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Properties

Label 20.0.16082319642...0000.9
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $102.40$
Ramified primes $2, 5, 7, 11$
Class number $2738912$ (GRH)
Class group $[2, 2, 684728]$ (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![180439414849, -105352883318, 146193237687, -91142019246, 70607861373, -34050121102, 18430542913, -6903574292, 2830018544, -833653436, 269460830, -62765934, 16319348, -2976898, 626716, -86628, 14679, -1416, 189, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 189*x^18 - 1416*x^17 + 14679*x^16 - 86628*x^15 + 626716*x^14 - 2976898*x^13 + 16319348*x^12 - 62765934*x^11 + 269460830*x^10 - 833653436*x^9 + 2830018544*x^8 - 6903574292*x^7 + 18430542913*x^6 - 34050121102*x^5 + 70607861373*x^4 - 91142019246*x^3 + 146193237687*x^2 - 105352883318*x + 180439414849)
 
gp: K = bnfinit(x^20 - 10*x^19 + 189*x^18 - 1416*x^17 + 14679*x^16 - 86628*x^15 + 626716*x^14 - 2976898*x^13 + 16319348*x^12 - 62765934*x^11 + 269460830*x^10 - 833653436*x^9 + 2830018544*x^8 - 6903574292*x^7 + 18430542913*x^6 - 34050121102*x^5 + 70607861373*x^4 - 91142019246*x^3 + 146193237687*x^2 - 105352883318*x + 180439414849, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 189 x^{18} - 1416 x^{17} + 14679 x^{16} - 86628 x^{15} + 626716 x^{14} - 2976898 x^{13} + 16319348 x^{12} - 62765934 x^{11} + 269460830 x^{10} - 833653436 x^{9} + 2830018544 x^{8} - 6903574292 x^{7} + 18430542913 x^{6} - 34050121102 x^{5} + 70607861373 x^{4} - 91142019246 x^{3} + 146193237687 x^{2} - 105352883318 x + 180439414849 \)

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:  \(16082319642092842466929804994560000000000=2^{20}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $102.40$
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:  $2, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1540=2^{2}\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1540}(1,·)$, $\chi_{1540}(1539,·)$, $\chi_{1540}(391,·)$, $\chi_{1540}(841,·)$, $\chi_{1540}(139,·)$, $\chi_{1540}(141,·)$, $\chi_{1540}(1231,·)$, $\chi_{1540}(1091,·)$, $\chi_{1540}(729,·)$, $\chi_{1540}(1371,·)$, $\chi_{1540}(1119,·)$, $\chi_{1540}(421,·)$, $\chi_{1540}(169,·)$, $\chi_{1540}(811,·)$, $\chi_{1540}(449,·)$, $\chi_{1540}(309,·)$, $\chi_{1540}(1399,·)$, $\chi_{1540}(1401,·)$, $\chi_{1540}(699,·)$, $\chi_{1540}(1149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{5} + \frac{2}{7} a^{3} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{6} - \frac{3}{7} a^{3} - \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{7} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{49} a^{8} + \frac{3}{49} a^{7} + \frac{2}{49} a^{6} + \frac{1}{49} a^{5} + \frac{2}{49} a^{4} + \frac{20}{49} a^{3} + \frac{9}{49} a^{2} + \frac{11}{49} a + \frac{15}{49}$, $\frac{1}{49} a^{9} + \frac{2}{49} a^{6} - \frac{1}{49} a^{5} + \frac{12}{49} a^{3} + \frac{12}{49} a^{2} + \frac{17}{49} a - \frac{3}{49}$, $\frac{1}{49} a^{10} + \frac{2}{49} a^{7} - \frac{1}{49} a^{6} - \frac{2}{49} a^{4} - \frac{9}{49} a^{3} - \frac{18}{49} a^{2} + \frac{18}{49} a - \frac{2}{7}$, $\frac{1}{49} a^{11} + \frac{3}{49} a^{6} + \frac{3}{49} a^{5} + \frac{1}{49} a^{4} + \frac{12}{49} a^{3} - \frac{8}{49} a + \frac{5}{49}$, $\frac{1}{343} a^{12} + \frac{1}{343} a^{11} + \frac{2}{343} a^{10} + \frac{3}{343} a^{9} - \frac{2}{343} a^{8} + \frac{15}{343} a^{7} + \frac{13}{343} a^{6} + \frac{13}{343} a^{5} - \frac{2}{343} a^{4} - \frac{171}{343} a^{3} - \frac{89}{343} a^{2} + \frac{13}{343} a + \frac{134}{343}$, $\frac{1}{343} a^{13} + \frac{1}{343} a^{11} + \frac{1}{343} a^{10} + \frac{2}{343} a^{9} + \frac{3}{343} a^{8} + \frac{5}{343} a^{7} - \frac{2}{49} a^{6} + \frac{13}{343} a^{5} - \frac{1}{343} a^{4} - \frac{16}{343} a^{3} - \frac{38}{343} a^{2} - \frac{110}{343} a + \frac{76}{343}$, $\frac{1}{343} a^{14} - \frac{1}{343} a^{7} - \frac{2}{49} a^{6} - \frac{3}{49} a^{5} + \frac{3}{49} a^{4} - \frac{8}{49} a^{3} - \frac{5}{49} a^{2} - \frac{9}{49} a - \frac{43}{343}$, $\frac{1}{343} a^{15} - \frac{1}{343} a^{8} - \frac{2}{49} a^{7} - \frac{3}{49} a^{6} + \frac{3}{49} a^{5} - \frac{1}{49} a^{4} - \frac{19}{49} a^{3} - \frac{16}{49} a^{2} + \frac{55}{343} a + \frac{1}{7}$, $\frac{1}{2401} a^{16} - \frac{1}{2401} a^{15} - \frac{1}{2401} a^{14} - \frac{3}{343} a^{11} + \frac{1}{343} a^{10} - \frac{8}{2401} a^{9} + \frac{22}{2401} a^{8} + \frac{113}{2401} a^{7} - \frac{1}{343} a^{6} + \frac{3}{343} a^{5} - \frac{16}{343} a^{4} - \frac{156}{343} a^{3} - \frac{379}{2401} a^{2} + \frac{274}{2401} a - \frac{839}{2401}$, $\frac{1}{2401} a^{17} - \frac{2}{2401} a^{15} - \frac{1}{2401} a^{14} + \frac{1}{343} a^{11} - \frac{8}{2401} a^{10} - \frac{3}{343} a^{9} - \frac{5}{2401} a^{8} + \frac{29}{2401} a^{7} - \frac{8}{343} a^{6} - \frac{23}{343} a^{5} + \frac{4}{343} a^{4} + \frac{818}{2401} a^{3} + \frac{40}{343} a^{2} - \frac{488}{2401} a - \frac{230}{2401}$, $\frac{1}{2266458277858949053687937217641} a^{18} - \frac{9}{2266458277858949053687937217641} a^{17} + \frac{37493094662843899416889849}{2266458277858949053687937217641} a^{16} - \frac{299944757302751195335118588}{2266458277858949053687937217641} a^{15} + \frac{30673619356399347311379533}{2266458277858949053687937217641} a^{14} - \frac{32110860488123986230946058}{46254250568549980687508922809} a^{13} - \frac{9512412605208582158235123}{14077380607819559339676628681} a^{12} + \frac{695483576099621313313907594}{98541664254736915377736400767} a^{11} + \frac{5543601922374095898440910881}{2266458277858949053687937217641} a^{10} - \frac{17519844618362698025127693276}{2266458277858949053687937217641} a^{9} - \frac{2600409905558749001062590012}{2266458277858949053687937217641} a^{8} - \frac{51233195584797330247017019247}{2266458277858949053687937217641} a^{7} - \frac{17334484177427297872859079703}{323779753979849864812562459663} a^{6} + \frac{9698717827261706762074191782}{323779753979849864812562459663} a^{5} - \frac{133758700946953353449808797112}{2266458277858949053687937217641} a^{4} - \frac{927082669389068048907347287128}{2266458277858949053687937217641} a^{3} - \frac{631541891233923844091990980645}{2266458277858949053687937217641} a^{2} - \frac{705353220189122442543396248190}{2266458277858949053687937217641} a - \frac{557363812685770772552324131766}{2266458277858949053687937217641}$, $\frac{1}{918164171811581991263179690283075341393} a^{19} + \frac{354737}{1607993295641999984699088774576314083} a^{18} + \frac{28398288979453257643489759775926436}{918164171811581991263179690283075341393} a^{17} - \frac{42586514761088803334241640564275671}{918164171811581991263179690283075341393} a^{16} + \frac{1524550117297602445740075619918222}{4613890310610964780216983368256660007} a^{15} + \frac{1146844037310536144044872171112126223}{918164171811581991263179690283075341393} a^{14} - \frac{5430005874530875866124411185106875}{5702883054730322927100494970702331313} a^{13} - \frac{13599309866822980056303388737761233}{39920181383112260489703464794916319191} a^{12} - \frac{7902214438750174016812004148055252111}{918164171811581991263179690283075341393} a^{11} + \frac{4420234071258299115981252597540682892}{918164171811581991263179690283075341393} a^{10} - \frac{7872440992087341269171783110738148411}{918164171811581991263179690283075341393} a^{9} + \frac{2833735353657978092682240588173926469}{918164171811581991263179690283075341393} a^{8} - \frac{32694072953158105957774525031768305704}{918164171811581991263179690283075341393} a^{7} - \frac{7593961061873986893952042497830458241}{131166310258797427323311384326153620199} a^{6} + \frac{22694688918345313097082073347923322667}{918164171811581991263179690283075341393} a^{5} - \frac{57458698541381686243988790166301512005}{918164171811581991263179690283075341393} a^{4} + \frac{224144419242317484752444249402254611582}{918164171811581991263179690283075341393} a^{3} + \frac{355351376435025643271849796429584938927}{918164171811581991263179690283075341393} a^{2} - \frac{251612712207911821778445393594045815240}{918164171811581991263179690283075341393} a + \frac{15680236753495372576503324150445299734}{39920181383112260489703464794916319191}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{684728}$, which has order $2738912$ (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

$C_2\times C_{10}$ (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-77}) \), \(\Q(\sqrt{-385}) \), \(\Q(\sqrt{5}, \sqrt{-77})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.40581147486860288.1, 10.0.126816085896438400000.1

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 R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
5Data not computed
7Data not computed
$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}$