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Properties

Label 20.0.12843535723...0000.5
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{10}\cdot 101^{18}$
Root discriminant $569.42$
Ramified primes $2, 5, 101$
Class number $1304901752$ (GRH)
Class group $[11, 154, 770308]$ (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![13016444524561, 0, -4338353882334, 0, 655617306707, 0, -62349833294, 0, 4263113739, 0, -222386744, 0, 8216560, 0, -189690, 0, 3793, 0, 102, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 102*x^18 + 3793*x^16 - 189690*x^14 + 8216560*x^12 - 222386744*x^10 + 4263113739*x^8 - 62349833294*x^6 + 655617306707*x^4 - 4338353882334*x^2 + 13016444524561)
 
gp: K = bnfinit(x^20 + 102*x^18 + 3793*x^16 - 189690*x^14 + 8216560*x^12 - 222386744*x^10 + 4263113739*x^8 - 62349833294*x^6 + 655617306707*x^4 - 4338353882334*x^2 + 13016444524561, 1)
 

Normalized defining polynomial

\( x^{20} + 102 x^{18} + 3793 x^{16} - 189690 x^{14} + 8216560 x^{12} - 222386744 x^{10} + 4263113739 x^{8} - 62349833294 x^{6} + 655617306707 x^{4} - 4338353882334 x^{2} + 13016444524561 \)

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:  \(12843535723167951800917506774806325511837450240000000000=2^{40}\cdot 5^{10}\cdot 101^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $569.42$
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:  $2, 5, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4040=2^{3}\cdot 5\cdot 101\)
Dirichlet character group:    $\lbrace$$\chi_{4040}(1,·)$, $\chi_{4040}(3499,·)$, $\chi_{4040}(2309,·)$, $\chi_{4040}(2561,·)$, $\chi_{4040}(1801,·)$, $\chi_{4040}(3339,·)$, $\chi_{4040}(3231,·)$, $\chi_{4040}(2829,·)$, $\chi_{4040}(589,·)$, $\chi_{4040}(3521,·)$, $\chi_{4040}(1431,·)$, $\chi_{4040}(219,·)$, $\chi_{4040}(1349,·)$, $\chi_{4040}(2721,·)$, $\chi_{4040}(2019,·)$, $\chi_{4040}(1509,·)$, $\chi_{4040}(3751,·)$, $\chi_{4040}(2539,·)$, $\chi_{4040}(671,·)$, $\chi_{4040}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{137523676} a^{16} + \frac{7336279}{68761838} a^{14} - \frac{1}{4} a^{13} + \frac{22803819}{137523676} a^{12} - \frac{1}{4} a^{11} - \frac{4680747}{34380919} a^{10} - \frac{1}{2} a^{9} + \frac{2378729}{137523676} a^{8} + \frac{1}{4} a^{7} + \frac{15787049}{68761838} a^{6} - \frac{1}{4} a^{5} - \frac{8405422}{34380919} a^{4} - \frac{1}{4} a^{3} - \frac{1148149}{3354236} a^{2} - \frac{1}{2} a - \frac{59381203}{137523676}$, $\frac{1}{137523676} a^{17} + \frac{7336279}{68761838} a^{15} - \frac{1}{4} a^{14} + \frac{22803819}{137523676} a^{13} - \frac{1}{4} a^{12} - \frac{4680747}{34380919} a^{11} + \frac{2378729}{137523676} a^{9} - \frac{1}{4} a^{8} + \frac{15787049}{68761838} a^{7} - \frac{1}{4} a^{6} - \frac{8405422}{34380919} a^{5} + \frac{1}{4} a^{4} - \frac{1148149}{3354236} a^{3} - \frac{59381203}{137523676} a - \frac{1}{2}$, $\frac{1}{1067549391481957607785176221215692111004172956} a^{18} - \frac{515386841611800551214622839730772064}{266887347870489401946294055303923027751043239} a^{16} + \frac{50518849284868819700173205355722841395728138}{266887347870489401946294055303923027751043239} a^{14} + \frac{33299479415562347898103451011292731232470082}{266887347870489401946294055303923027751043239} a^{12} - \frac{1}{4} a^{11} + \frac{4758707172714518693978010790253803956990077}{26037790036145307506955517590626636853760316} a^{10} + \frac{1}{4} a^{9} - \frac{93418996857087788210682452662121638403942936}{266887347870489401946294055303923027751043239} a^{8} - \frac{129836873756378189901895318648070356260873497}{266887347870489401946294055303923027751043239} a^{6} + \frac{1}{4} a^{5} + \frac{2930779384558632484407704730078203259923573}{15699255757087611879193767959054295750061367} a^{4} + \frac{1}{4} a^{3} - \frac{311566374318296601369220826263228394569518915}{1067549391481957607785176221215692111004172956} a^{2} + \frac{1}{4} a + \frac{83799900721585341079010827808933084997960795}{1067549391481957607785176221215692111004172956}$, $\frac{1}{3851537788619742598053200111364831684536296320018436} a^{19} - \frac{528592901529470911134667833979425147091747}{226561046389396623414894124197931275560958607059908} a^{17} - \frac{12806231217786399282207657374815077742967324517159}{226561046389396623414894124197931275560958607059908} a^{15} + \frac{35204883194325166002552026156602429258065592131608}{962884447154935649513300027841207921134074080004609} a^{13} - \frac{130996360171064247861938007493195770107995497190989}{962884447154935649513300027841207921134074080004609} a^{11} - \frac{1}{4} a^{10} + \frac{1745662361677409499501747795914623124580297011792755}{3851537788619742598053200111364831684536296320018436} a^{9} + \frac{1}{4} a^{8} - \frac{1187031785372557995528250803497981522888504662221173}{3851537788619742598053200111364831684536296320018436} a^{7} - \frac{109999956341431506544083018062933349831202544352407}{1925768894309871299026600055682415842268148160009218} a^{5} + \frac{1}{4} a^{4} - \frac{1501149628976013488962608995744784601521777041843871}{3851537788619742598053200111364831684536296320018436} a^{3} + \frac{1}{4} a^{2} + \frac{206550531101738024379382828792435849824082807776281}{3851537788619742598053200111364831684536296320018436} a + \frac{1}{4}$

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

Class group and class number

$C_{11}\times C_{154}\times C_{770308}$, which has order $1304901752$ (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:  \( 8119542085.512349 \) (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

\(\Q(\sqrt{-1010}) \), \(\Q(\sqrt{-101}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{10}, \sqrt{-101})\), 5.5.104060401.1, 10.0.111993371922878556262400000.1, 10.0.1119933719228785562624.1, 10.10.1108845266563154022400000.1

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
101Data not computed