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Properties

Label 20.0.82802905234...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{35}\cdot 11^{16}$
Root discriminant $394.37$
Ramified primes $2, 3, 5, 11$
Class number $2037640000$ (GRH)
Class group $[2, 2, 10, 10, 110, 46310]$ (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![66790722351376, -48578566298240, 18744110591600, -337208187360, -1461670305700, 422425247552, 44066619140, -33027911400, 3849422785, 1584756800, -330084201, -38258880, 16919275, 506880, -539050, -2552, 11495, 0, -145, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 145*x^18 + 11495*x^16 - 2552*x^15 - 539050*x^14 + 506880*x^13 + 16919275*x^12 - 38258880*x^11 - 330084201*x^10 + 1584756800*x^9 + 3849422785*x^8 - 33027911400*x^7 + 44066619140*x^6 + 422425247552*x^5 - 1461670305700*x^4 - 337208187360*x^3 + 18744110591600*x^2 - 48578566298240*x + 66790722351376)
 
gp: K = bnfinit(x^20 - 145*x^18 + 11495*x^16 - 2552*x^15 - 539050*x^14 + 506880*x^13 + 16919275*x^12 - 38258880*x^11 - 330084201*x^10 + 1584756800*x^9 + 3849422785*x^8 - 33027911400*x^7 + 44066619140*x^6 + 422425247552*x^5 - 1461670305700*x^4 - 337208187360*x^3 + 18744110591600*x^2 - 48578566298240*x + 66790722351376, 1)
 

Normalized defining polynomial

\( x^{20} - 145 x^{18} + 11495 x^{16} - 2552 x^{15} - 539050 x^{14} + 506880 x^{13} + 16919275 x^{12} - 38258880 x^{11} - 330084201 x^{10} + 1584756800 x^{9} + 3849422785 x^{8} - 33027911400 x^{7} + 44066619140 x^{6} + 422425247552 x^{5} - 1461670305700 x^{4} - 337208187360 x^{3} + 18744110591600 x^{2} - 48578566298240 x + 66790722351376 \)

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:  \(8280290523419410812039184570312500000000000000000000=2^{20}\cdot 3^{10}\cdot 5^{35}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $394.37$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3300=2^{2}\cdot 3\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3300}(1,·)$, $\chi_{3300}(323,·)$, $\chi_{3300}(2821,·)$, $\chi_{3300}(961,·)$, $\chi_{3300}(1609,·)$, $\chi_{3300}(203,·)$, $\chi_{3300}(1741,·)$, $\chi_{3300}(2447,·)$, $\chi_{3300}(1681,·)$, $\chi_{3300}(3227,·)$, $\chi_{3300}(1763,·)$, $\chi_{3300}(1489,·)$, $\chi_{3300}(2687,·)$, $\chi_{3300}(1607,·)$, $\chi_{3300}(2029,·)$, $\chi_{3300}(1967,·)$, $\chi_{3300}(2869,·)$, $\chi_{3300}(1849,·)$, $\chi_{3300}(1343,·)$, $\chi_{3300}(383,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{10} + \frac{2}{5} a^{5} - \frac{1}{2} a^{4} + \frac{2}{5}$, $\frac{1}{10} a^{11} + \frac{2}{5} a^{6} - \frac{1}{2} a^{5} + \frac{2}{5} a$, $\frac{1}{40} a^{12} - \frac{1}{40} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{2}{5} a^{7} + \frac{1}{8} a^{6} - \frac{1}{10} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{3}{20} a^{2} - \frac{1}{2} a - \frac{1}{10}$, $\frac{1}{3800} a^{13} + \frac{11}{3800} a^{12} - \frac{73}{3800} a^{11} - \frac{157}{3800} a^{10} + \frac{6}{95} a^{9} + \frac{427}{1900} a^{8} - \frac{1091}{3800} a^{7} - \frac{677}{3800} a^{6} + \frac{727}{3800} a^{5} - \frac{369}{760} a^{4} - \frac{102}{475} a^{3} + \frac{927}{1900} a^{2} + \frac{231}{475} a - \frac{257}{950}$, $\frac{1}{3800} a^{14} - \frac{1}{950} a^{12} - \frac{3}{100} a^{11} - \frac{123}{3800} a^{10} + \frac{3}{100} a^{9} + \frac{183}{760} a^{8} + \frac{9}{50} a^{7} - \frac{379}{950} a^{6} + \frac{21}{100} a^{5} + \frac{1429}{3800} a^{4} + \frac{7}{20} a^{3} + \frac{607}{1900} a^{2} - \frac{21}{50} a - \frac{213}{950}$, $\frac{1}{7600} a^{15} - \frac{7}{760} a^{12} + \frac{69}{1520} a^{11} - \frac{67}{3800} a^{10} - \frac{1}{304} a^{9} + \frac{3}{76} a^{8} + \frac{43}{190} a^{7} - \frac{267}{760} a^{6} - \frac{1743}{7600} a^{5} - \frac{7}{152} a^{4} + \frac{73}{152} a^{3} - \frac{89}{380} a^{2} + \frac{99}{380} a - \frac{162}{475}$, $\frac{1}{2424400} a^{16} + \frac{69}{1212200} a^{14} - \frac{3}{55100} a^{13} - \frac{28249}{2424400} a^{12} - \frac{473}{11020} a^{11} + \frac{1761}{44080} a^{10} - \frac{2973}{13775} a^{9} + \frac{189521}{1212200} a^{8} - \frac{3529}{55100} a^{7} + \frac{280943}{2424400} a^{6} + \frac{3569}{11020} a^{5} + \frac{98721}{1212200} a^{4} - \frac{6323}{13775} a^{3} - \frac{160019}{606100} a^{2} + \frac{6633}{27550} a + \frac{497}{1045}$, $\frac{1}{482455600} a^{17} - \frac{3}{25392400} a^{16} - \frac{2283}{48245560} a^{15} + \frac{3368}{30153475} a^{14} - \frac{15621}{482455600} a^{13} + \frac{2680053}{482455600} a^{12} + \frac{108511}{2308400} a^{11} + \frac{2183333}{43859600} a^{10} - \frac{38050329}{241227800} a^{9} - \frac{705091}{120613900} a^{8} - \frac{938243}{5078480} a^{7} + \frac{4673293}{19298224} a^{6} - \frac{80690999}{241227800} a^{5} + \frac{25945693}{120613900} a^{4} + \frac{20320497}{60306950} a^{3} - \frac{14372929}{30153475} a^{2} - \frac{3190392}{30153475} a + \frac{898011}{2079550}$, $\frac{1}{9166656400} a^{18} + \frac{1}{1833331280} a^{17} - \frac{1091}{9166656400} a^{16} + \frac{479377}{9166656400} a^{15} - \frac{16949}{366666256} a^{14} + \frac{533869}{9166656400} a^{13} - \frac{5131991}{4583328200} a^{12} + \frac{28497}{41666620} a^{11} + \frac{341870071}{9166656400} a^{10} + \frac{79405759}{1833331280} a^{9} - \frac{1544415781}{9166656400} a^{8} - \frac{252752243}{9166656400} a^{7} - \frac{1329341683}{9166656400} a^{6} + \frac{823882447}{1833331280} a^{5} - \frac{14302669}{48245560} a^{4} + \frac{81818821}{916665640} a^{3} - \frac{331250107}{2291664100} a^{2} - \frac{10692481}{2291664100} a - \frac{19204829}{39511450}$, $\frac{1}{2350445519217726802982618734989806969784383600005294123828777519715007999270399200} a^{19} + \frac{27640546070164345453424113658327002188401583019953746672309540223935583}{1175222759608863401491309367494903484892191800002647061914388759857503999635199600} a^{18} + \frac{2065846994375128371512386769234056035028945895317343103303925700403571707}{2350445519217726802982618734989806969784383600005294123828777519715007999270399200} a^{17} + \frac{8138657859686652577325681079885486239235265541742671976598109342825970551}{1175222759608863401491309367494903484892191800002647061914388759857503999635199600} a^{16} - \frac{51151918088421887175361682377508215919720096497734281950862003186537026385341}{2350445519217726802982618734989806969784383600005294123828777519715007999270399200} a^{15} - \frac{92862032269515996903831886279048567583684253451510988990795078645602019073611}{1175222759608863401491309367494903484892191800002647061914388759857503999635199600} a^{14} + \frac{216251877007901754592983870376121316235566504780534446948486893903029404893}{2474153178123922950508019721041902073457245894742414867188186862857903157126736} a^{13} - \frac{1613941641842049689045321121374519226303387390450355021093821590396780855797939}{293805689902215850372827341873725871223047950000661765478597189964375999908799900} a^{12} - \frac{55344075970319737803367998892690454374213047014093064588506787917805502276621973}{2350445519217726802982618734989806969784383600005294123828777519715007999270399200} a^{11} + \frac{1397100239001213845754358071989810878185388063792874883143444095616111470186771}{47008910384354536059652374699796139395687672000105882476575550394300159985407984} a^{10} + \frac{58110378934878881632204399872694571844330675318884144322447798552613301931449147}{470089103843545360596523746997961393956876720001058824765755503943001599854079840} a^{9} - \frac{183608403113301790688052645520524614626990663130329979661954824888918483994334991}{1175222759608863401491309367494903484892191800002647061914388759857503999635199600} a^{8} + \frac{370604640865541804847723127317459435897962002893082992857167175179507501053248113}{2350445519217726802982618734989806969784383600005294123828777519715007999270399200} a^{7} - \frac{28601891360618358360277265570473497564767537205649568241508999636670927013893153}{106838432691714854681028124317718498626562890909331551083126250896136727239563600} a^{6} - \frac{22717202662811767684481030301521240593553953638217904089517565584142298677523661}{293805689902215850372827341873725871223047950000661765478597189964375999908799900} a^{5} - \frac{2655702306055151471566636581614065514871443470130540755528479088286481617378073}{26709608172928713670257031079429624656640722727332887770781562724034181809890900} a^{4} + \frac{6545092127610330651825433088549526823706901468140062200469766910277869216679249}{146902844951107925186413670936862935611523975000330882739298594982187999954399950} a^{3} - \frac{20265988572649170490786372315206070648703533584223166832325697871707728400182921}{58761137980443170074565468374745174244609590000132353095719437992875199981759980} a^{2} + \frac{6457554942461759283672609326364727002118843832877291742793502160745794652605197}{293805689902215850372827341873725871223047950000661765478597189964375999908799900} a - \frac{1127321994077075976286835921558188397534851807134201875695934391185930226781508}{2532807671570826296317477085118326476060758189660877288608596465210137930248275}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}\times C_{10}\times C_{110}\times C_{46310}$, which has order $2037640000$ (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

\(\Q(\sqrt{5}) \), 4.0.18000.1, 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 R R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
3Data not computed
5Data not computed
$11$11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$