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Properties

Label 20.0.14359290582...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{35}\cdot 11^{16}$
Root discriminant $322.00$
Ramified primes $2, 5, 11$
Class number $195100960$ (GRH)
Class group $[2, 2, 2, 2, 12193810]$ (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![17554000481876, -17767331888240, 8276685709850, -627310736360, -830021195575, 318126118552, 7753231540, -28427326400, 4616589810, 1384479800, -411711326, -35460480, 20953775, 478280, -676800, -2552, 13970, 0, -170, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 170*x^18 + 13970*x^16 - 2552*x^15 - 676800*x^14 + 478280*x^13 + 20953775*x^12 - 35460480*x^11 - 411711326*x^10 + 1384479800*x^9 + 4616589810*x^8 - 28427326400*x^7 + 7753231540*x^6 + 318126118552*x^5 - 830021195575*x^4 - 627310736360*x^3 + 8276685709850*x^2 - 17767331888240*x + 17554000481876)
 
gp: K = bnfinit(x^20 - 170*x^18 + 13970*x^16 - 2552*x^15 - 676800*x^14 + 478280*x^13 + 20953775*x^12 - 35460480*x^11 - 411711326*x^10 + 1384479800*x^9 + 4616589810*x^8 - 28427326400*x^7 + 7753231540*x^6 + 318126118552*x^5 - 830021195575*x^4 - 627310736360*x^3 + 8276685709850*x^2 - 17767331888240*x + 17554000481876, 1)
 

Normalized defining polynomial

\( x^{20} - 170 x^{18} + 13970 x^{16} - 2552 x^{15} - 676800 x^{14} + 478280 x^{13} + 20953775 x^{12} - 35460480 x^{11} - 411711326 x^{10} + 1384479800 x^{9} + 4616589810 x^{8} - 28427326400 x^{7} + 7753231540 x^{6} + 318126118552 x^{5} - 830021195575 x^{4} - 627310736360 x^{3} + 8276685709850 x^{2} - 17767331888240 x + 17554000481876 \)

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:  \(143592905823663003125000000000000000000000000000000=2^{30}\cdot 5^{35}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $322.00$
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:  \(2200=2^{3}\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{2200}(1,·)$, $\chi_{2200}(797,·)$, $\chi_{2200}(961,·)$, $\chi_{2200}(1609,·)$, $\chi_{2200}(1037,·)$, $\chi_{2200}(1681,·)$, $\chi_{2200}(1849,·)$, $\chi_{2200}(477,·)$, $\chi_{2200}(1893,·)$, $\chi_{2200}(929,·)$, $\chi_{2200}(933,·)$, $\chi_{2200}(1489,·)$, $\chi_{2200}(1769,·)$, $\chi_{2200}(2157,·)$, $\chi_{2200}(1213,·)$, $\chi_{2200}(1973,·)$, $\chi_{2200}(641,·)$, $\chi_{2200}(1721,·)$, $\chi_{2200}(1853,·)$, $\chi_{2200}(317,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{10} + \frac{2}{5} a^{5} - \frac{1}{2} a^{4} + \frac{2}{5}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{2}{5} a$, $\frac{1}{760} a^{12} + \frac{9}{190} a^{11} - \frac{9}{190} a^{10} - \frac{7}{38} a^{9} + \frac{1}{152} a^{8} + \frac{8}{95} a^{7} + \frac{7}{380} a^{6} + \frac{39}{190} a^{5} + \frac{11}{152} a^{4} + \frac{4}{19} a^{3} - \frac{143}{380} a^{2} + \frac{13}{95} a + \frac{1}{10}$, $\frac{1}{3800} a^{13} + \frac{1}{3800} a^{12} + \frac{9}{475} a^{11} - \frac{43}{950} a^{10} - \frac{7}{760} a^{9} - \frac{871}{3800} a^{8} - \frac{353}{1900} a^{7} + \frac{289}{1900} a^{6} + \frac{67}{3800} a^{5} + \frac{331}{760} a^{4} + \frac{667}{1900} a^{3} - \frac{643}{1900} a^{2} - \frac{93}{950} a - \frac{13}{50}$, $\frac{1}{3800} a^{14} + \frac{1}{3800} a^{12} - \frac{13}{475} a^{11} - \frac{3}{3800} a^{10} - \frac{67}{475} a^{9} - \frac{37}{760} a^{8} + \frac{151}{950} a^{7} - \frac{351}{3800} a^{6} + \frac{116}{475} a^{5} - \frac{371}{3800} a^{4} - \frac{13}{95} a^{3} + \frac{17}{1900} a^{2} - \frac{132}{475} a + \frac{3}{50}$, $\frac{1}{110200} a^{15} + \frac{3}{55100} a^{14} - \frac{9}{110200} a^{13} - \frac{9}{55100} a^{12} - \frac{2287}{110200} a^{11} + \frac{2193}{55100} a^{10} - \frac{2351}{110200} a^{9} + \frac{11927}{55100} a^{8} - \frac{253}{5800} a^{7} - \frac{321}{1900} a^{6} + \frac{17807}{110200} a^{5} + \frac{10877}{55100} a^{4} + \frac{19887}{55100} a^{3} + \frac{367}{27550} a^{2} + \frac{8963}{27550} a - \frac{9}{25}$, $\frac{1}{2424400} a^{16} - \frac{161}{2424400} a^{14} - \frac{1}{110200} a^{13} + \frac{7}{151525} a^{12} - \frac{2151}{55100} a^{11} + \frac{2287}{220400} a^{10} + \frac{737}{5800} a^{9} - \frac{43457}{606100} a^{8} - \frac{1853}{55100} a^{7} + \frac{380597}{2424400} a^{6} + \frac{40099}{110200} a^{5} - \frac{724429}{2424400} a^{4} - \frac{9337}{55100} a^{3} - \frac{55701}{1212200} a^{2} - \frac{11619}{27550} a + \frac{47}{1100}$, $\frac{1}{2424400} a^{17} - \frac{7}{2424400} a^{15} + \frac{3}{27550} a^{14} + \frac{1}{1212200} a^{13} - \frac{49}{110200} a^{12} - \frac{10707}{220400} a^{11} - \frac{141}{27550} a^{10} - \frac{151187}{1212200} a^{9} - \frac{25431}{110200} a^{8} - \frac{369889}{2424400} a^{7} + \frac{839}{27550} a^{6} + \frac{15153}{83600} a^{5} + \frac{38803}{110200} a^{4} + \frac{163969}{1212200} a^{3} - \frac{16737}{55100} a^{2} - \frac{56097}{606100} a + \frac{3}{10}$, $\frac{1}{46063600} a^{18} - \frac{7}{46063600} a^{17} + \frac{3}{23031800} a^{16} - \frac{9}{2424400} a^{15} + \frac{1451}{46063600} a^{14} - \frac{1063}{23031800} a^{13} - \frac{1237}{2424400} a^{12} - \frac{387}{11600} a^{11} - \frac{178067}{9212720} a^{10} - \frac{5131633}{23031800} a^{9} - \frac{919571}{46063600} a^{8} + \frac{8753007}{46063600} a^{7} - \frac{99043}{575795} a^{6} + \frac{4396637}{46063600} a^{5} + \frac{15404919}{46063600} a^{4} - \frac{1609599}{4606360} a^{3} + \frac{1893879}{4606360} a^{2} - \frac{203339}{606100} a + \frac{251}{1100}$, $\frac{1}{475415392105599965758197009441500680883290974877436318858390295194868877699600} a^{19} + \frac{32338214645678848690851369102783799810611956519339358337866286062157}{4754153921055999657581970094415006808832909748774363188583902951948688776996} a^{18} - \frac{233885724674148260232523658645879762855233913776634925989436684618204}{5942692401319999571977462618018758511041137185967953985729878689935860971245} a^{17} + \frac{57841831728924924766418303357264037301076837928350075522290239617709231}{475415392105599965758197009441500680883290974877436318858390295194868877699600} a^{16} + \frac{363627587371444064309806245966595212055720309834166953981603267756992669}{95083078421119993151639401888300136176658194975487263771678059038973775539920} a^{15} + \frac{10947718572594591021533894304083861138774264430837467257819935882929666523}{475415392105599965758197009441500680883290974877436318858390295194868877699600} a^{14} + \frac{7325766283673720422918934193838546515488410452508652199254451791602800617}{95083078421119993151639401888300136176658194975487263771678059038973775539920} a^{13} + \frac{4514003184335102348707050841531864271037752793150991422340332446935950239}{12510931371199999098899921301092123181139236180985166285747113031443917834200} a^{12} + \frac{12473316329362380139188199136832889327696213061046985468368095084533489927903}{475415392105599965758197009441500680883290974877436318858390295194868877699600} a^{11} + \frac{9287669574722649365918811192323382459554511807182685561233478150127589165449}{475415392105599965758197009441500680883290974877436318858390295194868877699600} a^{10} - \frac{10134934069880132322034326149917692581306354315719084088988847436939792798639}{95083078421119993151639401888300136176658194975487263771678059038973775539920} a^{9} - \frac{3741397388226061647387549239299155768738772819877841254407414149409986954543}{47541539210559996575819700944150068088329097487743631885839029519486887769960} a^{8} - \frac{18527322467809562050127880372271815967749159076585421341532292135808116714441}{237707696052799982879098504720750340441645487438718159429195147597434438849800} a^{7} - \frac{57328934720587082923878157916381680778082542832896311701386144068624828789}{1000874509695999927911993704087369854491138894478813302859769042515513426736} a^{6} - \frac{167369236429776557494880671048164972807050436354857567137251297374317350349439}{475415392105599965758197009441500680883290974877436318858390295194868877699600} a^{5} - \frac{44534753053669135786889340051082659557206812923500756508449197454715406200189}{475415392105599965758197009441500680883290974877436318858390295194868877699600} a^{4} + \frac{2782476419518532771393091796337685929199625396414830083735271187790799859077}{9508307842111999315163940188830013617665819497548726377167805903897377553992} a^{3} + \frac{37076401558195671716627525045433305333861439936535317168371444578338227804779}{237707696052799982879098504720750340441645487438718159429195147597434438849800} a^{2} + \frac{2870254178519189765477329798525001326763764272702826368747946672654806313471}{6255465685599999549449960650546061590569618090492583142873556515721958917100} a + \frac{3009754118990865350997550059610061448483522804148737984345432941086787981}{11352932278765879400090672687016445717912192541728826030623514547589762100}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{12193810}$, which has order $195100960$ (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:  \( 4235385044.5954027 \) (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

\(\Q(\sqrt{5}) \), 4.0.8000.2, 5.5.5719140625.4, 10.10.163542847442626953125.4

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 $20$ R $20$ $20$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
5Data not computed
$11$11.10.8.4$x^{10} - 781 x^{5} + 290521$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.4$x^{10} - 781 x^{5} + 290521$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$