514622 x^{2} + 1072100788929 \)

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:  \(1150825008314476629127654256038421463040000000000=2^{40}\cdot 5^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $252.96$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1640=2^{3}\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{1640}(1,·)$, $\chi_{1640}(901,·)$, $\chi_{1640}(961,·)$, $\chi_{1640}(201,·)$, $\chi_{1640}(1419,·)$, $\chi_{1640}(1281,·)$, $\chi_{1640}(1261,·)$, $\chi_{1640}(1041,·)$, $\chi_{1640}(1179,·)$, $\chi_{1640}(1559,·)$, $\chi_{1640}(1499,·)$, $\chi_{1640}(1501,·)$, $\chi_{1640}(879,·)$, $\chi_{1640}(619,·)$, $\chi_{1640}(1581,·)$, $\chi_{1640}(1199,·)$, $\chi_{1640}(819,·)$, $\chi_{1640}(119,·)$, $\chi_{1640}(701,·)$, $\chi_{1640}(959,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{3} - \frac{1}{6} a$, $\frac{1}{12} a^{4} + \frac{1}{6} a^{2} - \frac{1}{4}$, $\frac{1}{12} a^{5} - \frac{1}{12} a$, $\frac{1}{72} a^{6} + \frac{1}{72} a^{4} + \frac{7}{72} a^{2} - \frac{1}{8}$, $\frac{1}{72} a^{7} + \frac{1}{72} a^{5} - \frac{5}{72} a^{3} + \frac{1}{24} a$, $\frac{1}{144} a^{8} - \frac{1}{24} a^{4} + \frac{2}{9} a^{2} - \frac{3}{16}$, $\frac{1}{1296} a^{9} + \frac{1}{216} a^{7} + \frac{1}{108} a^{5} + \frac{13}{648} a^{3} - \frac{5}{144} a$, $\frac{1}{2592} a^{10} - \frac{1}{864} a^{8} - \frac{1}{432} a^{6} - \frac{23}{1296} a^{4} - \frac{1}{96} a^{2} + \frac{1}{32}$, $\frac{1}{5184} a^{11} - \frac{1}{5184} a^{10} - \frac{1}{5184} a^{9} - \frac{5}{1728} a^{8} - \frac{5}{864} a^{7} + \frac{1}{864} a^{6} + \frac{79}{2592} a^{5} + \frac{77}{2592} a^{4} - \frac{227}{5184} a^{3} - \frac{61}{576} a^{2} - \frac{277}{576} a - \frac{27}{64}$, $\frac{1}{15552} a^{12} + \frac{1}{7776} a^{10} + \frac{5}{5184} a^{8} - \frac{19}{3888} a^{6} - \frac{41}{15552} a^{4} - \frac{35}{864} a^{2} + \frac{3}{64}$, $\frac{1}{31104} a^{13} - \frac{1}{31104} a^{12} + \frac{1}{15552} a^{11} - \frac{1}{15552} a^{10} - \frac{1}{3456} a^{9} - \frac{5}{10368} a^{8} + \frac{53}{7776} a^{7} + \frac{19}{7776} a^{6} - \frac{1193}{31104} a^{5} - \frac{1255}{31104} a^{4} - \frac{137}{5184} a^{3} - \frac{109}{1728} a^{2} + \frac{67}{1152} a + \frac{13}{128}$, $\frac{1}{31104} a^{14} - \frac{1}{31104} a^{12} - \frac{1}{10368} a^{10} - \frac{85}{31104} a^{8} - \frac{173}{31104} a^{6} - \frac{43}{3456} a^{4} - \frac{1}{384} a^{2} + \frac{3}{128}$, $\frac{1}{186624} a^{15} - \frac{1}{62208} a^{14} + \frac{1}{186624} a^{13} - \frac{1}{62208} a^{12} + \frac{13}{186624} a^{11} + \frac{11}{62208} a^{10} + \frac{53}{186624} a^{9} + \frac{19}{62208} a^{8} - \frac{397}{186624} a^{7} + \frac{253}{62208} a^{6} - \frac{157}{186624} a^{5} - \frac{83}{62208} a^{4} + \frac{143}{2304} a^{3} - \frac{1615}{6912} a^{2} - \frac{137}{2304} a + \frac{59}{256}$, $\frac{1}{13623552} a^{16} + \frac{107}{6811776} a^{14} + \frac{95}{6811776} a^{12} - \frac{713}{6811776} a^{10} + \frac{607}{425736} a^{8} - \frac{12911}{6811776} a^{6} - \frac{65557}{2270592} a^{4} - \frac{11465}{252288} a^{2} - \frac{5513}{18688}$, $\frac{1}{27247104} a^{17} - \frac{1}{27247104} a^{16} + \frac{17}{6811776} a^{15} + \frac{7}{851472} a^{14} + \frac{11}{6811776} a^{13} + \frac{31}{3405888} a^{12} + \frac{161}{2270592} a^{11} + \frac{233}{3405888} a^{10} + \frac{3215}{13623552} a^{9} + \frac{25547}{13623552} a^{8} - \frac{31385}{6811776} a^{7} - \frac{7283}{1702944} a^{6} + \frac{115007}{6811776} a^{5} - \frac{293}{21024} a^{4} - \frac{34243}{756864} a^{3} + \frac{4721}{42048} a^{2} - \frac{51223}{336384} a + \frac{3323}{37376}$, $\frac{1}{424964984153488589006959104} a^{18} - \frac{734207043769087607}{141654994717829529668986368} a^{16} + \frac{1293180401102293679}{129247257954224023420608} a^{14} + \frac{1370459886307254660161}{53120623019186073625869888} a^{12} + \frac{3874515833276453587747}{70827497358914764834493184} a^{10} + \frac{31380385901424368138921}{70827497358914764834493184} a^{8} + \frac{81310648886440190307623}{13280155754796518406467472} a^{6} - \frac{23523480524085521905079}{2951145723288115201437216} a^{4} - \frac{355829585375788168987115}{5246481285845538135888384} a^{2} + \frac{8168131656841230990071}{64771373899327631307264}$, $\frac{1}{10864654784868089266551916452864} a^{19} - \frac{1}{849929968306977178013918208} a^{18} - \frac{39522779186390568092141}{3621551594956029755517305484288} a^{17} - \frac{9663594845035647727}{283309989435659059337972736} a^{16} + \frac{4215176104869083655577}{6608670793715382765542528256} a^{15} - \frac{6646816284329064295}{516989031816896093682432} a^{14} - \frac{38925178858548826320596585}{2716163696217022316637979113216} a^{13} - \frac{4222606541769184105417}{212482492076744294503479552} a^{12} + \frac{52703105817871760570576617}{1810775797478014877758652742144} a^{11} - \frac{23786306450337521752357}{141654994717829529668986368} a^{10} - \frac{600989185348828943314108183}{1810775797478014877758652742144} a^{9} - \frac{50387567754159424329473}{141654994717829529668986368} a^{8} + \frac{12429241659240198479545994971}{2716163696217022316637979113216} a^{7} + \frac{1272385511106082585223303}{212482492076744294503479552} a^{6} + \frac{449674103194300893520019407}{301795966246335812959775457024} a^{5} + \frac{808365723946838979021779}{23609165786304921611497728} a^{4} + \frac{10901560962159165987443269441}{134131540553927027982122425344} a^{3} - \frac{1464264044051925532288249}{10492962571691076271776768} a^{2} + \frac{529702459727423085816291641}{1655944945110210222001511424} a - \frac{31566651840614820403349}{129542747798655262614528}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{382484080}$, which has order $1529936320$ (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:  \( 6411717617.202166 \) (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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-410}) \), \(\Q(\sqrt{82}) \), \(\Q(\sqrt{-5}, \sqrt{82})\), 5.5.2825761.1, 10.0.25551760733187200000.2, 10.0.33523910081941606400000.1, 10.10.10727651226221314048.1

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 ${\href{/LocalNumberField/3.1.0.1}{1} }^{20}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
5Data not computed
$41$41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$