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Properties

Label 20.0.11238525471...0000.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{10}\cdot 41^{18}$
Root discriminant $178.87$
Ramified primes $2, 5, 41$
Class number $60213680$ (GRH)
Class group $[2, 2, 15053420]$ (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![156709628451, -10933110054, 82545355413, -6430154154, 20342958970, -1566380644, 3115413077, -225430718, 332045847, -22501622, 26178289, -1687702, 1573687, -96362, 72661, -4024, 2550, -118, 65, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 65*x^18 - 118*x^17 + 2550*x^16 - 4024*x^15 + 72661*x^14 - 96362*x^13 + 1573687*x^12 - 1687702*x^11 + 26178289*x^10 - 22501622*x^9 + 332045847*x^8 - 225430718*x^7 + 3115413077*x^6 - 1566380644*x^5 + 20342958970*x^4 - 6430154154*x^3 + 82545355413*x^2 - 10933110054*x + 156709628451)
 
gp: K = bnfinit(x^20 - 2*x^19 + 65*x^18 - 118*x^17 + 2550*x^16 - 4024*x^15 + 72661*x^14 - 96362*x^13 + 1573687*x^12 - 1687702*x^11 + 26178289*x^10 - 22501622*x^9 + 332045847*x^8 - 225430718*x^7 + 3115413077*x^6 - 1566380644*x^5 + 20342958970*x^4 - 6430154154*x^3 + 82545355413*x^2 - 10933110054*x + 156709628451, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 65 x^{18} - 118 x^{17} + 2550 x^{16} - 4024 x^{15} + 72661 x^{14} - 96362 x^{13} + 1573687 x^{12} - 1687702 x^{11} + 26178289 x^{10} - 22501622 x^{9} + 332045847 x^{8} - 225430718 x^{7} + 3115413077 x^{6} - 1566380644 x^{5} + 20342958970 x^{4} - 6430154154 x^{3} + 82545355413 x^{2} - 10933110054 x + 156709628451 \)

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:  \(1123852547182106083132474859412520960000000000=2^{30}\cdot 5^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $178.87$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1640=2^{3}\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{1640}(1,·)$, $\chi_{1640}(939,·)$, $\chi_{1640}(1521,·)$, $\chi_{1640}(961,·)$, $\chi_{1640}(201,·)$, $\chi_{1640}(139,·)$, $\chi_{1640}(1281,·)$, $\chi_{1640}(81,·)$, $\chi_{1640}(441,·)$, $\chi_{1640}(1179,·)$, $\chi_{1640}(1499,·)$, $\chi_{1640}(739,·)$, $\chi_{1640}(1041,·)$, $\chi_{1640}(681,·)$, $\chi_{1640}(619,·)$, $\chi_{1640}(59,·)$, $\chi_{1640}(819,·)$, $\chi_{1640}(761,·)$, $\chi_{1640}(379,·)$, $\chi_{1640}(1419,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{10} - \frac{4}{9} a^{7} + \frac{2}{9} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{3} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{4}{9} a^{8} - \frac{1}{3} a^{7} - \frac{4}{9} a^{5} + \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} - \frac{4}{9} a^{6} + \frac{2}{9} a^{5} + \frac{2}{9} a^{4} + \frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{2241} a^{18} + \frac{43}{2241} a^{17} - \frac{31}{2241} a^{16} + \frac{47}{2241} a^{15} + \frac{49}{747} a^{14} + \frac{11}{2241} a^{13} - \frac{305}{2241} a^{12} - \frac{209}{2241} a^{11} - \frac{242}{2241} a^{10} + \frac{20}{2241} a^{9} + \frac{1048}{2241} a^{8} + \frac{76}{2241} a^{7} - \frac{18}{83} a^{6} + \frac{253}{2241} a^{5} + \frac{485}{2241} a^{4} + \frac{176}{2241} a^{3} - \frac{437}{2241} a^{2} - \frac{316}{747} a - \frac{67}{249}$, $\frac{1}{11019562701911506722448158399026384210369941708642350608608087024628610681} a^{19} - \frac{1926854803323229827254122880628672825254198207756102612131713358413326}{11019562701911506722448158399026384210369941708642350608608087024628610681} a^{18} + \frac{511570420482285941461830587254358325986632760925644822820449335443805361}{11019562701911506722448158399026384210369941708642350608608087024628610681} a^{17} - \frac{388990723237366625322313667035785011140252989483324291766667003381712421}{11019562701911506722448158399026384210369941708642350608608087024628610681} a^{16} + \frac{43747667483764084260890995508028130534858292204888519020496045713338229}{3673187567303835574149386133008794736789980569547450202869362341542870227} a^{15} - \frac{152766819866685032169448662317487671539816279929878703821615416933830091}{11019562701911506722448158399026384210369941708642350608608087024628610681} a^{14} + \frac{1410366912383596064014447886311497598138252407543172835496064105657473754}{11019562701911506722448158399026384210369941708642350608608087024628610681} a^{13} - \frac{1538508227140056250810819516721981142303025923199096001984220798038254934}{11019562701911506722448158399026384210369941708642350608608087024628610681} a^{12} - \frac{438049841888204105374105310277317355081419772369070699430465994089974751}{11019562701911506722448158399026384210369941708642350608608087024628610681} a^{11} + \frac{247222077215853038124405641277413125124209486313032408417173018442955378}{11019562701911506722448158399026384210369941708642350608608087024628610681} a^{10} + \frac{517784017816327220389861933047649681318586955834412427688448793322269224}{11019562701911506722448158399026384210369941708642350608608087024628610681} a^{9} + \frac{5310942133634194571392712984629432207051712772086249127209442561240582721}{11019562701911506722448158399026384210369941708642350608608087024628610681} a^{8} + \frac{47857722348118711282327804661853205872811669421317100479402562694698580}{136043983974216132375903190111436842103332613686942600106272679316402601} a^{7} - \frac{4428288317371974475505231965734819578497089961534616302621038555203589488}{11019562701911506722448158399026384210369941708642350608608087024628610681} a^{6} - \frac{4223520665567445473663808173095191961065164549931516951242586630402023569}{11019562701911506722448158399026384210369941708642350608608087024628610681} a^{5} - \frac{5395301060613528550031385732261866261649332808739765450969443247912211383}{11019562701911506722448158399026384210369941708642350608608087024628610681} a^{4} + \frac{3773037680716100811696288262774075917991569692228423833586150215857787411}{11019562701911506722448158399026384210369941708642350608608087024628610681} a^{3} - \frac{899655620192852749833426203518537656906808208800677781043609155564600070}{3673187567303835574149386133008794736789980569547450202869362341542870227} a^{2} + \frac{113550183167200321089150386723036891256995393686284516732395302303979630}{408131951922648397127709570334310526309997841060827800318818037949207803} a - \frac{74301128398716467994227616041828790945645557830402597174424008625100695}{408131951922648397127709570334310526309997841060827800318818037949207803}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{15053420}$, which has order $60213680$ (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:  \( 5104264.636551031 \) (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

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{-410}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-10}, \sqrt{41})\), 5.5.2825761.1, 10.10.327381934393961.1, 10.0.33523910081941606400000.1, 10.0.817656343461990400000.2

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 R ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$41$41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.1$x^{10} - 41$$10$$1$$9$$C_{10}$$[\ ]_{10}$