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Properties

Label 20.0.22306224130...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{35}\cdot 7^{10}\cdot 11^{16}$
Root discriminant $521.70$
Ramified primes $3, 5, 7, 11$
Class number $8332060820$ (GRH)
Class group $[2, 4166030410]$ (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![839461258717201, -200742904991115, 35137458539585, 8391148914665, 1545328775040, -396677379402, 90182043685, -7032212790, 6710256915, 641828715, -124930751, -42260205, 11968120, 975255, -238670, -33798, 7555, 330, -80, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 80*x^18 + 330*x^17 + 7555*x^16 - 33798*x^15 - 238670*x^14 + 975255*x^13 + 11968120*x^12 - 42260205*x^11 - 124930751*x^10 + 641828715*x^9 + 6710256915*x^8 - 7032212790*x^7 + 90182043685*x^6 - 396677379402*x^5 + 1545328775040*x^4 + 8391148914665*x^3 + 35137458539585*x^2 - 200742904991115*x + 839461258717201)
 
gp: K = bnfinit(x^20 - 5*x^19 - 80*x^18 + 330*x^17 + 7555*x^16 - 33798*x^15 - 238670*x^14 + 975255*x^13 + 11968120*x^12 - 42260205*x^11 - 124930751*x^10 + 641828715*x^9 + 6710256915*x^8 - 7032212790*x^7 + 90182043685*x^6 - 396677379402*x^5 + 1545328775040*x^4 + 8391148914665*x^3 + 35137458539585*x^2 - 200742904991115*x + 839461258717201, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 80 x^{18} + 330 x^{17} + 7555 x^{16} - 33798 x^{15} - 238670 x^{14} + 975255 x^{13} + 11968120 x^{12} - 42260205 x^{11} - 124930751 x^{10} + 641828715 x^{9} + 6710256915 x^{8} - 7032212790 x^{7} + 90182043685 x^{6} - 396677379402 x^{5} + 1545328775040 x^{4} + 8391148914665 x^{3} + 35137458539585 x^{2} - 200742904991115 x + 839461258717201 \)

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:  \(2230622413058508301319180354362470097839832305908203125=3^{10}\cdot 5^{35}\cdot 7^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $521.70$
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:  $3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(5775=3\cdot 5^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{5775}(1,·)$, $\chi_{5775}(64,·)$, $\chi_{5775}(608,·)$, $\chi_{5775}(841,·)$, $\chi_{5775}(1343,·)$, $\chi_{5775}(1534,·)$, $\chi_{5775}(1849,·)$, $\chi_{5775}(2269,·)$, $\chi_{5775}(2479,·)$, $\chi_{5775}(2731,·)$, $\chi_{5775}(2836,·)$, $\chi_{5775}(2897,·)$, $\chi_{5775}(3023,·)$, $\chi_{5775}(3128,·)$, $\chi_{5775}(3338,·)$, $\chi_{5775}(3842,·)$, $\chi_{5775}(4096,·)$, $\chi_{5775}(4262,·)$, $\chi_{5775}(5102,·)$, $\chi_{5775}(5732,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{7} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{13} - \frac{1}{25} a^{12} + \frac{2}{25} a^{11} + \frac{2}{25} a^{10} - \frac{2}{5} a^{9} + \frac{11}{25} a^{8} + \frac{4}{25} a^{7} + \frac{7}{25} a^{6} + \frac{2}{25} a^{5} - \frac{1}{5} a^{4} + \frac{4}{25} a^{3} - \frac{9}{25} a^{2} - \frac{12}{25} a - \frac{2}{25}$, $\frac{1}{25} a^{14} + \frac{1}{25} a^{12} - \frac{1}{25} a^{11} + \frac{2}{25} a^{10} + \frac{1}{25} a^{9} - \frac{2}{5} a^{8} + \frac{11}{25} a^{7} + \frac{4}{25} a^{6} + \frac{7}{25} a^{5} - \frac{1}{25} a^{4} - \frac{1}{5} a^{3} + \frac{4}{25} a^{2} - \frac{9}{25} a - \frac{12}{25}$, $\frac{1}{3475} a^{15} - \frac{13}{3475} a^{14} - \frac{27}{3475} a^{13} - \frac{111}{3475} a^{12} - \frac{146}{3475} a^{11} + \frac{329}{3475} a^{10} + \frac{1457}{3475} a^{9} + \frac{183}{3475} a^{8} - \frac{26}{3475} a^{7} + \frac{554}{3475} a^{6} - \frac{638}{3475} a^{5} - \frac{427}{3475} a^{4} - \frac{1093}{3475} a^{3} - \frac{959}{3475} a^{2} - \frac{1429}{3475} a - \frac{7}{25}$, $\frac{1}{38225} a^{16} - \frac{613}{38225} a^{14} + \frac{511}{38225} a^{13} - \frac{2284}{38225} a^{12} - \frac{2681}{38225} a^{11} - \frac{3579}{38225} a^{10} + \frac{15927}{38225} a^{9} + \frac{13751}{38225} a^{8} + \frac{216}{38225} a^{7} + \frac{18657}{38225} a^{6} - \frac{9694}{38225} a^{5} - \frac{667}{38225} a^{4} + \frac{11659}{38225} a^{3} + \frac{13209}{38225} a^{2} + \frac{14227}{38225} a + \frac{31}{275}$, $\frac{1}{38225} a^{17} + \frac{3}{38225} a^{15} + \frac{148}{38225} a^{14} - \frac{568}{38225} a^{13} + \frac{467}{7645} a^{12} - \frac{3304}{38225} a^{11} + \frac{3002}{38225} a^{10} + \frac{9153}{38225} a^{9} + \frac{8972}{38225} a^{8} + \frac{107}{275} a^{7} + \frac{16596}{38225} a^{6} + \frac{2336}{38225} a^{5} - \frac{6733}{38225} a^{4} - \frac{13312}{38225} a^{3} + \frac{7561}{38225} a^{2} - \frac{2896}{38225} a - \frac{1}{25}$, $\frac{1}{38225} a^{18} + \frac{1}{7645} a^{15} + \frac{72}{38225} a^{14} + \frac{76}{38225} a^{13} - \frac{397}{7645} a^{12} + \frac{2872}{38225} a^{11} + \frac{3423}{38225} a^{10} - \frac{13223}{38225} a^{9} + \frac{4024}{38225} a^{8} - \frac{3406}{7645} a^{7} - \frac{9008}{38225} a^{6} + \frac{14198}{38225} a^{5} - \frac{707}{38225} a^{4} + \frac{446}{1529} a^{3} - \frac{252}{1529} a^{2} - \frac{1937}{38225} a + \frac{83}{275}$, $\frac{1}{139924580325961928307572689265641807596073145079573813995558801862829031196218449856131074495401190622997539025} a^{19} + \frac{1142556211800983572034664724655216837691946401260660830691544884615681748213684180890144394711123852945712}{139924580325961928307572689265641807596073145079573813995558801862829031196218449856131074495401190622997539025} a^{18} - \frac{38690438092359537397999027758788581533266253814289440193804135207330459100610467993064284000445104295476}{139924580325961928307572689265641807596073145079573813995558801862829031196218449856131074495401190622997539025} a^{17} + \frac{329787960269881609959147544663093414189839261909536868340105414655940981531149360274727353851825913114372}{139924580325961928307572689265641807596073145079573813995558801862829031196218449856131074495401190622997539025} a^{16} + \frac{3762336971848705810070087228678624649452824417874150713827886356335796921537171557745937096463125336574222}{27984916065192385661514537853128361519214629015914762799111760372565806239243689971226214899080238124599507805} a^{15} - \frac{1466795833504633358775824313807805851781908417135641736452297494476141887934537825819457152363738605553307223}{139924580325961928307572689265641807596073145079573813995558801862829031196218449856131074495401190622997539025} a^{14} - \frac{1060056970985707765860383574958500724470943541306135899155447557952805411509936471555870517478210509611508563}{139924580325961928307572689265641807596073145079573813995558801862829031196218449856131074495401190622997539025} a^{13} - \frac{161966348375384240462171285398525980270508011591594352964703229713840686645751173982511066000504826728814263}{2544083278653853241955867077557123774474057183264978436282887306596891476294880906475110445370930738599955255} a^{12} - \frac{11502219778324826447853573498475712112035691173652034261437145339423155619724276076622782699974287746526705717}{139924580325961928307572689265641807596073145079573813995558801862829031196218449856131074495401190622997539025} a^{11} - \frac{6074037738756635371887981113728196255275721533371857711630515451060646918337687896409537342290806983631470846}{139924580325961928307572689265641807596073145079573813995558801862829031196218449856131074495401190622997539025} a^{10} - \frac{7716759214209072275321641456172288190268932460540226854547072330253447394914985329121983465001262787190915938}{27984916065192385661514537853128361519214629015914762799111760372565806239243689971226214899080238124599507805} a^{9} - \frac{31057206998484399218626865633763604983903506026857193473821978933925168635273364925039210950585448666956107677}{139924580325961928307572689265641807596073145079573813995558801862829031196218449856131074495401190622997539025} a^{8} - \frac{39901068309391078192253022642291973924490997479985240090134930220671080117208872000932535738321476233705175963}{139924580325961928307572689265641807596073145079573813995558801862829031196218449856131074495401190622997539025} a^{7} - \frac{457754390476431290123126356584347106606218546823702343386411387903126687312105637411549362893107407639287366}{12720416393269266209779335387785618872370285916324892181414436532984457381474404532375552226854653692999776275} a^{6} - \frac{34620271909502781472025790633322861039189093194163720038306364297690443614453137883577049429431181594284608476}{139924580325961928307572689265641807596073145079573813995558801862829031196218449856131074495401190622997539025} a^{5} + \frac{625415078067492627452917267168358004175593456571479100372708342322168452974099580599769095189054775023463654}{139924580325961928307572689265641807596073145079573813995558801862829031196218449856131074495401190622997539025} a^{4} + \frac{6305444886135270306861725358675187879822638559106117888948889043916463303458480843290640342174944853782279096}{27984916065192385661514537853128361519214629015914762799111760372565806239243689971226214899080238124599507805} a^{3} - \frac{64542232282239522764547262547104795729060491964675273245761663781122773062195943398664903818188909611792910586}{139924580325961928307572689265641807596073145079573813995558801862829031196218449856131074495401190622997539025} a^{2} - \frac{69474385433005335440197685732374889283117663608097133261005031246700766690759313767765853521463448852050942456}{139924580325961928307572689265641807596073145079573813995558801862829031196218449856131074495401190622997539025} a + \frac{1354422317286538238258283050544556296930910490307381519984704471325932844352238061895016763590402501491699}{24552479439544117969393347826924338935966510805329674328050324945223553464856720452032123968310438782768475}$

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

Class group and class number

$C_{2}\times C_{4166030410}$, which has order $8332060820$ (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:  \( 860169583.1577827 \) (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

\(\Q(\sqrt{5}) \), 4.0.55125.1, 5.5.5719140625.1, 10.10.163542847442626953125.2

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 $20$ R R R R $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
5Data not computed
7Data not computed
$11$11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$