↑

Properties

Label 20.0.11238525471...0000.3
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 $10516192$ (GRH)
Class group $[2, 5258096]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11215873819, 4741521926, -1615161017, -8278160402, 9723639107, -8232883868, 5520922330, -2848052980, 1342124467, -492640862, 175790321, -48289358, 13428857, -2767220, 604922, -90300, 15393, -1506, 199, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 199*x^18 - 1506*x^17 + 15393*x^16 - 90300*x^15 + 604922*x^14 - 2767220*x^13 + 13428857*x^12 - 48289358*x^11 + 175790321*x^10 - 492640862*x^9 + 1342124467*x^8 - 2848052980*x^7 + 5520922330*x^6 - 8232883868*x^5 + 9723639107*x^4 - 8278160402*x^3 - 1615161017*x^2 + 4741521926*x + 11215873819)
 
gp: K = bnfinit(x^20 - 10*x^19 + 199*x^18 - 1506*x^17 + 15393*x^16 - 90300*x^15 + 604922*x^14 - 2767220*x^13 + 13428857*x^12 - 48289358*x^11 + 175790321*x^10 - 492640862*x^9 + 1342124467*x^8 - 2848052980*x^7 + 5520922330*x^6 - 8232883868*x^5 + 9723639107*x^4 - 8278160402*x^3 - 1615161017*x^2 + 4741521926*x + 11215873819, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 199 x^{18} - 1506 x^{17} + 15393 x^{16} - 90300 x^{15} + 604922 x^{14} - 2767220 x^{13} + 13428857 x^{12} - 48289358 x^{11} + 175790321 x^{10} - 492640862 x^{9} + 1342124467 x^{8} - 2848052980 x^{7} + 5520922330 x^{6} - 8232883868 x^{5} + 9723639107 x^{4} - 8278160402 x^{3} - 1615161017 x^{2} + 4741521926 x + 11215873819 \)

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$$\chi_{1640}(1,·)$, $\chi_{1640}(1091,·)$, $\chi_{1640}(961,·)$, $\chi_{1640}(1609,·)$, $\chi_{1640}(1419,·)$, $\chi_{1640}(1281,·)$, $\chi_{1640}(1041,·)$, $\chi_{1640}(851,·)$, $\chi_{1640}(1179,·)$, $\chi_{1640}(1369,·)$, $\chi_{1640}(819,·)$, $\chi_{1640}(201,·)$, $\chi_{1640}(1499,·)$, $\chi_{1640}(329,·)$, $\chi_{1640}(491,·)$, $\chi_{1640}(291,·)$, $\chi_{1640}(529,·)$, $\chi_{1640}(619,·)$, $\chi_{1640}(1171,·)$, $\chi_{1640}(1289,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{4} + \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{5} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{18} a^{6} - \frac{1}{18} a^{4} - \frac{1}{9} a^{3} + \frac{7}{18} a^{2} - \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{18} a^{7} - \frac{1}{18} a^{5} + \frac{1}{18} a^{4} + \frac{1}{18} a^{3} - \frac{5}{18} a^{2} - \frac{4}{9} a - \frac{1}{6}$, $\frac{1}{36} a^{8} - \frac{1}{18} a^{5} - \frac{1}{12} a^{4} + \frac{1}{18} a^{3} + \frac{2}{9} a^{2} - \frac{7}{18} a - \frac{11}{36}$, $\frac{1}{324} a^{9} - \frac{1}{54} a^{7} + \frac{1}{54} a^{6} - \frac{5}{108} a^{5} - \frac{2}{27} a^{4} + \frac{10}{81} a^{3} + \frac{2}{27} a^{2} + \frac{13}{108} a + \frac{13}{162}$, $\frac{1}{324} a^{10} + \frac{1}{108} a^{8} + \frac{1}{54} a^{7} + \frac{1}{108} a^{6} + \frac{1}{27} a^{5} - \frac{5}{324} a^{4} - \frac{4}{27} a^{3} + \frac{7}{108} a^{2} + \frac{67}{162} a - \frac{5}{12}$, $\frac{1}{324} a^{11} - \frac{1}{108} a^{8} + \frac{1}{108} a^{7} - \frac{1}{54} a^{6} + \frac{11}{162} a^{5} - \frac{7}{108} a^{4} + \frac{1}{12} a^{3} - \frac{41}{162} a^{2} - \frac{1}{9} a + \frac{7}{108}$, $\frac{1}{1944} a^{12} + \frac{1}{1944} a^{10} + \frac{1}{972} a^{9} + \frac{1}{81} a^{8} - \frac{1}{162} a^{7} - \frac{35}{1944} a^{6} - \frac{5}{324} a^{5} + \frac{8}{243} a^{4} - \frac{16}{243} a^{3} + \frac{275}{648} a^{2} - \frac{473}{972} a - \frac{941}{1944}$, $\frac{1}{1944} a^{13} + \frac{1}{1944} a^{11} + \frac{1}{972} a^{10} - \frac{1}{162} a^{8} + \frac{1}{1944} a^{7} + \frac{7}{324} a^{6} - \frac{29}{486} a^{5} + \frac{31}{486} a^{4} - \frac{1}{72} a^{3} - \frac{383}{972} a^{2} + \frac{499}{1944} a + \frac{47}{162}$, $\frac{1}{1944} a^{14} + \frac{1}{972} a^{11} - \frac{1}{1944} a^{10} - \frac{1}{972} a^{9} - \frac{23}{1944} a^{8} - \frac{1}{108} a^{7} - \frac{1}{216} a^{6} - \frac{13}{972} a^{5} - \frac{55}{1944} a^{4} - \frac{79}{972} a^{3} + \frac{143}{972} a^{2} + \frac{341}{972} a + \frac{281}{1944}$, $\frac{1}{5832} a^{15} - \frac{1}{5832} a^{13} + \frac{1}{5832} a^{12} + \frac{1}{1458} a^{11} + \frac{1}{5832} a^{10} + \frac{5}{5832} a^{9} + \frac{13}{972} a^{8} - \frac{2}{729} a^{7} + \frac{7}{1944} a^{6} - \frac{155}{5832} a^{5} + \frac{145}{2916} a^{4} + \frac{145}{1944} a^{3} + \frac{2813}{5832} a^{2} - \frac{323}{2916} a + \frac{797}{5832}$, $\frac{1}{2554416} a^{16} - \frac{1}{319302} a^{15} - \frac{77}{1277208} a^{14} - \frac{2}{53217} a^{13} - \frac{17}{319302} a^{12} - \frac{58}{159651} a^{11} + \frac{59}{53217} a^{10} + \frac{443}{638604} a^{9} - \frac{18955}{2554416} a^{8} + \frac{3011}{159651} a^{7} - \frac{542}{159651} a^{6} - \frac{8195}{212868} a^{5} - \frac{29371}{638604} a^{4} + \frac{44341}{319302} a^{3} + \frac{150005}{1277208} a^{2} - \frac{88421}{638604} a + \frac{1137725}{2554416}$, $\frac{1}{2554416} a^{17} - \frac{109}{1277208} a^{15} - \frac{7}{1277208} a^{14} + \frac{205}{1277208} a^{13} + \frac{17}{70956} a^{12} - \frac{325}{1277208} a^{11} - \frac{1583}{1277208} a^{10} - \frac{239}{283824} a^{9} + \frac{1771}{141912} a^{8} - \frac{4133}{1277208} a^{7} - \frac{13559}{1277208} a^{6} - \frac{8059}{159651} a^{5} + \frac{2459}{141912} a^{4} + \frac{47797}{319302} a^{3} - \frac{16004}{53217} a^{2} - \frac{749609}{2554416} a - \frac{565159}{1277208}$, $\frac{1}{53276266093003595414791344} a^{18} - \frac{1}{5919585121444843934976816} a^{17} - \frac{1181532230835945175}{8879377682167265902465224} a^{16} + \frac{1575376307781260239}{1479896280361210983744204} a^{15} + \frac{93808096979924679875}{1479896280361210983744204} a^{14} + \frac{462226526752090946407}{8879377682167265902465224} a^{13} + \frac{8462334329763211031}{13319066523250898853697836} a^{12} + \frac{440305721422496068877}{1109922210270908237808153} a^{11} - \frac{23371085351458774262813}{17758755364334531804930448} a^{10} + \frac{33468547322112349485073}{53276266093003595414791344} a^{9} + \frac{2159999853854438693321}{246649380060201830624034} a^{8} + \frac{25249993449849499864351}{1109922210270908237808153} a^{7} + \frac{54251626319375266144601}{6659533261625449426848918} a^{6} + \frac{211114002936593248110427}{8879377682167265902465224} a^{5} + \frac{393569405526049184215571}{8879377682167265902465224} a^{4} - \frac{748568608319284131629897}{13319066523250898853697836} a^{3} + \frac{2431901990884005196005055}{17758755364334531804930448} a^{2} + \frac{934562665133170600925911}{5919585121444843934976816} a - \frac{884480287198454153125217}{13319066523250898853697836}$, $\frac{1}{75502964478206416411056048342768} a^{19} + \frac{708589}{75502964478206416411056048342768} a^{18} - \frac{186826476959187288236255}{6291913706517201367588004028564} a^{17} + \frac{2338818141834179393894507}{12583827413034402735176008057128} a^{16} + \frac{5034399136819900081679749}{349550761473177853754889112698} a^{15} - \frac{973741190272201480232809289}{12583827413034402735176008057128} a^{14} + \frac{7158614673011734745250158047}{37751482239103208205528024171384} a^{13} - \frac{688135105588888383658610047}{18875741119551604102764012085692} a^{12} + \frac{2966475511582336945109060969}{2796406091785422830039112901584} a^{11} - \frac{102261496525860473254008968717}{75502964478206416411056048342768} a^{10} - \frac{2110299447876176283751936123}{37751482239103208205528024171384} a^{9} - \frac{47939304839479749611657122087}{12583827413034402735176008057128} a^{8} - \frac{623663603950322246625939940027}{37751482239103208205528024171384} a^{7} - \frac{741382463919564050738643895897}{37751482239103208205528024171384} a^{6} + \frac{152999367926763879635811405251}{4194609137678134245058669352376} a^{5} - \frac{1164388011870568412423126130377}{18875741119551604102764012085692} a^{4} + \frac{1339212139316841610509145131859}{75502964478206416411056048342768} a^{3} - \frac{9818077668224727519834196624181}{25167654826068805470352016114256} a^{2} + \frac{1532295487104158187427827726595}{18875741119551604102764012085692} a - \frac{9397135633900341911815622309191}{37751482239103208205528024171384}$

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

Class group and class number

$C_{2}\times C_{5258096}$, which has order $10516192$ (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:  \( 41023218.25673422 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-410}) \), \(\Q(\sqrt{-82}) \), \(\Q(\sqrt{5}, \sqrt{-82})\), 5.5.2825761.1, 10.10.24952891341003125.1, 10.0.33523910081941606400000.1, 10.0.10727651226221314048.1

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ 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
2Data not computed
5Data not computed
$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}$