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Properties

Label 20.20.9775641609...9552.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{55}\cdot 3^{10}\cdot 11^{16}$
Root discriminant $79.34$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6247, 391236, -111054, -3602816, 1920559, 7942856, -3351642, -7029912, 2504547, 3078868, -977916, -732552, 215144, 97888, -27190, -7236, 1927, 272, -70, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 70*x^18 + 272*x^17 + 1927*x^16 - 7236*x^15 - 27190*x^14 + 97888*x^13 + 215144*x^12 - 732552*x^11 - 977916*x^10 + 3078868*x^9 + 2504547*x^8 - 7029912*x^7 - 3351642*x^6 + 7942856*x^5 + 1920559*x^4 - 3602816*x^3 - 111054*x^2 + 391236*x + 6247)
 
gp: K = bnfinit(x^20 - 4*x^19 - 70*x^18 + 272*x^17 + 1927*x^16 - 7236*x^15 - 27190*x^14 + 97888*x^13 + 215144*x^12 - 732552*x^11 - 977916*x^10 + 3078868*x^9 + 2504547*x^8 - 7029912*x^7 - 3351642*x^6 + 7942856*x^5 + 1920559*x^4 - 3602816*x^3 - 111054*x^2 + 391236*x + 6247, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 70 x^{18} + 272 x^{17} + 1927 x^{16} - 7236 x^{15} - 27190 x^{14} + 97888 x^{13} + 215144 x^{12} - 732552 x^{11} - 977916 x^{10} + 3078868 x^{9} + 2504547 x^{8} - 7029912 x^{7} - 3351642 x^{6} + 7942856 x^{5} + 1920559 x^{4} - 3602816 x^{3} - 111054 x^{2} + 391236 x + 6247 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(97756416090547441671665711234313879552=2^{55}\cdot 3^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.34$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(528=2^{4}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{528}(1,·)$, $\chi_{528}(515,·)$, $\chi_{528}(179,·)$, $\chi_{528}(289,·)$, $\chi_{528}(265,·)$, $\chi_{528}(203,·)$, $\chi_{528}(323,·)$, $\chi_{528}(25,·)$, $\chi_{528}(155,·)$, $\chi_{528}(361,·)$, $\chi_{528}(97,·)$, $\chi_{528}(419,·)$, $\chi_{528}(433,·)$, $\chi_{528}(169,·)$, $\chi_{528}(49,·)$, $\chi_{528}(467,·)$, $\chi_{528}(251,·)$, $\chi_{528}(313,·)$, $\chi_{528}(59,·)$, $\chi_{528}(443,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{7667572812246861} a^{18} - \frac{49976307178983}{2555857604082287} a^{17} - \frac{987140118967208}{7667572812246861} a^{16} + \frac{283407588278473}{2555857604082287} a^{15} + \frac{213143044832530}{2555857604082287} a^{14} + \frac{1079225814519094}{7667572812246861} a^{13} + \frac{284221522590708}{2555857604082287} a^{12} + \frac{22279529154227}{7667572812246861} a^{11} + \frac{233161550722880}{7667572812246861} a^{10} - \frac{1826868411170816}{7667572812246861} a^{9} - \frac{961865513810371}{2555857604082287} a^{8} - \frac{2101808009652335}{7667572812246861} a^{7} + \frac{1032586883293381}{2555857604082287} a^{6} - \frac{748428593863327}{2555857604082287} a^{5} + \frac{467920960232212}{7667572812246861} a^{4} - \frac{84815097156069}{2555857604082287} a^{3} + \frac{1088887364701601}{2555857604082287} a^{2} - \frac{436277306991420}{2555857604082287} a - \frac{10128437185090}{2555857604082287}$, $\frac{1}{9156706613327710347585019237737174531} a^{19} + \frac{8565928665353580614}{3052235537775903449195006412579058177} a^{18} + \frac{765711734908120684621121798681220061}{9156706613327710347585019237737174531} a^{17} - \frac{138300654114939989562429414152951002}{9156706613327710347585019237737174531} a^{16} - \frac{518529189154502822714159682055288529}{9156706613327710347585019237737174531} a^{15} - \frac{953045954797331171722021700248001620}{9156706613327710347585019237737174531} a^{14} + \frac{480438393465311452266287015789825964}{3052235537775903449195006412579058177} a^{13} + \frac{6661768264948585171098190036642240}{3052235537775903449195006412579058177} a^{12} - \frac{641256236999363530185490212255333080}{9156706613327710347585019237737174531} a^{11} - \frac{174701340868185329335410728514641020}{9156706613327710347585019237737174531} a^{10} - \frac{407621417858781904000328253389413732}{3052235537775903449195006412579058177} a^{9} - \frac{3003026091404914731632494375349752517}{9156706613327710347585019237737174531} a^{8} - \frac{4388898186937770758435800767915479416}{9156706613327710347585019237737174531} a^{7} - \frac{3805186103375349929588254887156266791}{9156706613327710347585019237737174531} a^{6} - \frac{398179379435314964369791453618788119}{3052235537775903449195006412579058177} a^{5} + \frac{4101777118495935189924039078955193942}{9156706613327710347585019237737174531} a^{4} - \frac{291780151840420038945587040814465676}{9156706613327710347585019237737174531} a^{3} - \frac{2665752635868562492494437535290532079}{9156706613327710347585019237737174531} a^{2} + \frac{1106967476808666693987163688586490042}{3052235537775903449195006412579058177} a - \frac{2224939838229534484625054437571437396}{9156706613327710347585019237737174531}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $19$
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:  \( 2842376749290 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.18432.1, \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1

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 $20$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ $20$ $20$

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
3Data not computed
11Data not computed