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Properties

Label 20.0.61917364224...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 3^{10}\cdot 5^{35}$
Root discriminant $194.80$
Ramified primes $2, 3, 5$
Class number $25589152$ (GRH)
Class group $[2, 4, 3198644]$ (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![17469056160, 0, 49191753600, 0, 50624676000, 0, 23234688000, 0, 4844788200, 0, 528359760, 0, 32464800, 0, 1150200, 0, 23040, 0, 240, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 240*x^18 + 23040*x^16 + 1150200*x^14 + 32464800*x^12 + 528359760*x^10 + 4844788200*x^8 + 23234688000*x^6 + 50624676000*x^4 + 49191753600*x^2 + 17469056160)
 
gp: K = bnfinit(x^20 + 240*x^18 + 23040*x^16 + 1150200*x^14 + 32464800*x^12 + 528359760*x^10 + 4844788200*x^8 + 23234688000*x^6 + 50624676000*x^4 + 49191753600*x^2 + 17469056160, 1)
 

Normalized defining polynomial

\( x^{20} + 240 x^{18} + 23040 x^{16} + 1150200 x^{14} + 32464800 x^{12} + 528359760 x^{10} + 4844788200 x^{8} + 23234688000 x^{6} + 50624676000 x^{4} + 49191753600 x^{2} + 17469056160 \)

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:  \(6191736422400000000000000000000000000000000000=2^{55}\cdot 3^{10}\cdot 5^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $194.80$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1200=2^{4}\cdot 3\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{1200}(1,·)$, $\chi_{1200}(203,·)$, $\chi_{1200}(961,·)$, $\chi_{1200}(649,·)$, $\chi_{1200}(1163,·)$, $\chi_{1200}(721,·)$, $\chi_{1200}(227,·)$, $\chi_{1200}(409,·)$, $\chi_{1200}(923,·)$, $\chi_{1200}(947,·)$, $\chi_{1200}(481,·)$, $\chi_{1200}(1187,·)$, $\chi_{1200}(1129,·)$, $\chi_{1200}(683,·)$, $\chi_{1200}(241,·)$, $\chi_{1200}(467,·)$, $\chi_{1200}(707,·)$, $\chi_{1200}(169,·)$, $\chi_{1200}(889,·)$, $\chi_{1200}(443,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{18} a^{4}$, $\frac{1}{18} a^{5}$, $\frac{1}{54} a^{6}$, $\frac{1}{54} a^{7}$, $\frac{1}{324} a^{8}$, $\frac{1}{324} a^{9}$, $\frac{1}{972} a^{10}$, $\frac{1}{972} a^{11}$, $\frac{1}{5832} a^{12}$, $\frac{1}{5832} a^{13}$, $\frac{1}{17496} a^{14}$, $\frac{1}{17496} a^{15}$, $\frac{1}{1763491824} a^{16} - \frac{14}{36739413} a^{14} + \frac{6977}{97971768} a^{12} - \frac{8015}{16328628} a^{10} + \frac{3409}{5442876} a^{8} - \frac{1201}{907146} a^{6} - \frac{406}{151191} a^{4} - \frac{4151}{50397} a^{2} + \frac{3440}{16799}$, $\frac{1}{75830148432} a^{17} + \frac{50285}{12638358072} a^{15} + \frac{208565}{4212786024} a^{13} + \frac{2939}{16328628} a^{11} - \frac{97385}{234043668} a^{9} - \frac{337181}{39007278} a^{7} + \frac{167584}{6501213} a^{5} + \frac{197437}{2167071} a^{3} - \frac{231746}{722357} a$, $\frac{1}{1160077071008557531728} a^{18} - \frac{26093869937}{193346178501426255288} a^{16} - \frac{432991267203419}{21482908722380695032} a^{14} - \frac{1575224903581}{62450316053432253} a^{12} - \frac{5849309706043}{66305273834508318} a^{10} - \frac{516776816875795}{596747464510574862} a^{8} - \frac{169040864479861}{22101757944836106} a^{6} + \frac{33112735296538}{3683626324139351} a^{4} + \frac{443896162578566}{11050878972418053} a^{2} + \frac{3564747576974}{85665728468357}$, $\frac{1}{1160077071008557531728} a^{19} - \frac{11048143}{3580484787063449172} a^{17} + \frac{1265286663912293}{64448726167142085096} a^{15} - \frac{35447407943221}{795663286014099816} a^{13} - \frac{619289168241457}{1790242393531724586} a^{11} + \frac{588973167421}{513994370810142} a^{9} - \frac{148813148174912}{99457910751762477} a^{7} - \frac{730364258516831}{66305273834508318} a^{5} - \frac{179591243653859}{3683626324139351} a^{3} - \frac{613617648262845}{3683626324139351} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{3198644}$, which has order $25589152$ (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:  \( 19344397.966990974 \) (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

\(\Q(\sqrt{10}) \), 4.0.2304000.2, 5.5.390625.1, 10.10.25000000000000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
$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}$
5Data not computed