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Properties

Label 20.0.92738759037...5681.4
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $39.66$
Ramified primes $3, 7, 11$
Class number $66$ (GRH)
Class group $[66]$ (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![9765625, 1953125, 2343750, 859375, 640625, 300000, 188125, 97625, 57150, 30955, 17621, -6191, 2286, -781, 301, -96, 41, -11, 6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 6*x^18 - 11*x^17 + 41*x^16 - 96*x^15 + 301*x^14 - 781*x^13 + 2286*x^12 - 6191*x^11 + 17621*x^10 + 30955*x^9 + 57150*x^8 + 97625*x^7 + 188125*x^6 + 300000*x^5 + 640625*x^4 + 859375*x^3 + 2343750*x^2 + 1953125*x + 9765625)
 
gp: K = bnfinit(x^20 - x^19 + 6*x^18 - 11*x^17 + 41*x^16 - 96*x^15 + 301*x^14 - 781*x^13 + 2286*x^12 - 6191*x^11 + 17621*x^10 + 30955*x^9 + 57150*x^8 + 97625*x^7 + 188125*x^6 + 300000*x^5 + 640625*x^4 + 859375*x^3 + 2343750*x^2 + 1953125*x + 9765625, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 6 x^{18} - 11 x^{17} + 41 x^{16} - 96 x^{15} + 301 x^{14} - 781 x^{13} + 2286 x^{12} - 6191 x^{11} + 17621 x^{10} + 30955 x^{9} + 57150 x^{8} + 97625 x^{7} + 188125 x^{6} + 300000 x^{5} + 640625 x^{4} + 859375 x^{3} + 2343750 x^{2} + 1953125 x + 9765625 \)

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:  \(92738759037689478010716606945681=3^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.66$
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$$\chi_{231}(64,·)$, $\chi_{231}(1,·)$, $\chi_{231}(146,·)$, $\chi_{231}(83,·)$, $\chi_{231}(20,·)$, $\chi_{231}(85,·)$, $\chi_{231}(230,·)$, $\chi_{231}(167,·)$, $\chi_{231}(104,·)$, $\chi_{231}(41,·)$, $\chi_{231}(106,·)$, $\chi_{231}(43,·)$, $\chi_{231}(211,·)$, $\chi_{231}(62,·)$, $\chi_{231}(169,·)$, $\chi_{231}(148,·)$, $\chi_{231}(188,·)$, $\chi_{231}(125,·)$, $\chi_{231}(190,·)$, $\chi_{231}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{88105} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{6191}{17621}$, $\frac{1}{440525} a^{12} - \frac{1}{440525} a^{11} + \frac{11}{25} a^{10} + \frac{9}{25} a^{9} - \frac{4}{25} a^{8} - \frac{1}{25} a^{7} + \frac{6}{25} a^{6} - \frac{11}{25} a^{5} - \frac{9}{25} a^{4} + \frac{4}{25} a^{3} + \frac{1}{25} a^{2} + \frac{6191}{88105} a + \frac{2286}{17621}$, $\frac{1}{2202625} a^{13} - \frac{1}{2202625} a^{12} + \frac{6}{2202625} a^{11} + \frac{34}{125} a^{10} + \frac{21}{125} a^{9} + \frac{24}{125} a^{8} - \frac{44}{125} a^{7} + \frac{39}{125} a^{6} - \frac{9}{125} a^{5} - \frac{46}{125} a^{4} + \frac{1}{125} a^{3} + \frac{6191}{440525} a^{2} + \frac{2286}{88105} a + \frac{781}{17621}$, $\frac{1}{11013125} a^{14} - \frac{1}{11013125} a^{13} + \frac{6}{11013125} a^{12} - \frac{11}{11013125} a^{11} + \frac{21}{625} a^{10} + \frac{149}{625} a^{9} - \frac{44}{625} a^{8} + \frac{164}{625} a^{7} + \frac{241}{625} a^{6} - \frac{46}{625} a^{5} + \frac{1}{625} a^{4} + \frac{6191}{2202625} a^{3} + \frac{2286}{440525} a^{2} + \frac{781}{88105} a + \frac{301}{17621}$, $\frac{1}{55065625} a^{15} - \frac{1}{55065625} a^{14} + \frac{6}{55065625} a^{13} - \frac{11}{55065625} a^{12} + \frac{41}{55065625} a^{11} - \frac{476}{3125} a^{10} + \frac{581}{3125} a^{9} + \frac{164}{3125} a^{8} - \frac{384}{3125} a^{7} + \frac{1204}{3125} a^{6} + \frac{1}{3125} a^{5} + \frac{6191}{11013125} a^{4} + \frac{2286}{2202625} a^{3} + \frac{781}{440525} a^{2} + \frac{301}{88105} a + \frac{96}{17621}$, $\frac{1}{275328125} a^{16} - \frac{1}{275328125} a^{15} + \frac{6}{275328125} a^{14} - \frac{11}{275328125} a^{13} + \frac{41}{275328125} a^{12} - \frac{96}{275328125} a^{11} - \frac{2544}{15625} a^{10} + \frac{164}{15625} a^{9} + \frac{2741}{15625} a^{8} - \frac{1921}{15625} a^{7} + \frac{1}{15625} a^{6} + \frac{6191}{55065625} a^{5} + \frac{2286}{11013125} a^{4} + \frac{781}{2202625} a^{3} + \frac{301}{440525} a^{2} + \frac{96}{88105} a + \frac{41}{17621}$, $\frac{1}{1376640625} a^{17} - \frac{1}{1376640625} a^{16} + \frac{6}{1376640625} a^{15} - \frac{11}{1376640625} a^{14} + \frac{41}{1376640625} a^{13} - \frac{96}{1376640625} a^{12} + \frac{301}{1376640625} a^{11} + \frac{164}{78125} a^{10} - \frac{12884}{78125} a^{9} + \frac{13704}{78125} a^{8} + \frac{1}{78125} a^{7} + \frac{6191}{275328125} a^{6} + \frac{2286}{55065625} a^{5} + \frac{781}{11013125} a^{4} + \frac{301}{2202625} a^{3} + \frac{96}{440525} a^{2} + \frac{41}{88105} a + \frac{11}{17621}$, $\frac{1}{6883203125} a^{18} - \frac{1}{6883203125} a^{17} + \frac{6}{6883203125} a^{16} - \frac{11}{6883203125} a^{15} + \frac{41}{6883203125} a^{14} - \frac{96}{6883203125} a^{13} + \frac{301}{6883203125} a^{12} - \frac{781}{6883203125} a^{11} + \frac{65241}{390625} a^{10} - \frac{64421}{390625} a^{9} + \frac{1}{390625} a^{8} + \frac{6191}{1376640625} a^{7} + \frac{2286}{275328125} a^{6} + \frac{781}{55065625} a^{5} + \frac{301}{11013125} a^{4} + \frac{96}{2202625} a^{3} + \frac{41}{440525} a^{2} + \frac{11}{88105} a + \frac{6}{17621}$, $\frac{1}{34416015625} a^{19} - \frac{1}{34416015625} a^{18} + \frac{6}{34416015625} a^{17} - \frac{11}{34416015625} a^{16} + \frac{41}{34416015625} a^{15} - \frac{96}{34416015625} a^{14} + \frac{301}{34416015625} a^{13} - \frac{781}{34416015625} a^{12} + \frac{2286}{34416015625} a^{11} + \frac{326204}{1953125} a^{10} + \frac{1}{1953125} a^{9} + \frac{6191}{6883203125} a^{8} + \frac{2286}{1376640625} a^{7} + \frac{781}{275328125} a^{6} + \frac{301}{55065625} a^{5} + \frac{96}{11013125} a^{4} + \frac{41}{2202625} a^{3} + \frac{11}{440525} a^{2} + \frac{6}{88105} a + \frac{1}{17621}$

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

Class group and class number

$C_{66}$, which has order $66$ (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:  \( -\frac{1}{440525} a^{13} - \frac{136681}{440525} a^{2} \) (order $22$)
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:  \( 5868059.79956 \) (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

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-231}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{21})\), \(\Q(\zeta_{11})^+\), 10.10.875463320250981.1, 10.0.9630096522760791.1, \(\Q(\zeta_{11})\)

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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7Data not computed
11Data not computed