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Properties

Label 20.0.20220451366...0000.5
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 7^{10}\cdot 31^{18}$
Root discriminant $260.19$
Ramified primes $2, 5, 7, 31$
Class number $175480832$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 2, 342736]$ (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![235337644525, -102141624450, 123739534495, -46132578390, 30688118609, -9949519754, 4714916309, -1320905036, 490379664, -118463028, 35955246, -7433118, 1890972, -336218, 74548, -11924, 2383, -372, 73, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 73*x^18 - 372*x^17 + 2383*x^16 - 11924*x^15 + 74548*x^14 - 336218*x^13 + 1890972*x^12 - 7433118*x^11 + 35955246*x^10 - 118463028*x^9 + 490379664*x^8 - 1320905036*x^7 + 4714916309*x^6 - 9949519754*x^5 + 30688118609*x^4 - 46132578390*x^3 + 123739534495*x^2 - 102141624450*x + 235337644525)
 
gp: K = bnfinit(x^20 - 10*x^19 + 73*x^18 - 372*x^17 + 2383*x^16 - 11924*x^15 + 74548*x^14 - 336218*x^13 + 1890972*x^12 - 7433118*x^11 + 35955246*x^10 - 118463028*x^9 + 490379664*x^8 - 1320905036*x^7 + 4714916309*x^6 - 9949519754*x^5 + 30688118609*x^4 - 46132578390*x^3 + 123739534495*x^2 - 102141624450*x + 235337644525, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 73 x^{18} - 372 x^{17} + 2383 x^{16} - 11924 x^{15} + 74548 x^{14} - 336218 x^{13} + 1890972 x^{12} - 7433118 x^{11} + 35955246 x^{10} - 118463028 x^{9} + 490379664 x^{8} - 1320905036 x^{7} + 4714916309 x^{6} - 9949519754 x^{5} + 30688118609 x^{4} - 46132578390 x^{3} + 123739534495 x^{2} - 102141624450 x + 235337644525 \)

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:  \(2022045136625475860721323111217817692160000000000=2^{20}\cdot 5^{10}\cdot 7^{10}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $260.19$
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:  $2, 5, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4340=2^{2}\cdot 5\cdot 7\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{4340}(1,·)$, $\chi_{4340}(139,·)$, $\chi_{4340}(3009,·)$, $\chi_{4340}(841,·)$, $\chi_{4340}(1611,·)$, $\chi_{4340}(1331,·)$, $\chi_{4340}(771,·)$, $\chi_{4340}(3991,·)$, $\chi_{4340}(281,·)$, $\chi_{4340}(2379,·)$, $\chi_{4340}(4059,·)$, $\chi_{4340}(349,·)$, $\chi_{4340}(1961,·)$, $\chi_{4340}(3499,·)$, $\chi_{4340}(1751,·)$, $\chi_{4340}(2589,·)$, $\chi_{4340}(3569,·)$, $\chi_{4340}(4339,·)$, $\chi_{4340}(2729,·)$, $\chi_{4340}(4201,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{25} a^{12} - \frac{1}{25} a^{11} - \frac{1}{25} a^{10} + \frac{2}{25} a^{8} - \frac{4}{25} a^{7} + \frac{1}{5} a^{6} - \frac{8}{25} a^{5} - \frac{12}{25} a^{4} - \frac{7}{25} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{25} a^{13} - \frac{2}{25} a^{11} - \frac{1}{25} a^{10} + \frac{2}{25} a^{9} - \frac{2}{25} a^{8} + \frac{1}{25} a^{7} - \frac{3}{25} a^{6} + \frac{1}{5} a^{5} + \frac{6}{25} a^{4} - \frac{2}{25} a^{3} - \frac{1}{5} a$, $\frac{1}{25} a^{14} + \frac{2}{25} a^{11} - \frac{2}{25} a^{9} - \frac{1}{25} a^{7} + \frac{1}{5} a^{6} + \frac{9}{25} a^{4} + \frac{11}{25} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{25} a^{15} + \frac{2}{25} a^{11} - \frac{7}{25} a^{7} - \frac{2}{5} a^{5} + \frac{4}{25} a^{3} + \frac{2}{5} a$, $\frac{1}{7625} a^{16} - \frac{8}{7625} a^{15} + \frac{27}{1525} a^{14} + \frac{22}{1525} a^{13} + \frac{1}{125} a^{12} + \frac{56}{1525} a^{11} - \frac{714}{7625} a^{10} + \frac{21}{1525} a^{9} - \frac{99}{7625} a^{8} - \frac{91}{305} a^{7} + \frac{394}{1525} a^{6} - \frac{2862}{7625} a^{5} + \frac{2641}{7625} a^{4} + \frac{138}{305} a^{3} - \frac{124}{1525} a^{2} - \frac{87}{305} a + \frac{9}{305}$, $\frac{1}{7625} a^{17} + \frac{71}{7625} a^{15} - \frac{6}{1525} a^{14} + \frac{26}{7625} a^{13} - \frac{147}{7625} a^{12} + \frac{306}{7625} a^{11} - \frac{727}{7625} a^{10} - \frac{174}{7625} a^{9} - \frac{17}{7625} a^{8} - \frac{13}{1525} a^{7} + \frac{3443}{7625} a^{6} - \frac{147}{1525} a^{5} + \frac{788}{7625} a^{4} + \frac{699}{1525} a^{3} - \frac{512}{1525} a^{2} - \frac{138}{305} a + \frac{72}{305}$, $\frac{1}{20164684756091138563005189571893625} a^{18} - \frac{9}{20164684756091138563005189571893625} a^{17} - \frac{459096369282710939817035537069}{20164684756091138563005189571893625} a^{16} + \frac{3672770954261687518536284296756}{20164684756091138563005189571893625} a^{15} + \frac{309906697587797202553326907511521}{20164684756091138563005189571893625} a^{14} + \frac{186141795916776678112949420852644}{20164684756091138563005189571893625} a^{13} + \frac{284693868194037870956568250668529}{20164684756091138563005189571893625} a^{12} - \frac{1541133306932480631553728290963261}{20164684756091138563005189571893625} a^{11} + \frac{2012073619643646753314061223915289}{20164684756091138563005189571893625} a^{10} - \frac{403772784928446441390255754324576}{20164684756091138563005189571893625} a^{9} - \frac{472984370427328180610200090763772}{20164684756091138563005189571893625} a^{8} - \frac{4475824658220193519679293128498667}{20164684756091138563005189571893625} a^{7} + \frac{5222131954352292850714445423167728}{20164684756091138563005189571893625} a^{6} - \frac{4937773756252787650870384869760612}{20164684756091138563005189571893625} a^{5} - \frac{665222464665120477077220293950767}{20164684756091138563005189571893625} a^{4} + \frac{174068672413450316957387069836347}{806587390243645542520207582875745} a^{3} + \frac{18441871819770600451451893458419}{108998295978871019259487511199425} a^{2} - \frac{131396534633879697540090603293197}{806587390243645542520207582875745} a + \frac{360335466085230048277871286895342}{806587390243645542520207582875745}$, $\frac{1}{9487308321525122823074533725317694078196375} a^{19} + \frac{1271598}{51282747683919582827429912028744292314575} a^{18} + \frac{233226388339220072279720025618931373728}{9487308321525122823074533725317694078196375} a^{17} + \frac{398049593975245060999549664064524913329}{9487308321525122823074533725317694078196375} a^{16} - \frac{97386290398037350763908411044884138438399}{9487308321525122823074533725317694078196375} a^{15} - \frac{102411536643451228458150284879068027373897}{9487308321525122823074533725317694078196375} a^{14} + \frac{12976300950821315370808037598681327245718}{9487308321525122823074533725317694078196375} a^{13} - \frac{3500606111844045579268330166487865949551}{256413738419597914137149560143721461572875} a^{12} + \frac{326166157910629102830513582882935472979233}{9487308321525122823074533725317694078196375} a^{11} - \frac{599008567523375640238828622581055687750127}{9487308321525122823074533725317694078196375} a^{10} - \frac{208578669408248032863288322089319508270328}{9487308321525122823074533725317694078196375} a^{9} - \frac{465684079329956838652670474799103425339857}{9487308321525122823074533725317694078196375} a^{8} - \frac{530779231227420164138125532030444696689988}{1897461664305024564614906745063538815639275} a^{7} + \frac{2972888144143812601395982074124335207058819}{9487308321525122823074533725317694078196375} a^{6} + \frac{68169657579157548545091594949433896264012}{155529644615165947919254651234716296363875} a^{5} - \frac{841466588392566332422040389306610378306902}{1897461664305024564614906745063538815639275} a^{4} - \frac{133878898954013288725358608633502351818839}{1897461664305024564614906745063538815639275} a^{3} - \frac{176520199855014164235321360588642408566278}{379492332861004912922981349012707763127855} a^{2} - \frac{25573930343043921444366397304307295605886}{75898466572200982584596269802541552625571} a - \frac{17377941535530286265032753318306632596625}{75898466572200982584596269802541552625571}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{342736}$, which has order $175480832$ (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:  \( 190570522.65858147 \) (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

\(\Q(\sqrt{31}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-1085}) \), \(\Q(\sqrt{31}, \sqrt{-35})\), 5.5.923521.1, 10.10.27074173092527104.1, 10.0.44795436457096521875.1, 10.0.1421986334894071990400000.1

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 R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ 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
$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}$
7Data not computed
31Data not computed