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Properties

Label 20.0.76643602510...2361.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 7^{10}\cdot 11^{16}$
Root discriminant $31.20$
Ramified primes $3, 7, 11$
Class number $5$ (GRH)
Class group $[5]$ (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![1024, -1536, -4608, -9856, 39296, 18272, -42976, 6480, 29156, -7082, -11777, 355, 3770, -9, -734, 43, 104, 0, -13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 13*x^18 + 104*x^16 + 43*x^15 - 734*x^14 - 9*x^13 + 3770*x^12 + 355*x^11 - 11777*x^10 - 7082*x^9 + 29156*x^8 + 6480*x^7 - 42976*x^6 + 18272*x^5 + 39296*x^4 - 9856*x^3 - 4608*x^2 - 1536*x + 1024)
 
gp: K = bnfinit(x^20 - x^19 - 13*x^18 + 104*x^16 + 43*x^15 - 734*x^14 - 9*x^13 + 3770*x^12 + 355*x^11 - 11777*x^10 - 7082*x^9 + 29156*x^8 + 6480*x^7 - 42976*x^6 + 18272*x^5 + 39296*x^4 - 9856*x^3 - 4608*x^2 - 1536*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 13 x^{18} + 104 x^{16} + 43 x^{15} - 734 x^{14} - 9 x^{13} + 3770 x^{12} + 355 x^{11} - 11777 x^{10} - 7082 x^{9} + 29156 x^{8} + 6480 x^{7} - 42976 x^{6} + 18272 x^{5} + 39296 x^{4} - 9856 x^{3} - 4608 x^{2} - 1536 x + 1024 \)

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:  \(766436025104871719096831462361=3^{10}\cdot 7^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.20$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(231=3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{231}(64,·)$, $\chi_{231}(1,·)$, $\chi_{231}(71,·)$, $\chi_{231}(202,·)$, $\chi_{231}(146,·)$, $\chi_{231}(20,·)$, $\chi_{231}(218,·)$, $\chi_{231}(155,·)$, $\chi_{231}(92,·)$, $\chi_{231}(223,·)$, $\chi_{231}(97,·)$, $\chi_{231}(34,·)$, $\chi_{231}(104,·)$, $\chi_{231}(169,·)$, $\chi_{231}(113,·)$, $\chi_{231}(181,·)$, $\chi_{231}(148,·)$, $\chi_{231}(188,·)$, $\chi_{231}(125,·)$, $\chi_{231}(190,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} + \frac{3}{8} a^{8} + \frac{1}{4} a^{7} + \frac{3}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{3}{16} a^{9} + \frac{1}{8} a^{8} - \frac{5}{16} a^{7} - \frac{3}{8} a^{6} + \frac{7}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{8} a^{3}$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{14} - \frac{1}{32} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} + \frac{3}{32} a^{10} + \frac{1}{16} a^{9} - \frac{5}{32} a^{8} + \frac{5}{16} a^{7} + \frac{7}{32} a^{6} + \frac{15}{32} a^{5} + \frac{5}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{64} a^{16} - \frac{1}{64} a^{15} - \frac{1}{64} a^{14} - \frac{1}{16} a^{13} + \frac{1}{16} a^{12} + \frac{3}{64} a^{11} + \frac{1}{32} a^{10} - \frac{5}{64} a^{9} - \frac{11}{32} a^{8} + \frac{7}{64} a^{7} - \frac{17}{64} a^{6} + \frac{5}{32} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{13952} a^{17} - \frac{107}{13952} a^{16} + \frac{89}{13952} a^{15} - \frac{217}{6976} a^{14} - \frac{27}{3488} a^{13} + \frac{1203}{13952} a^{12} - \frac{603}{3488} a^{11} + \frac{3255}{13952} a^{10} - \frac{1415}{3488} a^{9} - \frac{1421}{13952} a^{8} + \frac{1393}{13952} a^{7} - \frac{455}{3488} a^{6} - \frac{735}{3488} a^{5} - \frac{215}{1744} a^{4} - \frac{21}{218} a^{3} - \frac{39}{109} a^{2} - \frac{77}{218} a - \frac{10}{109}$, $\frac{1}{5639071393024} a^{18} + \frac{171881021}{5639071393024} a^{17} + \frac{11000040525}{5639071393024} a^{16} + \frac{37737849765}{2819535696512} a^{15} - \frac{19138663505}{704883924128} a^{14} + \frac{13447414613}{245177017088} a^{13} + \frac{149920643919}{1409767848256} a^{12} + \frac{1181170729467}{5639071393024} a^{11} - \frac{552859782523}{1409767848256} a^{10} - \frac{42735556223}{245177017088} a^{9} - \frac{2334822740159}{5639071393024} a^{8} + \frac{70833943923}{176220981032} a^{7} - \frac{152775288149}{704883924128} a^{6} + \frac{9274740009}{22027622629} a^{5} - \frac{7497265145}{88110490516} a^{4} + \frac{4614670009}{176220981032} a^{3} - \frac{9512766487}{22027622629} a^{2} + \frac{242945916}{22027622629} a - \frac{111343436}{22027622629}$, $\frac{1}{159621012634948438888415744} a^{19} + \frac{5190359519875}{159621012634948438888415744} a^{18} - \frac{822767574996976152817}{159621012634948438888415744} a^{17} + \frac{82986320550690545284435}{39905253158737109722103936} a^{16} - \frac{307181320434110256292057}{19952626579368554861051968} a^{15} - \frac{2908170585300521978800965}{159621012634948438888415744} a^{14} + \frac{113773596235570870163149}{1856058286452888824283904} a^{13} - \frac{19074367071501874110913729}{159621012634948438888415744} a^{12} + \frac{19851382968686657793032411}{79810506317474219444207872} a^{11} + \frac{13767830798073119492728987}{159621012634948438888415744} a^{10} + \frac{75014515109347288814339243}{159621012634948438888415744} a^{9} - \frac{7374247331994989075794735}{79810506317474219444207872} a^{8} - \frac{3640208466643549802448769}{39905253158737109722103936} a^{7} + \frac{15460784910514591308709}{232007285806611103035488} a^{6} + \frac{855462558647649911707511}{9976313289684277430525984} a^{5} + \frac{1977235091004486838249813}{4988156644842138715262992} a^{4} + \frac{40589967828301921108183}{311759790302633669703937} a^{3} + \frac{395472907337739637960637}{1247039161210534678815748} a^{2} - \frac{45532400216670903949842}{311759790302633669703937} a - \frac{145867160066094093366010}{311759790302633669703937}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( -\frac{5535229389885625535}{10531174548719960340992} a^{19} + \frac{2013572255781494657}{10531174548719960340992} a^{18} + \frac{74736202242098525493}{10531174548719960340992} a^{17} + \frac{22837991672698904805}{5265587274359980170496} a^{16} - \frac{141303934624722806829}{2632793637179990085248} a^{15} - \frac{594074380580586622541}{10531174548719960340992} a^{14} + \frac{11150429988410841785}{30613879502092907968} a^{13} + \frac{2524732247358610335655}{10531174548719960340992} a^{12} - \frac{2541423820665460344355}{1316396818589995042624} a^{11} - \frac{14691831125782921942333}{10531174548719960340992} a^{10} + \frac{61151550721798694391685}{10531174548719960340992} a^{9} + \frac{4859548331739653152251}{658198409294997521312} a^{8} - \frac{32062117902631407624095}{2632793637179990085248} a^{7} - \frac{366011264946183654677}{30613879502092907968} a^{6} + \frac{1559571278045433832019}{82274801161874690164} a^{5} + \frac{109122794520016912161}{41137400580937345082} a^{4} - \frac{3935364285206466681573}{164549602323749380328} a^{3} - \frac{626146622318748687023}{82274801161874690164} a^{2} + \frac{39223546541648365258}{20568700290468672541} a + \frac{43859219384923174195}{20568700290468672541} \) (order $6$)
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:  \( 2934029.89978 \) (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

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\zeta_{11})^+\), 10.10.875463320250981.1, 10.0.52089208083.1, 10.0.3602729712967.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$7$7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$