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Properties

Label 20.0.66362369058...0000.5
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}$
Root discriminant $309.81$
Ramified primes $2, 3, 5, 41$
Class number $297504768$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 4, 290532]$ (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![246562051689, 66897276912, 134963891166, 24488206434, 25772416231, 1840387582, 1831089339, -217328252, 53569228, -22197228, 8761372, 299032, 509781, -64298, -18998, -3198, 2077, 118, -57, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 57*x^18 + 118*x^17 + 2077*x^16 - 3198*x^15 - 18998*x^14 - 64298*x^13 + 509781*x^12 + 299032*x^11 + 8761372*x^10 - 22197228*x^9 + 53569228*x^8 - 217328252*x^7 + 1831089339*x^6 + 1840387582*x^5 + 25772416231*x^4 + 24488206434*x^3 + 134963891166*x^2 + 66897276912*x + 246562051689)
 
gp: K = bnfinit(x^20 - 2*x^19 - 57*x^18 + 118*x^17 + 2077*x^16 - 3198*x^15 - 18998*x^14 - 64298*x^13 + 509781*x^12 + 299032*x^11 + 8761372*x^10 - 22197228*x^9 + 53569228*x^8 - 217328252*x^7 + 1831089339*x^6 + 1840387582*x^5 + 25772416231*x^4 + 24488206434*x^3 + 134963891166*x^2 + 66897276912*x + 246562051689, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 57 x^{18} + 118 x^{17} + 2077 x^{16} - 3198 x^{15} - 18998 x^{14} - 64298 x^{13} + 509781 x^{12} + 299032 x^{11} + 8761372 x^{10} - 22197228 x^{9} + 53569228 x^{8} - 217328252 x^{7} + 1831089339 x^{6} + 1840387582 x^{5} + 25772416231 x^{4} + 24488206434 x^{3} + 134963891166 x^{2} + 66897276912 x + 246562051689 \)

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:  \(66362369058556182102889507973449950167040000000000=2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $309.81$
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:  $2, 3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4920=2^{3}\cdot 3\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4920}(1,·)$, $\chi_{4920}(961,·)$, $\chi_{4920}(269,·)$, $\chi_{4920}(461,·)$, $\chi_{4920}(4369,·)$, $\chi_{4920}(4301,·)$, $\chi_{4920}(409,·)$, $\chi_{4920}(3481,·)$, $\chi_{4920}(221,·)$, $\chi_{4920}(2669,·)$, $\chi_{4920}(4321,·)$, $\chi_{4920}(1829,·)$, $\chi_{4920}(4561,·)$, $\chi_{4920}(1589,·)$, $\chi_{4920}(1009,·)$, $\chi_{4920}(1229,·)$, $\chi_{4920}(821,·)$, $\chi_{4920}(1849,·)$, $\chi_{4920}(769,·)$, $\chi_{4920}(1781,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{2}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} + \frac{2}{9} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{27} a^{12} - \frac{1}{27} a^{10} - \frac{1}{9} a^{8} - \frac{1}{27} a^{6} + \frac{1}{27} a^{4} + \frac{1}{9} a^{2} + \frac{1}{3}$, $\frac{1}{27} a^{13} - \frac{1}{27} a^{11} - \frac{1}{27} a^{7} - \frac{8}{27} a^{5} - \frac{1}{3} a$, $\frac{1}{81} a^{14} - \frac{1}{81} a^{13} - \frac{2}{81} a^{11} - \frac{4}{81} a^{10} - \frac{1}{27} a^{9} - \frac{4}{81} a^{8} - \frac{8}{81} a^{7} + \frac{2}{81} a^{5} + \frac{31}{81} a^{4} - \frac{8}{27} a^{3} - \frac{5}{27} a^{2} + \frac{1}{9}$, $\frac{1}{81} a^{15} - \frac{1}{81} a^{13} + \frac{1}{81} a^{12} + \frac{1}{27} a^{11} - \frac{1}{81} a^{10} + \frac{2}{81} a^{9} + \frac{2}{27} a^{8} - \frac{8}{81} a^{7} - \frac{1}{81} a^{6} + \frac{8}{27} a^{5} + \frac{28}{81} a^{4} + \frac{2}{27} a^{3} + \frac{7}{27} a^{2} + \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{243} a^{16} + \frac{1}{81} a^{13} - \frac{2}{81} a^{11} + \frac{10}{243} a^{10} + \frac{4}{81} a^{9} - \frac{1}{81} a^{8} - \frac{4}{81} a^{7} - \frac{1}{9} a^{6} + \frac{20}{81} a^{5} - \frac{83}{243} a^{4} + \frac{23}{81} a^{3} - \frac{23}{81} a^{2} + \frac{4}{27} a - \frac{11}{27}$, $\frac{1}{729} a^{17} + \frac{1}{729} a^{16} - \frac{1}{243} a^{15} - \frac{1}{81} a^{12} - \frac{17}{729} a^{11} + \frac{37}{729} a^{10} + \frac{4}{243} a^{9} + \frac{20}{243} a^{8} - \frac{7}{81} a^{7} + \frac{4}{81} a^{6} - \frac{164}{729} a^{5} + \frac{52}{729} a^{4} + \frac{11}{27} a^{3} - \frac{44}{243} a^{2} + \frac{17}{81} a + \frac{28}{81}$, $\frac{1}{6974343} a^{18} + \frac{1199}{6974343} a^{17} - \frac{4127}{6974343} a^{16} - \frac{14014}{2324781} a^{15} + \frac{2179}{774927} a^{14} - \frac{29}{3189} a^{13} + \frac{81190}{6974343} a^{12} - \frac{338308}{6974343} a^{11} + \frac{173521}{6974343} a^{10} - \frac{27665}{774927} a^{9} + \frac{68912}{2324781} a^{8} - \frac{93925}{774927} a^{7} + \frac{1146130}{6974343} a^{6} + \frac{2462216}{6974343} a^{5} + \frac{1578469}{6974343} a^{4} + \frac{34501}{2324781} a^{3} + \frac{616504}{2324781} a^{2} + \frac{73736}{258309} a + \frac{235195}{774927}$, $\frac{1}{118909700362265111109473956048279214801820547726230994111744312819715450283} a^{19} + \frac{2379582073433408237750251854685883184600762971765467536302379461341}{39636566787421703703157985349426404933940182575410331370581437606571816761} a^{18} + \frac{8158798806467551706626430232521734178504033508089079221171971505553014}{39636566787421703703157985349426404933940182575410331370581437606571816761} a^{17} - \frac{94802683704669475457372999970504958424189633950214357239001589117315470}{118909700362265111109473956048279214801820547726230994111744312819715450283} a^{16} + \frac{28918387410711564784866983558425680858182386443086001172129155393989351}{39636566787421703703157985349426404933940182575410331370581437606571816761} a^{15} - \frac{11898406887919989707479653958198344321712476875904357265621591560156851}{13212188929140567901052661783142134977980060858470110456860479202190605587} a^{14} - \frac{69776772946452083495559012579057807101882463115417849281915328054726439}{118909700362265111109473956048279214801820547726230994111744312819715450283} a^{13} - \frac{719826975060794134728262158598483128999629201132883300147431071126009883}{39636566787421703703157985349426404933940182575410331370581437606571816761} a^{12} + \frac{1900847250117471770954898853088306555128114419449793953889095860529312194}{39636566787421703703157985349426404933940182575410331370581437606571816761} a^{11} + \frac{1679005029854478403043806279767322252635135406832000653355970077074287748}{118909700362265111109473956048279214801820547726230994111744312819715450283} a^{10} + \frac{986128868593149433844251069111388946821187164814585693509694810990149006}{39636566787421703703157985349426404933940182575410331370581437606571816761} a^{9} - \frac{6252835682573138167374965061894666143753084638302416837304225906000716200}{39636566787421703703157985349426404933940182575410331370581437606571816761} a^{8} - \frac{9485668152567969406891252587757402777264917524662846019349682919283093320}{118909700362265111109473956048279214801820547726230994111744312819715450283} a^{7} - \frac{4244813882744338009587124584458403593175052728774600353661380012563324808}{39636566787421703703157985349426404933940182575410331370581437606571816761} a^{6} - \frac{1291792754693797263170099519492547801883449161611487789187979550691498752}{13212188929140567901052661783142134977980060858470110456860479202190605587} a^{5} - \frac{9390526951818721038379553122151491546750002744507718009177758579916879526}{118909700362265111109473956048279214801820547726230994111744312819715450283} a^{4} + \frac{1966517648816295718414225666361567303965347855053219827605679558725639260}{39636566787421703703157985349426404933940182575410331370581437606571816761} a^{3} - \frac{1211383670352345311412976403390202296540603215615679816863871757981547796}{39636566787421703703157985349426404933940182575410331370581437606571816761} a^{2} + \frac{4917068757256719913359106619029675692801112954001319818546028892020822286}{13212188929140567901052661783142134977980060858470110456860479202190605587} a - \frac{4442944962074649603752673153653975876068489401385771084313748064662061355}{13212188929140567901052661783142134977980060858470110456860479202190605587}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{290532}$, which has order $297504768$ (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:  \( 3541438824.6395073 \) (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

\(\Q(\sqrt{205}) \), \(\Q(\sqrt{-1230}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-6}, \sqrt{205})\), 5.5.2825761.1, 10.10.1023068544981128125.1, 10.0.8146310149911810355200000.1, 10.0.63580957267604373504.1

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 R R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
41Data not computed