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Properties

Label 20.0.34906338751...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{18}\cdot 19^{10}$
Root discriminant $168.71$
Ramified primes $2, 5, 11, 19$
Class number $12576080$ (GRH)
Class group $[2, 2, 3144020]$ (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![316393965625, 970468750, 6443371875, -14154588750, 5527504125, 1499675750, 64802725, -416447300, 26828720, 40780300, 1816046, -4500870, 28580, 269870, 4060, -14772, 495, 420, -15, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 - 15*x^18 + 420*x^17 + 495*x^16 - 14772*x^15 + 4060*x^14 + 269870*x^13 + 28580*x^12 - 4500870*x^11 + 1816046*x^10 + 40780300*x^9 + 26828720*x^8 - 416447300*x^7 + 64802725*x^6 + 1499675750*x^5 + 5527504125*x^4 - 14154588750*x^3 + 6443371875*x^2 + 970468750*x + 316393965625)
 
gp: K = bnfinit(x^20 - 10*x^19 - 15*x^18 + 420*x^17 + 495*x^16 - 14772*x^15 + 4060*x^14 + 269870*x^13 + 28580*x^12 - 4500870*x^11 + 1816046*x^10 + 40780300*x^9 + 26828720*x^8 - 416447300*x^7 + 64802725*x^6 + 1499675750*x^5 + 5527504125*x^4 - 14154588750*x^3 + 6443371875*x^2 + 970468750*x + 316393965625, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} - 15 x^{18} + 420 x^{17} + 495 x^{16} - 14772 x^{15} + 4060 x^{14} + 269870 x^{13} + 28580 x^{12} - 4500870 x^{11} + 1816046 x^{10} + 40780300 x^{9} + 26828720 x^{8} - 416447300 x^{7} + 64802725 x^{6} + 1499675750 x^{5} + 5527504125 x^{4} - 14154588750 x^{3} + 6443371875 x^{2} + 970468750 x + 316393965625 \)

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:  \(349063387514017843536891023700797440000000000=2^{20}\cdot 5^{10}\cdot 11^{18}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $168.71$
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:  $2, 5, 11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4180=2^{2}\cdot 5\cdot 11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4180}(1,·)$, $\chi_{4180}(3079,·)$, $\chi_{4180}(1101,·)$, $\chi_{4180}(4141,·)$, $\chi_{4180}(4179,·)$, $\chi_{4180}(1559,·)$, $\chi_{4180}(1179,·)$, $\chi_{4180}(2279,·)$, $\chi_{4180}(3041,·)$, $\chi_{4180}(2659,·)$, $\chi_{4180}(39,·)$, $\chi_{4180}(2281,·)$, $\chi_{4180}(1899,·)$, $\chi_{4180}(1901,·)$, $\chi_{4180}(1521,·)$, $\chi_{4180}(1139,·)$, $\chi_{4180}(3381,·)$, $\chi_{4180}(3001,·)$, $\chi_{4180}(799,·)$, $\chi_{4180}(2621,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{6} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{8} + \frac{1}{25} a^{7} + \frac{1}{25} a^{6} + \frac{1}{25} a^{5} + \frac{1}{25} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{10} - \frac{1}{25} a^{5} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{11} - \frac{1}{25} a^{6} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{125} a^{12} - \frac{1}{125} a^{11} + \frac{4}{125} a^{7} + \frac{6}{125} a^{6} - \frac{2}{25} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{125} a^{13} - \frac{1}{125} a^{11} - \frac{1}{125} a^{8} + \frac{1}{25} a^{7} + \frac{1}{125} a^{6} + \frac{2}{25} a^{5} + \frac{2}{25} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{125} a^{14} - \frac{1}{125} a^{11} - \frac{1}{125} a^{9} + \frac{11}{125} a^{6} + \frac{1}{25} a^{5} + \frac{2}{25} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{125} a^{15} - \frac{1}{125} a^{11} - \frac{1}{125} a^{10} - \frac{2}{25} a^{7} + \frac{11}{125} a^{6} + \frac{2}{25} a^{5} - \frac{2}{25} a^{4}$, $\frac{1}{625} a^{16} + \frac{2}{625} a^{15} - \frac{2}{625} a^{14} - \frac{1}{625} a^{13} + \frac{4}{625} a^{11} - \frac{7}{625} a^{10} - \frac{3}{625} a^{9} - \frac{4}{625} a^{8} - \frac{11}{125} a^{7} - \frac{12}{125} a^{6} - \frac{2}{25} a^{5} + \frac{2}{25} a^{4} + \frac{1}{5} a^{2}$, $\frac{1}{14375} a^{17} + \frac{3}{14375} a^{16} + \frac{2}{2875} a^{15} - \frac{43}{14375} a^{14} + \frac{29}{14375} a^{13} - \frac{6}{14375} a^{12} - \frac{193}{14375} a^{11} + \frac{6}{2875} a^{10} + \frac{133}{14375} a^{9} - \frac{264}{14375} a^{8} - \frac{181}{2875} a^{7} - \frac{214}{2875} a^{6} + \frac{54}{575} a^{5} + \frac{42}{575} a^{4} - \frac{2}{5} a^{3} + \frac{6}{23} a^{2} + \frac{3}{23} a$, $\frac{1}{6404930524829445714402480302350625} a^{18} - \frac{9}{6404930524829445714402480302350625} a^{17} - \frac{151374005250270264835407406599}{256197220993177828576099212094025} a^{16} - \frac{20964643148581512748138361098801}{6404930524829445714402480302350625} a^{15} + \frac{18628076360291941112367952953123}{6404930524829445714402480302350625} a^{14} + \frac{1181444336854567917471271534364}{1280986104965889142880496060470125} a^{13} + \frac{5343704473035396735047717432814}{6404930524829445714402480302350625} a^{12} + \frac{156765430391447136532577513217}{11139009608399036025047791830175} a^{11} - \frac{80670232429808310590451021143094}{6404930524829445714402480302350625} a^{10} - \frac{38393413264709674672730714515018}{6404930524829445714402480302350625} a^{9} - \frac{70686954748659071766704391823766}{6404930524829445714402480302350625} a^{8} + \frac{15443289071941120277513296974746}{256197220993177828576099212094025} a^{7} - \frac{31031528159821956325650457938424}{1280986104965889142880496060470125} a^{6} + \frac{3159643446390397634176857626538}{256197220993177828576099212094025} a^{5} + \frac{3255921837283108419048182509791}{256197220993177828576099212094025} a^{4} - \frac{9592045435549525463652694218573}{51239444198635565715219842418805} a^{3} + \frac{4436594340067155368673641362}{445560384335961441001911673207} a^{2} + \frac{1341434218310883989359364192394}{10247888839727113143043968483761} a + \frac{38295508466738966740010801390}{445560384335961441001911673207}$, $\frac{1}{160580274128474298300121767735779249145625} a^{19} + \frac{545029}{6981751049064099926092250771120836919375} a^{18} - \frac{3563312361864713190413433231783412407}{160580274128474298300121767735779249145625} a^{17} - \frac{17151233932317177320135642401525704138}{160580274128474298300121767735779249145625} a^{16} + \frac{345053097030138907564853310785234456861}{160580274128474298300121767735779249145625} a^{15} - \frac{113782449683091463445999180822965943412}{160580274128474298300121767735779249145625} a^{14} - \frac{119166174616693109833189950693089514848}{32116054825694859660024353547155849829125} a^{13} + \frac{285434304301326808782453076729438948177}{160580274128474298300121767735779249145625} a^{12} + \frac{1239687984665056795881451272880677641798}{160580274128474298300121767735779249145625} a^{11} - \frac{63861033106170127325833093255638822622}{6981751049064099926092250771120836919375} a^{10} + \frac{1890269674309168045946324232492637817941}{160580274128474298300121767735779249145625} a^{9} + \frac{2793046989426765697135001402340197012328}{160580274128474298300121767735779249145625} a^{8} - \frac{3192834222022306186425141964930151658053}{32116054825694859660024353547155849829125} a^{7} - \frac{2924219804349159371195834460037996911716}{32116054825694859660024353547155849829125} a^{6} + \frac{7080755028124130866220227914987295037}{6423210965138971932004870709431169965825} a^{5} + \frac{261737265121114935524773025678936441696}{6423210965138971932004870709431169965825} a^{4} - \frac{290998825664513950704022648308443758117}{1284642193027794386400974141886233993165} a^{3} + \frac{50167148110746693699108698174204686853}{256928438605558877280194828377246798633} a^{2} - \frac{86884604346271044341946548578047344151}{256928438605558877280194828377246798633} a - \frac{2363298209738324558853700762930296691}{11170801678502559881747601233793339071}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{3144020}$, which has order $12576080$ (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:  \( 11184526.893275889 \) (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

\(\Q(\sqrt{-1045}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-19}, \sqrt{55})\), \(\Q(\zeta_{11})^+\), 10.0.18683238143159708800000.1, 10.10.7545432611200000.1, 10.0.530773810885219.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$19$19.10.5.2$x^{10} - 130321 x^{2} + 12380495$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
19.10.5.2$x^{10} - 130321 x^{2} + 12380495$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$