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Properties

Label 20.0.12336847311...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{15}\cdot 41^{16}$
Root discriminant $319.57$
Ramified primes $2, 3, 5, 41$
Class number $2026729024$ (GRH)
Class group $[2, 2, 2, 2, 2, 63335282]$ (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![6333479962731, 1044505487556, 2440534826736, 259762732920, 417171166533, 25464090456, 41927345826, 433584384, 2738480335, -119745724, 139147532, -11075856, 6148162, -515036, 217234, -16484, 5419, -372, 92, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 92*x^18 - 372*x^17 + 5419*x^16 - 16484*x^15 + 217234*x^14 - 515036*x^13 + 6148162*x^12 - 11075856*x^11 + 139147532*x^10 - 119745724*x^9 + 2738480335*x^8 + 433584384*x^7 + 41927345826*x^6 + 25464090456*x^5 + 417171166533*x^4 + 259762732920*x^3 + 2440534826736*x^2 + 1044505487556*x + 6333479962731)
 
gp: K = bnfinit(x^20 - 4*x^19 + 92*x^18 - 372*x^17 + 5419*x^16 - 16484*x^15 + 217234*x^14 - 515036*x^13 + 6148162*x^12 - 11075856*x^11 + 139147532*x^10 - 119745724*x^9 + 2738480335*x^8 + 433584384*x^7 + 41927345826*x^6 + 25464090456*x^5 + 417171166533*x^4 + 259762732920*x^3 + 2440534826736*x^2 + 1044505487556*x + 6333479962731, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 92 x^{18} - 372 x^{17} + 5419 x^{16} - 16484 x^{15} + 217234 x^{14} - 515036 x^{13} + 6148162 x^{12} - 11075856 x^{11} + 139147532 x^{10} - 119745724 x^{9} + 2738480335 x^{8} + 433584384 x^{7} + 41927345826 x^{6} + 25464090456 x^{5} + 417171166533 x^{4} + 259762732920 x^{3} + 2440534826736 x^{2} + 1044505487556 x + 6333479962731 \)

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:  \(123368473115995282017566753371226112000000000000000=2^{30}\cdot 3^{10}\cdot 5^{15}\cdot 41^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $319.57$
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:  $2, 3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4920=2^{3}\cdot 3\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4920}(1,·)$, $\chi_{4920}(83,·)$, $\chi_{4920}(3563,·)$, $\chi_{4920}(961,·)$, $\chi_{4920}(1609,·)$, $\chi_{4920}(2027,·)$, $\chi_{4920}(529,·)$, $\chi_{4920}(467,·)$, $\chi_{4920}(1369,·)$, $\chi_{4920}(3481,·)$, $\chi_{4920}(2929,·)$, $\chi_{4920}(1043,·)$, $\chi_{4920}(4321,·)$, $\chi_{4920}(4643,·)$, $\chi_{4920}(4561,·)$, $\chi_{4920}(1067,·)$, $\chi_{4920}(4547,·)$, $\chi_{4920}(1969,·)$, $\chi_{4920}(4403,·)$, $\chi_{4920}(707,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{6} a^{4} + \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2} a$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{10} - \frac{1}{18} a^{6} + \frac{1}{18} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{54} a^{13} - \frac{1}{54} a^{12} - \frac{2}{27} a^{11} - \frac{1}{27} a^{10} + \frac{1}{18} a^{9} + \frac{1}{18} a^{8} - \frac{1}{54} a^{7} + \frac{1}{54} a^{6} + \frac{2}{27} a^{5} + \frac{10}{27} a^{4} + \frac{1}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{54} a^{14} + \frac{1}{54} a^{12} + \frac{1}{18} a^{11} + \frac{2}{27} a^{10} - \frac{1}{18} a^{9} - \frac{7}{54} a^{8} - \frac{1}{54} a^{6} + \frac{5}{18} a^{5} + \frac{23}{54} a^{4} + \frac{7}{18} a^{3} + \frac{4}{9} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{54} a^{15} + \frac{1}{54} a^{12} - \frac{1}{54} a^{11} + \frac{1}{27} a^{10} - \frac{1}{54} a^{9} - \frac{1}{18} a^{8} - \frac{1}{54} a^{6} - \frac{13}{27} a^{5} - \frac{10}{27} a^{4} + \frac{1}{6} a^{3} - \frac{1}{9} a^{2} - \frac{1}{6}$, $\frac{1}{12798} a^{16} - \frac{7}{4266} a^{15} + \frac{40}{6399} a^{14} - \frac{17}{6399} a^{13} + \frac{43}{2133} a^{12} - \frac{163}{6399} a^{11} + \frac{64}{6399} a^{10} - \frac{314}{2133} a^{9} + \frac{770}{6399} a^{8} - \frac{775}{6399} a^{7} + \frac{565}{4266} a^{6} + \frac{3053}{12798} a^{5} - \frac{29}{2133} a^{4} + \frac{325}{2133} a^{3} + \frac{13}{474} a^{2} + \frac{367}{1422} a + \frac{64}{237}$, $\frac{1}{1394982} a^{17} + \frac{5}{697491} a^{16} + \frac{5591}{1394982} a^{15} + \frac{3157}{1394982} a^{14} + \frac{12713}{1394982} a^{13} + \frac{2651}{697491} a^{12} + \frac{7501}{232497} a^{11} - \frac{54559}{1394982} a^{10} + \frac{45757}{1394982} a^{9} + \frac{167771}{1394982} a^{8} + \frac{25526}{697491} a^{7} + \frac{26851}{697491} a^{6} - \frac{271933}{1394982} a^{5} - \frac{27262}{77499} a^{4} + \frac{97055}{464994} a^{3} - \frac{1993}{17222} a^{2} - \frac{28603}{77499} a + \frac{8}{79}$, $\frac{1}{21654505391159123408759314878} a^{18} - \frac{1948743892578507799157}{21654505391159123408759314878} a^{17} - \frac{340107476786882696847767}{21654505391159123408759314878} a^{16} - \frac{172226951185278768407188463}{21654505391159123408759314878} a^{15} + \frac{24485228861402054173196018}{3609084231859853901459885813} a^{14} + \frac{114216790434465485965598141}{21654505391159123408759314878} a^{13} - \frac{25783088507276671814508877}{7218168463719707802919771626} a^{12} + \frac{1047186495726588258427793437}{21654505391159123408759314878} a^{11} + \frac{131005926735701478538352440}{10827252695579561704379657439} a^{10} - \frac{123253180875077803502135126}{10827252695579561704379657439} a^{9} + \frac{264935998990564752459710914}{3609084231859853901459885813} a^{8} - \frac{1536203427558320880629494}{10827252695579561704379657439} a^{7} + \frac{1746981833585285934137324593}{10827252695579561704379657439} a^{6} - \frac{3760959419932646234829221725}{10827252695579561704379657439} a^{5} - \frac{108956087437641875761625965}{7218168463719707802919771626} a^{4} + \frac{1080142300658033675891746670}{3609084231859853901459885813} a^{3} + \frac{118881519677861259120418153}{802018718191078644768863514} a^{2} + \frac{909612993474056186749003109}{2406056154573235934306590542} a - \frac{2167354952706604893966817}{7357969891661271970356546}$, $\frac{1}{48925885495115619463730481951651556213508085991048537328810226} a^{19} + \frac{126575748731078659001125555198469}{48925885495115619463730481951651556213508085991048537328810226} a^{18} - \frac{5836406547989999692368178417530324000100513967026195099}{48925885495115619463730481951651556213508085991048537328810226} a^{17} - \frac{96062190282351709017826880631636495246411692350759435664}{2718104749728645525762804552869530900750449221724918740489457} a^{16} + \frac{298718223047270340488519828886144886726509891010933467519487}{48925885495115619463730481951651556213508085991048537328810226} a^{15} + \frac{394075982675442389913113576443449084839236471293142798473763}{48925885495115619463730481951651556213508085991048537328810226} a^{14} + \frac{247878553510852989850775547192710272426284146853654209332825}{48925885495115619463730481951651556213508085991048537328810226} a^{13} + \frac{135443413827712295451969716358771517846344887578588349779147}{48925885495115619463730481951651556213508085991048537328810226} a^{12} + \frac{2565099539000839812884199167859448286509188035776883044936831}{48925885495115619463730481951651556213508085991048537328810226} a^{11} + \frac{398979516941743481557427458229520679851920163203464818765787}{5436209499457291051525609105739061801500898443449837480978914} a^{10} + \frac{4112663859142216245064915643358022499025221037425752850521877}{48925885495115619463730481951651556213508085991048537328810226} a^{9} - \frac{2746721999879419033512998554453647331962954671461827835763113}{48925885495115619463730481951651556213508085991048537328810226} a^{8} + \frac{3831336117582186009545523343783667440427241732771119340429662}{24462942747557809731865240975825778106754042995524268664405113} a^{7} + \frac{718337812083888165527144991128202647666967493042129851233881}{5436209499457291051525609105739061801500898443449837480978914} a^{6} + \frac{742412810870169737069649604357336365371034915858292091374265}{1812069833152430350508536368579687267166966147816612493659638} a^{5} - \frac{2106932893788633069665704895483889866039024604095026974206665}{5436209499457291051525609105739061801500898443449837480978914} a^{4} + \frac{722635607985663729978763613579572709160679305662196246554909}{2718104749728645525762804552869530900750449221724918740489457} a^{3} - \frac{217959418654447464820982209632720888298098992158802139953789}{604023277717476783502845456193229089055655382605537497886546} a^{2} + \frac{555080424579548648653025619515433813431727364421020116936055}{1812069833152430350508536368579687267166966147816612493659638} a + \frac{601158800074217550351073581621331824692918741862223212691}{1847165986903598726308395890499171526164083738854854733598}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{63335282}$, which has order $2026729024$ (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:  \( 41023218.25673422 \) (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

\(\Q(\sqrt{5}) \), 4.0.72000.2, 5.5.2825761.1, 10.10.24952891341003125.1

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 $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5Data not computed
$41$41.10.8.1$x^{10} - 27101 x^{5} + 418286592$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
41.10.8.1$x^{10} - 27101 x^{5} + 418286592$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$