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Properties

Label 20.0.23389771273...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{15}\cdot 7^{10}\cdot 11^{16}$
Root discriminant $104.34$
Ramified primes $3, 5, 7, 11$
Class number $3116884$ (GRH)
Class group $[2, 1558442]$ (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![258891665341, -60274636071, 157681514772, -35276815488, 45302317311, -9291183528, 7982392620, -1434111949, 948074777, -142439068, 78907466, -9377568, 4703832, -406236, 200941, -11044, 5949, -167, 111, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 111*x^18 - 167*x^17 + 5949*x^16 - 11044*x^15 + 200941*x^14 - 406236*x^13 + 4703832*x^12 - 9377568*x^11 + 78907466*x^10 - 142439068*x^9 + 948074777*x^8 - 1434111949*x^7 + 7982392620*x^6 - 9291183528*x^5 + 45302317311*x^4 - 35276815488*x^3 + 157681514772*x^2 - 60274636071*x + 258891665341)
 
gp: K = bnfinit(x^20 - x^19 + 111*x^18 - 167*x^17 + 5949*x^16 - 11044*x^15 + 200941*x^14 - 406236*x^13 + 4703832*x^12 - 9377568*x^11 + 78907466*x^10 - 142439068*x^9 + 948074777*x^8 - 1434111949*x^7 + 7982392620*x^6 - 9291183528*x^5 + 45302317311*x^4 - 35276815488*x^3 + 157681514772*x^2 - 60274636071*x + 258891665341, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 111 x^{18} - 167 x^{17} + 5949 x^{16} - 11044 x^{15} + 200941 x^{14} - 406236 x^{13} + 4703832 x^{12} - 9377568 x^{11} + 78907466 x^{10} - 142439068 x^{9} + 948074777 x^{8} - 1434111949 x^{7} + 7982392620 x^{6} - 9291183528 x^{5} + 45302317311 x^{4} - 35276815488 x^{3} + 157681514772 x^{2} - 60274636071 x + 258891665341 \)

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:  \(23389771273952384005640608592559814453125=3^{10}\cdot 5^{15}\cdot 7^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $104.34$
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:  \(1155=3\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1155}(64,·)$, $\chi_{1155}(1,·)$, $\chi_{1155}(1028,·)$, $\chi_{1155}(841,·)$, $\chi_{1155}(587,·)$, $\chi_{1155}(526,·)$, $\chi_{1155}(1112,·)$, $\chi_{1155}(1114,·)$, $\chi_{1155}(797,·)$, $\chi_{1155}(608,·)$, $\chi_{1155}(482,·)$, $\chi_{1155}(421,·)$, $\chi_{1155}(169,·)$, $\chi_{1155}(818,·)$, $\chi_{1155}(694,·)$, $\chi_{1155}(713,·)$, $\chi_{1155}(631,·)$, $\chi_{1155}(377,·)$, $\chi_{1155}(379,·)$, $\chi_{1155}(188,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{41} a^{15} + \frac{14}{41} a^{14} + \frac{14}{41} a^{13} - \frac{16}{41} a^{12} + \frac{18}{41} a^{11} - \frac{20}{41} a^{10} - \frac{4}{41} a^{9} - \frac{19}{41} a^{8} + \frac{7}{41} a^{7} - \frac{2}{41} a^{6} + \frac{5}{41} a^{5} - \frac{17}{41} a^{4} + \frac{12}{41} a^{3} + \frac{20}{41} a^{2} + \frac{9}{41} a + \frac{6}{41}$, $\frac{1}{41} a^{16} - \frac{18}{41} a^{14} - \frac{7}{41} a^{13} - \frac{4}{41} a^{12} + \frac{15}{41} a^{11} - \frac{11}{41} a^{10} - \frac{4}{41} a^{9} - \frac{14}{41} a^{8} - \frac{18}{41} a^{7} - \frac{8}{41} a^{6} - \frac{5}{41} a^{5} + \frac{4}{41} a^{4} + \frac{16}{41} a^{3} + \frac{16}{41} a^{2} + \frac{3}{41} a - \frac{2}{41}$, $\frac{1}{41} a^{17} - \frac{1}{41} a^{14} + \frac{2}{41} a^{13} + \frac{14}{41} a^{12} - \frac{15}{41} a^{11} + \frac{5}{41} a^{10} - \frac{4}{41} a^{9} + \frac{9}{41} a^{8} - \frac{5}{41} a^{7} + \frac{12}{41} a^{5} - \frac{3}{41} a^{4} - \frac{14}{41} a^{3} - \frac{6}{41} a^{2} - \frac{4}{41} a - \frac{15}{41}$, $\frac{1}{41} a^{18} + \frac{16}{41} a^{14} - \frac{13}{41} a^{13} + \frac{10}{41} a^{12} - \frac{18}{41} a^{11} + \frac{17}{41} a^{10} + \frac{5}{41} a^{9} + \frac{17}{41} a^{8} + \frac{7}{41} a^{7} + \frac{10}{41} a^{6} + \frac{2}{41} a^{5} + \frac{10}{41} a^{4} + \frac{6}{41} a^{3} + \frac{16}{41} a^{2} - \frac{6}{41} a + \frac{6}{41}$, $\frac{1}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{19} - \frac{341222871557378117439273433516610235767362700134801342895376511600011541649933326957}{31671403019261322864468461593224589874309833708538599074727932714134122528329194795791} a^{18} + \frac{8787076499115659231629689618862314289388641470856397051551871108968356610992114702002}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{17} - \frac{5892774181420813487390880431566096425969906418257851664944093099571547189451780750619}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{16} + \frac{1032307122876941580119330023998087717178123439182424502438620447023821885562239562330}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{15} - \frac{429788033130738845836428625937093472963427625406820082774643342854222846572634355594314}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{14} - \frac{356298152580083649839500824309724826884180787031333321295316044089689173682639748928249}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{13} + \frac{407381292304958398026893248406000997447973551434371579430670342662114596329802412407091}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{12} + \frac{115279059770365098082145240376318407747138981574102808054623272962093796549224820087719}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{11} - \frac{189416185327819952472987814744821271853970302755287194873243535666053015469471894978245}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{10} + \frac{272234865878350028890439895294482184610304847984600846426324297812092215259771777133059}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{9} + \frac{374184649750799908879774884568658556735336819348380040209230525243061375647175283323291}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{8} + \frac{2254961296224618617075215752610162665320140144552773238326572172185767343212584414924}{31671403019261322864468461593224589874309833708538599074727932714134122528329194795791} a^{7} - \frac{485626042928514389704020541962852148581891182985024903796117087721336701454829702483283}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{6} + \frac{42707966270551176527284070010021527006926337081829369313827843719205587995542613413673}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{5} + \frac{167216279293230605960346892587185923513171811273848254222695275195376750224967628600699}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{4} + \frac{624108803627354596836943212017262966316695336628757122499528689854641337597903339641701}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{3} + \frac{268826580221126294881370083180881794072073210016601492461843568288247225332572073417719}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a^{2} - \frac{441224566379937967164412550952285086423312377230992563201687764662328393444921059002162}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431} a + \frac{307809085326485779050968124263758948493575966109152785719327894449607365518279830444184}{1298527523789714237443206925322208184846703182050082562063845241279499023661496986627431}$

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

Class group and class number

$C_{2}\times C_{1558442}$, which has order $3116884$ (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:  \( 140644.5991815038 \) (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, \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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 $20$ R R R R $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$