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Properties

Label 20.0.44322362984...0000.8
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 $200228864$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 2, 44, 8888]$ (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![1042471630225, 0, 541362815880, 0, 117549415351, 0, 9912462280, 0, 46016871, 0, -35281220, 0, -1396144, 0, -36520, 0, 3221, 0, 80, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 80*x^18 + 3221*x^16 - 36520*x^14 - 1396144*x^12 - 35281220*x^10 + 46016871*x^8 + 9912462280*x^6 + 117549415351*x^4 + 541362815880*x^2 + 1042471630225)
 
gp: K = bnfinit(x^20 + 80*x^18 + 3221*x^16 - 36520*x^14 - 1396144*x^12 - 35281220*x^10 + 46016871*x^8 + 9912462280*x^6 + 117549415351*x^4 + 541362815880*x^2 + 1042471630225, 1)
 

Normalized defining polynomial

\( x^{20} + 80 x^{18} + 3221 x^{16} - 36520 x^{14} - 1396144 x^{12} - 35281220 x^{10} + 46016871 x^{8} + 9912462280 x^{6} + 117549415351 x^{4} + 541362815880 x^{2} + 1042471630225 \)

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$$\chi_{3720}(1,·)$, $\chi_{3720}(3011,·)$, $\chi_{3720}(709,·)$, $\chi_{3720}(3719,·)$, $\chi_{3720}(841,·)$, $\chi_{3720}(2891,·)$, $\chi_{3720}(1549,·)$, $\chi_{3720}(2509,·)$, $\chi_{3720}(721,·)$, $\chi_{3720}(2999,·)$, $\chi_{3720}(959,·)$, $\chi_{3720}(2651,·)$, $\chi_{3720}(481,·)$, $\chi_{3720}(1211,·)$, $\chi_{3720}(3239,·)$, $\chi_{3720}(1069,·)$, $\chi_{3720}(2761,·)$, $\chi_{3720}(2171,·)$, $\chi_{3720}(829,·)$, $\chi_{3720}(2879,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{9} + \frac{1}{5} a^{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{6} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{7} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{8} + \frac{2}{5} a^{4}$, $\frac{1}{185} a^{17} - \frac{11}{185} a^{15} + \frac{4}{185} a^{13} + \frac{1}{37} a^{11} - \frac{71}{185} a^{9} - \frac{32}{185} a^{7} + \frac{71}{185} a^{5} - \frac{62}{185} a^{3} - \frac{6}{37} a$, $\frac{1}{1822478021047919430853927219804692700998964118710805} a^{18} - \frac{18705411964705666531485207273384054154737567854621}{1822478021047919430853927219804692700998964118710805} a^{16} + \frac{54014665221785544963600334769352049082364854505249}{1822478021047919430853927219804692700998964118710805} a^{14} - \frac{66520060846261525660776574217153504825021991703}{27201164493252528818715331638876010462671106249415} a^{12} - \frac{14309769166539922314893537794394520691058810126207}{1822478021047919430853927219804692700998964118710805} a^{10} - \frac{167700323229726088894134623197414178563355019906180}{364495604209583886170785443960938540199792823742161} a^{8} - \frac{354109735349180812838047801900523900817453053110902}{1822478021047919430853927219804692700998964118710805} a^{6} + \frac{720414107601045184233983318864941283644258768470652}{1822478021047919430853927219804692700998964118710805} a^{4} - \frac{803562825429855712499307244971411373676946686391071}{1822478021047919430853927219804692700998964118710805} a^{2} - \frac{4807530540871566319626996232211475824611615333718}{9851232546204969896507714701646987572967373614653}$, $\frac{1}{50291280990817336694414121630510495084066414855824663975} a^{19} + \frac{4955369381366640712595686539354416733400828364045396}{10058256198163467338882824326102099016813282971164932795} a^{17} - \frac{4446900720249709880407480400850398955251692725909620134}{50291280990817336694414121630510495084066414855824663975} a^{15} + \frac{5803744778925458415499511393779045409227353775706571}{150123226838260706550489915314956701743481835390521385} a^{13} + \frac{4989250776814348718832878801973040578671937949440646826}{50291280990817336694414121630510495084066414855824663975} a^{11} - \frac{4996594632967835997958848938722129747336739643888531944}{10058256198163467338882824326102099016813282971164932795} a^{9} + \frac{13077068434258832512072084649202160656325214314036214721}{50291280990817336694414121630510495084066414855824663975} a^{7} + \frac{4591106105343385365930419872900557289296655603491539998}{10058256198163467338882824326102099016813282971164932795} a^{5} + \frac{4395496134336915855032102024944328085826899908951188586}{50291280990817336694414121630510495084066414855824663975} a^{3} + \frac{4516592541339874039655167976415148683624884237803823628}{10058256198163467338882824326102099016813282971164932795} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{44}\times C_{8888}$, which has order $200228864$ (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:  \( 3425669156.8503156 \) (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

\(\Q(\sqrt{-310}) \), \(\Q(\sqrt{-465}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{6}, \sqrt{-310})\), 5.5.923521.1, 10.0.2707417309252710400000.1, 10.0.20559450192137769600000.1, 10.10.6791250644112605184.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ 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$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
31Data not computed