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Properties

Label 20.0.16998350586...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{38}\cdot 5^{22}\cdot 11^{10}$
Root discriminant $72.70$
Ramified primes $2, 5, 11$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![103789316951, -244797510, 681245765, -1120914930, 1217545455, -892866972, 473490030, -154608420, 43483815, -3221790, 644073, 770, 45, -540, 90, 228, -105, -30, 35, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 + 644073*x^10 - 3221790*x^9 + 43483815*x^8 - 154608420*x^7 + 473490030*x^6 - 892866972*x^5 + 1217545455*x^4 - 1120914930*x^3 + 681245765*x^2 - 244797510*x + 103789316951)
 
gp: K = bnfinit(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 + 644073*x^10 - 3221790*x^9 + 43483815*x^8 - 154608420*x^7 + 473490030*x^6 - 892866972*x^5 + 1217545455*x^4 - 1120914930*x^3 + 681245765*x^2 - 244797510*x + 103789316951, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 35 x^{18} - 30 x^{17} - 105 x^{16} + 228 x^{15} + 90 x^{14} - 540 x^{13} + 45 x^{12} + 770 x^{11} + 644073 x^{10} - 3221790 x^{9} + 43483815 x^{8} - 154608420 x^{7} + 473490030 x^{6} - 892866972 x^{5} + 1217545455 x^{4} - 1120914930 x^{3} + 681245765 x^{2} - 244797510 x + 103789316951 \)

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:  \(16998350586511360000000000000000000000=2^{38}\cdot 5^{22}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.70$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{11} a^{4} - \frac{2}{11} a^{3} - \frac{1}{11} a^{2} + \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{5} - \frac{5}{11} a^{3} + \frac{5}{11} a + \frac{2}{11}$, $\frac{1}{11} a^{6} + \frac{1}{11} a^{3} + \frac{1}{11} a + \frac{5}{11}$, $\frac{1}{11} a^{7} + \frac{2}{11} a^{3} + \frac{2}{11} a^{2} + \frac{3}{11} a - \frac{1}{11}$, $\frac{1}{121} a^{8} - \frac{4}{121} a^{7} + \frac{2}{121} a^{6} - \frac{3}{121} a^{5} - \frac{5}{121} a^{4} + \frac{47}{121} a^{3} + \frac{2}{121} a^{2} - \frac{51}{121} a - \frac{21}{121}$, $\frac{1}{121} a^{9} - \frac{3}{121} a^{7} + \frac{5}{121} a^{6} + \frac{5}{121} a^{5} + \frac{5}{121} a^{4} + \frac{25}{121} a^{3} + \frac{1}{121} a^{2} - \frac{5}{121} a + \frac{48}{121}$, $\frac{1}{605} a^{10} + \frac{3}{121} a^{7} - \frac{26}{605} a^{5} + \frac{2}{121} a^{4} + \frac{57}{121} a^{3} + \frac{9}{121} a^{2} - \frac{32}{121} a - \frac{184}{605}$, $\frac{1}{605} a^{11} + \frac{1}{121} a^{7} - \frac{1}{605} a^{6} - \frac{5}{121} a^{4} - \frac{5}{11} a^{3} + \frac{17}{121} a^{2} + \frac{31}{605} a + \frac{30}{121}$, $\frac{1}{6655} a^{12} + \frac{1}{1331} a^{11} - \frac{2}{6655} a^{10} + \frac{2}{1331} a^{9} + \frac{5}{1331} a^{8} + \frac{269}{6655} a^{7} + \frac{28}{1331} a^{6} + \frac{127}{6655} a^{5} - \frac{17}{1331} a^{4} + \frac{614}{1331} a^{3} - \frac{2829}{6655} a^{2} - \frac{67}{1331} a - \frac{142}{6655}$, $\frac{1}{6655} a^{13} - \frac{1}{1331} a^{11} - \frac{2}{6655} a^{10} - \frac{5}{1331} a^{9} - \frac{21}{6655} a^{8} - \frac{32}{1331} a^{7} + \frac{57}{1331} a^{6} - \frac{258}{6655} a^{5} - \frac{16}{1331} a^{4} + \frac{1071}{6655} a^{3} - \frac{153}{1331} a^{2} + \frac{47}{1331} a - \frac{1182}{6655}$, $\frac{1}{6655} a^{14} + \frac{1}{6655} a^{11} - \frac{2}{6655} a^{10} - \frac{26}{6655} a^{9} + \frac{4}{1331} a^{8} + \frac{29}{1331} a^{7} + \frac{299}{6655} a^{6} - \frac{138}{6655} a^{5} - \frac{234}{6655} a^{4} + \frac{79}{1331} a^{3} + \frac{661}{1331} a^{2} - \frac{2879}{6655} a + \frac{1248}{6655}$, $\frac{1}{6655} a^{15} + \frac{4}{6655} a^{11} - \frac{2}{6655} a^{10} + \frac{2}{1331} a^{9} + \frac{2}{1331} a^{8} + \frac{50}{1331} a^{7} + \frac{96}{6655} a^{6} + \frac{2}{6655} a^{5} - \frac{47}{1331} a^{4} - \frac{580}{1331} a^{3} + \frac{331}{1331} a^{2} - \frac{3246}{6655} a - \frac{2971}{6655}$, $\frac{1}{366025} a^{16} - \frac{8}{366025} a^{15} - \frac{24}{366025} a^{14} - \frac{2}{33275} a^{13} - \frac{4}{366025} a^{12} - \frac{274}{366025} a^{11} - \frac{251}{366025} a^{10} + \frac{419}{366025} a^{9} - \frac{103}{366025} a^{8} - \frac{3741}{366025} a^{7} - \frac{11526}{366025} a^{6} - \frac{3884}{366025} a^{5} - \frac{15404}{366025} a^{4} - \frac{12472}{33275} a^{3} + \frac{35011}{366025} a^{2} + \frac{51367}{366025} a + \frac{54891}{366025}$, $\frac{1}{366025} a^{17} + \frac{2}{33275} a^{15} + \frac{6}{366025} a^{14} - \frac{3}{73205} a^{13} + \frac{24}{366025} a^{12} + \frac{252}{366025} a^{11} - \frac{214}{366025} a^{10} + \frac{829}{366025} a^{9} - \frac{6}{6655} a^{8} + \frac{13766}{366025} a^{7} + \frac{8738}{366025} a^{6} + \frac{10449}{366025} a^{5} - \frac{879}{366025} a^{4} + \frac{3238}{73205} a^{3} + \frac{35557}{73205} a^{2} + \frac{115697}{366025} a + \frac{75413}{366025}$, $\frac{1}{8625100756002484009554064525} a^{18} - \frac{9}{8625100756002484009554064525} a^{17} + \frac{2654835896175976296733}{8625100756002484009554064525} a^{16} - \frac{4247737433881562074732}{1725020151200496801910812905} a^{15} - \frac{7487060246466429553178}{784100068727498546323096775} a^{14} - \frac{347852507007859063235713}{8625100756002484009554064525} a^{13} + \frac{90380080297263641779497}{8625100756002484009554064525} a^{12} + \frac{6901541357461763060494674}{8625100756002484009554064525} a^{11} + \frac{144784642544013531552374}{8625100756002484009554064525} a^{10} - \frac{17258250075044134424959107}{8625100756002484009554064525} a^{9} + \frac{2032927809202495989574228}{784100068727498546323096775} a^{8} - \frac{240997383543904772564506112}{8625100756002484009554064525} a^{7} - \frac{388126879237963092209437099}{8625100756002484009554064525} a^{6} + \frac{238279968220688654600595151}{8625100756002484009554064525} a^{5} + \frac{4259812688298036816636157}{8625100756002484009554064525} a^{4} + \frac{3152701558913381187855571483}{8625100756002484009554064525} a^{3} + \frac{1138195085206076595284317233}{8625100756002484009554064525} a^{2} + \frac{2369656851402421728933498597}{8625100756002484009554064525} a - \frac{3827858353808000996845649466}{8625100756002484009554064525}$, $\frac{1}{2844865050038812502714602347082347825} a^{19} + \frac{8679883}{149729739475726973827084334056965675} a^{18} - \frac{403454696872387156464496283962}{568973010007762500542920469416469565} a^{17} - \frac{1846156802148800662289140451518}{2844865050038812502714602347082347825} a^{16} - \frac{28217927364386518240026939596363}{2844865050038812502714602347082347825} a^{15} - \frac{35138382564145229345827827312961}{2844865050038812502714602347082347825} a^{14} + \frac{117154244209210335300358361348026}{2844865050038812502714602347082347825} a^{13} + \frac{95484219880652875163561461202339}{2844865050038812502714602347082347825} a^{12} - \frac{2057303114760032701727592357972851}{2844865050038812502714602347082347825} a^{11} - \frac{82079822354937538566013370715026}{2844865050038812502714602347082347825} a^{10} + \frac{5024848559152737186346665913343231}{2844865050038812502714602347082347825} a^{9} + \frac{6881477467426698240598837337563834}{2844865050038812502714602347082347825} a^{8} + \frac{114102514344317900095296025671216731}{2844865050038812502714602347082347825} a^{7} + \frac{98550331463992037810509907821601716}{2844865050038812502714602347082347825} a^{6} - \frac{92588617544198996705844444522768454}{2844865050038812502714602347082347825} a^{5} - \frac{2357987691024603186964193172328921}{568973010007762500542920469416469565} a^{4} - \frac{51464616164205970967091303060238078}{149729739475726973827084334056965675} a^{3} + \frac{427174315845196884927158247690896734}{2844865050038812502714602347082347825} a^{2} - \frac{422299569918402311670074736467305604}{2844865050038812502714602347082347825} a - \frac{68442045891827315469801891753821596}{149729739475726973827084334056965675}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{5}, \sqrt{-22})\), 5.1.50000.1, 10.2.12500000000.1, 10.0.824581120000000000.1, 10.0.4122905600000000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$11$11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$