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Properties

Label 20.0.44322362984...000.18
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 31^{18}$
Root discriminant $340.67$
Ramified primes $2, 3, 5, 31$
Class number $2682273792$ (GRH)
Class group $[2, 2, 2, 4, 4, 44, 44, 10824]$ (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![239981769909025, 0, 9957442719440, 0, -4378808955609, 0, -192927061440, 0, 47796205399, 0, 5504137940, 0, 250897376, 0, 5921760, 0, 74069, 0, 440, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 440*x^18 + 74069*x^16 + 5921760*x^14 + 250897376*x^12 + 5504137940*x^10 + 47796205399*x^8 - 192927061440*x^6 - 4378808955609*x^4 + 9957442719440*x^2 + 239981769909025)
 
gp: K = bnfinit(x^20 + 440*x^18 + 74069*x^16 + 5921760*x^14 + 250897376*x^12 + 5504137940*x^10 + 47796205399*x^8 - 192927061440*x^6 - 4378808955609*x^4 + 9957442719440*x^2 + 239981769909025, 1)
 

Normalized defining polynomial

\( x^{20} + 440 x^{18} + 74069 x^{16} + 5921760 x^{14} + 250897376 x^{12} + 5504137940 x^{10} + 47796205399 x^{8} - 192927061440 x^{6} - 4378808955609 x^{4} + 9957442719440 x^{2} + 239981769909025 \)

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:  \(443223629840246396758117587996045905756160000000000=2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $340.67$
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:  $2, 3, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3720=2^{3}\cdot 3\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3720}(1,·)$, $\chi_{3720}(2819,·)$, $\chi_{3720}(901,·)$, $\chi_{3720}(3719,·)$, $\chi_{3720}(841,·)$, $\chi_{3720}(2999,·)$, $\chi_{3720}(2701,·)$, $\chi_{3720}(721,·)$, $\chi_{3720}(1859,·)$, $\chi_{3720}(2581,·)$, $\chi_{3720}(959,·)$, $\chi_{3720}(1861,·)$, $\chi_{3720}(481,·)$, $\chi_{3720}(1379,·)$, $\chi_{3720}(2341,·)$, $\chi_{3720}(3239,·)$, $\chi_{3720}(1139,·)$, $\chi_{3720}(2761,·)$, $\chi_{3720}(1019,·)$, $\chi_{3720}(2879,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{9} + \frac{1}{5} a^{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{6} - \frac{2}{5} a^{2}$, $\frac{1}{185} a^{15} - \frac{14}{185} a^{13} - \frac{1}{185} a^{11} - \frac{63}{185} a^{9} - \frac{1}{37} a^{7} + \frac{66}{185} a^{5} - \frac{13}{185} a^{3} + \frac{1}{37} a$, $\frac{1}{12395} a^{16} - \frac{62}{2479} a^{14} + \frac{1146}{12395} a^{12} + \frac{165}{2479} a^{10} + \frac{3954}{12395} a^{8} + \frac{546}{2479} a^{6} + \frac{2059}{12395} a^{4} - \frac{480}{2479} a^{2} - \frac{26}{67}$, $\frac{1}{12395} a^{17} + \frac{5}{2479} a^{15} - \frac{213}{2479} a^{13} + \frac{98}{2479} a^{11} + \frac{202}{12395} a^{9} + \frac{211}{2479} a^{7} + \frac{1858}{12395} a^{5} + \frac{1128}{2479} a^{3} - \frac{627}{2479} a$, $\frac{1}{635719622153346885241495545380929848315711572118319791995} a^{18} - \frac{3889635906625314946338297585429354311579134910656844}{127143924430669377048299109076185969663142314423663958399} a^{16} + \frac{36217206759968217837367111900916876423581209744020633714}{635719622153346885241495545380929848315711572118319791995} a^{14} - \frac{20845729608174773145679174044212503707063775605209653623}{635719622153346885241495545380929848315711572118319791995} a^{12} + \frac{1455734332233801515183767354517030625129012364076251812}{635719622153346885241495545380929848315711572118319791995} a^{10} + \frac{132538414506359282542231652736266779076884162104890386279}{635719622153346885241495545380929848315711572118319791995} a^{8} + \frac{175765927036331614319230374515689394863626139364847576376}{635719622153346885241495545380929848315711572118319791995} a^{6} + \frac{170454072561253397267403275958327291007803027166200730202}{635719622153346885241495545380929848315711572118319791995} a^{4} + \frac{265406227982640841590736477428646802006891966830039821347}{635719622153346885241495545380929848315711572118319791995} a^{2} + \frac{12803434190134934446591704810039732428944985150364447}{92873575186756301715339013203934236423040404984414871}$, $\frac{1}{266166270001274040647335562417814613542063699572358722111426575} a^{19} + \frac{857538029073569770105610783023039585880758811007752265476}{53233254000254808129467112483562922708412739914471744422285315} a^{17} - \frac{249856472879231382786322774335233524801501636328775552914991}{266166270001274040647335562417814613542063699572358722111426575} a^{15} + \frac{3303472180807716701581155996853132401139402821559621051583308}{53233254000254808129467112483562922708412739914471744422285315} a^{13} + \frac{13076616579249008996579358306221845815325850132574959824425401}{266166270001274040647335562417814613542063699572358722111426575} a^{11} - \frac{22757910724917657697235348116094213019658851768066138491913558}{53233254000254808129467112483562922708412739914471744422285315} a^{9} - \frac{85565444688632080043615674162377334748824220813809173373984696}{266166270001274040647335562417814613542063699572358722111426575} a^{7} - \frac{2779384124185036294087723573244484088436430010172162368761626}{53233254000254808129467112483562922708412739914471744422285315} a^{5} - \frac{955100452993100472447021805582765090378836347595760837898864}{266166270001274040647335562417814613542063699572358722111426575} a^{3} - \frac{99924366859705117083770275838730171011995323444719764922628}{1438736594601481300796408445501700613740884862553290389791495} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{44}\times C_{44}\times C_{10824}$, which has order $2682273792$ (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:  \( 24785765.76033278 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-465}) \), \(\Q(\sqrt{-930}) \), \(\Q(\sqrt{2}, \sqrt{-465})\), 5.5.923521.1, 10.10.27947533514866688.1, 10.0.20559450192137769600000.1, 10.0.657902406148408627200000.1

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 R R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$31.10.9.1$x^{10} - 31$$10$$1$$9$$C_{10}$$[\ ]_{10}$
31.10.9.1$x^{10} - 31$$10$$1$$9$$C_{10}$$[\ ]_{10}$