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Properties

Label 20.0.78483048970...0000.5
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{16}$
Root discriminant $139.55$
Ramified primes $2, 3, 5, 7, 11$
Class number $5304200$ (GRH)
Class group $[10, 530420]$ (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![116484273949, -35059541744, 62190491667, -15745676396, 14891005039, -3323353912, 2200421129, -448347204, 230191550, -43438336, 18194876, -3183174, 1124878, -181742, 54976, -7844, 2033, -248, 57, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 57*x^18 - 248*x^17 + 2033*x^16 - 7844*x^15 + 54976*x^14 - 181742*x^13 + 1124878*x^12 - 3183174*x^11 + 18194876*x^10 - 43438336*x^9 + 230191550*x^8 - 448347204*x^7 + 2200421129*x^6 - 3323353912*x^5 + 14891005039*x^4 - 15745676396*x^3 + 62190491667*x^2 - 35059541744*x + 116484273949)
 
gp: K = bnfinit(x^20 - 6*x^19 + 57*x^18 - 248*x^17 + 2033*x^16 - 7844*x^15 + 54976*x^14 - 181742*x^13 + 1124878*x^12 - 3183174*x^11 + 18194876*x^10 - 43438336*x^9 + 230191550*x^8 - 448347204*x^7 + 2200421129*x^6 - 3323353912*x^5 + 14891005039*x^4 - 15745676396*x^3 + 62190491667*x^2 - 35059541744*x + 116484273949, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 57 x^{18} - 248 x^{17} + 2033 x^{16} - 7844 x^{15} + 54976 x^{14} - 181742 x^{13} + 1124878 x^{12} - 3183174 x^{11} + 18194876 x^{10} - 43438336 x^{9} + 230191550 x^{8} - 448347204 x^{7} + 2200421129 x^{6} - 3323353912 x^{5} + 14891005039 x^{4} - 15745676396 x^{3} + 62190491667 x^{2} - 35059541744 x + 116484273949 \)

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:  \(7848304897073886403551554174576640000000000=2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $139.55$
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:  $2, 3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4620=2^{2}\cdot 3\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4620}(1,·)$, $\chi_{4620}(839,·)$, $\chi_{4620}(1609,·)$, $\chi_{4620}(3851,·)$, $\chi_{4620}(911,·)$, $\chi_{4620}(3359,·)$, $\chi_{4620}(1681,·)$, $\chi_{4620}(71,·)$, $\chi_{4620}(421,·)$, $\chi_{4620}(4129,·)$, $\chi_{4620}(419,·)$, $\chi_{4620}(1189,·)$, $\chi_{4620}(1259,·)$, $\chi_{4620}(2029,·)$, $\chi_{4620}(4271,·)$, $\chi_{4620}(2099,·)$, $\chi_{4620}(2869,·)$, $\chi_{4620}(841,·)$, $\chi_{4620}(2171,·)$, $\chi_{4620}(2941,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{47} a^{15} + \frac{19}{47} a^{14} - \frac{5}{47} a^{13} - \frac{16}{47} a^{12} - \frac{3}{47} a^{11} - \frac{20}{47} a^{10} + \frac{12}{47} a^{9} - \frac{10}{47} a^{8} + \frac{19}{47} a^{7} + \frac{8}{47} a^{6} + \frac{11}{47} a^{5} + \frac{19}{47} a^{3} - \frac{7}{47} a^{2} + \frac{14}{47} a + \frac{11}{47}$, $\frac{1}{47} a^{16} + \frac{10}{47} a^{14} - \frac{15}{47} a^{13} + \frac{19}{47} a^{12} - \frac{10}{47} a^{11} + \frac{16}{47} a^{10} - \frac{3}{47} a^{9} + \frac{21}{47} a^{8} + \frac{23}{47} a^{7} - \frac{21}{47} a^{5} + \frac{19}{47} a^{4} + \frac{8}{47} a^{3} + \frac{6}{47} a^{2} - \frac{20}{47} a - \frac{21}{47}$, $\frac{1}{47} a^{17} - \frac{17}{47} a^{14} + \frac{22}{47} a^{13} + \frac{9}{47} a^{12} - \frac{1}{47} a^{11} + \frac{9}{47} a^{10} - \frac{5}{47} a^{9} - \frac{18}{47} a^{8} - \frac{2}{47} a^{7} - \frac{7}{47} a^{6} + \frac{3}{47} a^{5} + \frac{8}{47} a^{4} + \frac{4}{47} a^{3} + \frac{3}{47} a^{2} - \frac{20}{47} a - \frac{16}{47}$, $\frac{1}{46028486232806859832209431} a^{18} + \frac{206384742137446348424043}{46028486232806859832209431} a^{17} - \frac{119304569842732119955873}{46028486232806859832209431} a^{16} + \frac{315541415097104695561685}{46028486232806859832209431} a^{15} + \frac{13860183636485318698516739}{46028486232806859832209431} a^{14} + \frac{4519189871121442269074933}{46028486232806859832209431} a^{13} - \frac{664001442054279126458915}{46028486232806859832209431} a^{12} - \frac{21538250135896818666027593}{46028486232806859832209431} a^{11} - \frac{1359503400290545776075717}{46028486232806859832209431} a^{10} - \frac{20740824990680990333560003}{46028486232806859832209431} a^{9} - \frac{18962040808546457559989236}{46028486232806859832209431} a^{8} + \frac{387198215493347492534419}{979329494315039570898073} a^{7} + \frac{14710639914181888232808518}{46028486232806859832209431} a^{6} - \frac{1230025673367344221626693}{46028486232806859832209431} a^{5} + \frac{15378526073688787738152416}{46028486232806859832209431} a^{4} - \frac{2621856672546804162743550}{46028486232806859832209431} a^{3} + \frac{6728964452939485744694883}{46028486232806859832209431} a^{2} - \frac{22169639561909443794610918}{46028486232806859832209431} a - \frac{14631970404777506511117374}{46028486232806859832209431}$, $\frac{1}{338921050491194002048245144111950359982946342654278559417959} a^{19} - \frac{3552838746264766092252328159081858}{338921050491194002048245144111950359982946342654278559417959} a^{18} + \frac{583806110167611052851101992044050462014248435548165420865}{338921050491194002048245144111950359982946342654278559417959} a^{17} - \frac{3592390657336827456889450070852846516965722268362080854618}{338921050491194002048245144111950359982946342654278559417959} a^{16} + \frac{347236057155152630186612072705091808251204485662264518874}{338921050491194002048245144111950359982946342654278559417959} a^{15} + \frac{84971452728573405838945209419718163533626121247527538301839}{338921050491194002048245144111950359982946342654278559417959} a^{14} + \frac{73383090789897995380913110323633411733196840131510211597208}{338921050491194002048245144111950359982946342654278559417959} a^{13} + \frac{27919873407798153509358606331640219734443811287240776863283}{338921050491194002048245144111950359982946342654278559417959} a^{12} + \frac{1283885057732925926789960970279109279737523361963160274819}{3109367435699027541727019670751838164981159106919986783651} a^{11} - \frac{141769861053385412696527173258723270767914091629405985370957}{338921050491194002048245144111950359982946342654278559417959} a^{10} + \frac{106859445728964626826025458792624450921362401978751690375320}{338921050491194002048245144111950359982946342654278559417959} a^{9} - \frac{1050361870256492767173291728931428417984032896142777277323}{3109367435699027541727019670751838164981159106919986783651} a^{8} + \frac{36829448043168251146048471386010776893099586527846409242962}{338921050491194002048245144111950359982946342654278559417959} a^{7} - \frac{42411374310985085481118881121056066858394235539268701819619}{338921050491194002048245144111950359982946342654278559417959} a^{6} + \frac{31331955758810561164519206872922939481453325236273304907392}{338921050491194002048245144111950359982946342654278559417959} a^{5} - \frac{92093759349564001087813237681729433315014619176190297751515}{338921050491194002048245144111950359982946342654278559417959} a^{4} + \frac{17224063889204505875986188395753870455789342896273629739358}{338921050491194002048245144111950359982946342654278559417959} a^{3} - \frac{1907167990427852110120334285963523962649784117558186392152}{338921050491194002048245144111950359982946342654278559417959} a^{2} + \frac{69910119314205764292880945641460807413348224467777280892545}{338921050491194002048245144111950359982946342654278559417959} a + \frac{83651778315825605655743847096726943419656939483031490367871}{338921050491194002048245144111950359982946342654278559417959}$

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

Class group and class number

$C_{10}\times C_{530420}$, which has order $5304200$ (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:  \( 1746210.0427691017 \) (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

\(\Q(\sqrt{-105}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{3}, \sqrt{-35})\), \(\Q(\zeta_{11})^+\), 10.0.2801482624803139200000.1, 10.10.53339349076992.1, 10.0.11258530353021875.4

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 R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5Data not computed
7Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$