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Properties

Label 20.0.38545427153...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{32}\cdot 11^{16}$
Root discriminant $601.59$
Ramified primes $2, 5, 11$
Class number $1105253605$ (GRH)
Class group $[1105253605]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3430132247057, 6538230383080, 4724072441100, 613825940640, -965387193390, -480743795532, 40389419300, 87627581140, 12788452045, -7860541700, -2040141230, 444269100, 190656040, -8873040, -5437050, 73436, 69250, -220, -420, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 420*x^18 - 220*x^17 + 69250*x^16 + 73436*x^15 - 5437050*x^14 - 8873040*x^13 + 190656040*x^12 + 444269100*x^11 - 2040141230*x^10 - 7860541700*x^9 + 12788452045*x^8 + 87627581140*x^7 + 40389419300*x^6 - 480743795532*x^5 - 965387193390*x^4 + 613825940640*x^3 + 4724072441100*x^2 + 6538230383080*x + 3430132247057)
 
gp: K = bnfinit(x^20 - 420*x^18 - 220*x^17 + 69250*x^16 + 73436*x^15 - 5437050*x^14 - 8873040*x^13 + 190656040*x^12 + 444269100*x^11 - 2040141230*x^10 - 7860541700*x^9 + 12788452045*x^8 + 87627581140*x^7 + 40389419300*x^6 - 480743795532*x^5 - 965387193390*x^4 + 613825940640*x^3 + 4724072441100*x^2 + 6538230383080*x + 3430132247057, 1)
 

Normalized defining polynomial

\( x^{20} - 420 x^{18} - 220 x^{17} + 69250 x^{16} + 73436 x^{15} - 5437050 x^{14} - 8873040 x^{13} + 190656040 x^{12} + 444269100 x^{11} - 2040141230 x^{10} - 7860541700 x^{9} + 12788452045 x^{8} + 87627581140 x^{7} + 40389419300 x^{6} - 480743795532 x^{5} - 965387193390 x^{4} + 613825940640 x^{3} + 4724072441100 x^{2} + 6538230383080 x + 3430132247057 \)

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:  \(38545427153140033834188800000000000000000000000000000000=2^{55}\cdot 5^{32}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $601.59$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4400=2^{4}\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4400}(1,·)$, $\chi_{4400}(1611,·)$, $\chi_{4400}(2891,·)$, $\chi_{4400}(2161,·)$, $\chi_{4400}(3721,·)$, $\chi_{4400}(3851,·)$, $\chi_{4400}(81,·)$, $\chi_{4400}(3811,·)$, $\chi_{4400}(691,·)$, $\chi_{4400}(1731,·)$, $\chi_{4400}(1241,·)$, $\chi_{4400}(2201,·)$, $\chi_{4400}(3931,·)$, $\chi_{4400}(3171,·)$, $\chi_{4400}(3441,·)$, $\chi_{4400}(2281,·)$, $\chi_{4400}(971,·)$, $\chi_{4400}(1521,·)$, $\chi_{4400}(1651,·)$, $\chi_{4400}(4361,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} + \frac{2}{17} a^{11} - \frac{8}{17} a^{10} + \frac{4}{17} a^{9} - \frac{8}{17} a^{8} - \frac{3}{17} a^{7} + \frac{7}{17} a^{6} - \frac{7}{17} a^{5} - \frac{2}{17} a^{4} - \frac{8}{17} a^{3} - \frac{8}{17} a^{2} - \frac{6}{17} a$, $\frac{1}{17} a^{13} + \frac{5}{17} a^{11} + \frac{3}{17} a^{10} + \frac{1}{17} a^{9} - \frac{4}{17} a^{8} - \frac{4}{17} a^{7} - \frac{4}{17} a^{6} - \frac{5}{17} a^{5} - \frac{4}{17} a^{4} + \frac{8}{17} a^{3} - \frac{7}{17} a^{2} - \frac{5}{17} a$, $\frac{1}{17} a^{14} - \frac{7}{17} a^{11} + \frac{7}{17} a^{10} - \frac{7}{17} a^{9} + \frac{2}{17} a^{8} - \frac{6}{17} a^{7} - \frac{6}{17} a^{6} - \frac{3}{17} a^{5} + \frac{1}{17} a^{4} - \frac{1}{17} a^{3} + \frac{1}{17} a^{2} - \frac{4}{17} a$, $\frac{1}{17} a^{15} + \frac{4}{17} a^{11} + \frac{5}{17} a^{10} - \frac{4}{17} a^{9} + \frac{6}{17} a^{8} + \frac{7}{17} a^{7} - \frac{5}{17} a^{6} + \frac{3}{17} a^{5} + \frac{2}{17} a^{4} - \frac{4}{17} a^{3} + \frac{8}{17} a^{2} - \frac{8}{17} a$, $\frac{1}{4777} a^{16} + \frac{77}{4777} a^{15} + \frac{15}{4777} a^{14} + \frac{37}{4777} a^{13} + \frac{56}{4777} a^{12} - \frac{999}{4777} a^{11} - \frac{142}{4777} a^{10} - \frac{1896}{4777} a^{9} + \frac{1074}{4777} a^{8} + \frac{1262}{4777} a^{7} - \frac{1616}{4777} a^{6} + \frac{1985}{4777} a^{5} - \frac{240}{4777} a^{4} + \frac{619}{4777} a^{3} + \frac{1019}{4777} a^{2} - \frac{100}{281} a + \frac{78}{281}$, $\frac{1}{4777} a^{17} - \frac{13}{4777} a^{15} + \frac{6}{4777} a^{14} + \frac{1}{281} a^{13} + \frac{28}{4777} a^{12} - \frac{2180}{4777} a^{11} - \frac{2202}{4777} a^{10} + \frac{1227}{4777} a^{9} - \frac{2194}{4777} a^{8} - \frac{721}{4777} a^{7} + \frac{1653}{4777} a^{6} + \frac{903}{4777} a^{5} + \frac{553}{4777} a^{4} - \frac{841}{4777} a^{3} + \frac{1327}{4777} a^{2} + \frac{999}{4777} a - \frac{105}{281}$, $\frac{1}{17566474777293208806319669594555040857} a^{18} - \frac{10710102825825174463460613132365}{1033322045723129929783509976150296521} a^{17} + \frac{54200950870303180728870253697637}{1033322045723129929783509976150296521} a^{16} + \frac{514602999138589776902867538443672595}{17566474777293208806319669594555040857} a^{15} - \frac{16365204151184826193582130734281235}{17566474777293208806319669594555040857} a^{14} + \frac{224763904543643169966350299781639102}{17566474777293208806319669594555040857} a^{13} - \frac{358847565240926288556666496554408516}{17566474777293208806319669594555040857} a^{12} + \frac{5544576045951982937234782118247126915}{17566474777293208806319669594555040857} a^{11} + \frac{2991475444326733347773965573749695355}{17566474777293208806319669594555040857} a^{10} + \frac{179272438883061373250697441342147152}{1033322045723129929783509976150296521} a^{9} + \frac{6485193967128770861816488464339555353}{17566474777293208806319669594555040857} a^{8} - \frac{6408759941824522809270028685693855843}{17566474777293208806319669594555040857} a^{7} - \frac{8584478885407933120185797878111039106}{17566474777293208806319669594555040857} a^{6} + \frac{6362313692245391982322412364666533555}{17566474777293208806319669594555040857} a^{5} + \frac{7339983520022977031595827128482672712}{17566474777293208806319669594555040857} a^{4} + \frac{2374103656482087287930141239079270452}{17566474777293208806319669594555040857} a^{3} + \frac{458460728916177356743057450103040549}{1033322045723129929783509976150296521} a^{2} + \frac{2707204303577768969004600366166713948}{17566474777293208806319669594555040857} a + \frac{226348275476724370708012846572610659}{1033322045723129929783509976150296521}$, $\frac{1}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{19} - \frac{1149763217533720086392413902700365613}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{18} - \frac{15902943611997038036265197782028301134202043657537154251289510716891747}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{17} - \frac{14896232705669169271366022964564248373510246962214348774472865240519231}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{16} - \frac{492609195237759740843887825610890740982815162293956524720733077897246457}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{15} - \frac{4088568817749572971927483695703330157856224513686759761924530351379296422}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{14} - \frac{2630835601815402073008991438330700653526409869473496343667188712247393609}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{13} + \frac{3142490498863271919723409947324404021166928211304662497033118088982102686}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{12} + \frac{39930731166700149687713512876319537525905254318290662088468230192614114429}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{11} - \frac{55288824773414346038866270953051087411362316097660164109390809153971796374}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{10} + \frac{21452840569137959953287229717195447509132049108767735752372994451308744785}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{9} + \frac{38487249300095935888330955006476376600788108978486838163581174115735054636}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{8} + \frac{49620339465701173334219945114937607908541881873481628241786422253113212791}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{7} + \frac{17919610045937129130209401314047778112800589280754192749738512246947675876}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{6} - \frac{70724372228776787223707293870289539358339367056960490122403747676152391805}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{5} + \frac{71883400319601562111029558520325939686957735747040454516538735287500893935}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{4} + \frac{5228248852711565320341435259030692767714293872678962841625973978354037027}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{3} - \frac{51366062407591997762019597143735262826841688985522565477279709969889785328}{161186714182018987234752840905203025894024052142872706632819036436172610793} a^{2} + \frac{3783747782747742934408942670945287913270521650434504471266755044131090514}{9481571422471705131456049465011942699648473655463100390165825672716035929} a - \frac{247604083977490011707926385190944158892932628837348248602539579223124654}{557739495439512066556238203824231923508733744439005905303872098395060937}$

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

Class group and class number

$C_{1105253605}$, which has order $1105253605$ (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:  \( 59880888363.52121 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

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

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.2048.2, 5.5.5719140625.3, 10.10.1071794405000000000000000.3

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 $20$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R $20$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
2Data not computed
5Data not computed
11Data not computed