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Properties

Label 20.20.4618412125...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{16}\cdot 3^{18}\cdot 5^{39}$
Root discriminant $107.95$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_5:F_5$ (as 20T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3600, -54000, 877500, -2079000, -421875, 5261850, -2860875, -3948750, 3435750, 1080000, -1431615, -127800, 300375, 6600, -34875, -120, 2250, 0, -75, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 75*x^18 + 2250*x^16 - 120*x^15 - 34875*x^14 + 6600*x^13 + 300375*x^12 - 127800*x^11 - 1431615*x^10 + 1080000*x^9 + 3435750*x^8 - 3948750*x^7 - 2860875*x^6 + 5261850*x^5 - 421875*x^4 - 2079000*x^3 + 877500*x^2 - 54000*x - 3600)
 
gp: K = bnfinit(x^20 - 75*x^18 + 2250*x^16 - 120*x^15 - 34875*x^14 + 6600*x^13 + 300375*x^12 - 127800*x^11 - 1431615*x^10 + 1080000*x^9 + 3435750*x^8 - 3948750*x^7 - 2860875*x^6 + 5261850*x^5 - 421875*x^4 - 2079000*x^3 + 877500*x^2 - 54000*x - 3600, 1)
 

Normalized defining polynomial

\( x^{20} - 75 x^{18} + 2250 x^{16} - 120 x^{15} - 34875 x^{14} + 6600 x^{13} + 300375 x^{12} - 127800 x^{11} - 1431615 x^{10} + 1080000 x^{9} + 3435750 x^{8} - 3948750 x^{7} - 2860875 x^{6} + 5261850 x^{5} - 421875 x^{4} - 2079000 x^{3} + 877500 x^{2} - 54000 x - 3600 \)

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:  \(46184121251106262207031250000000000000000=2^{16}\cdot 3^{18}\cdot 5^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $107.95$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{15} a^{10}$, $\frac{1}{15} a^{11}$, $\frac{1}{15} a^{12}$, $\frac{1}{15} a^{13}$, $\frac{1}{15} a^{14}$, $\frac{1}{15} a^{15}$, $\frac{1}{15} a^{16}$, $\frac{1}{30} a^{17} - \frac{1}{30} a^{15} - \frac{1}{30} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{85800} a^{18} - \frac{47}{8580} a^{17} - \frac{1}{440} a^{16} - \frac{139}{8580} a^{15} + \frac{217}{8580} a^{14} + \frac{23}{4290} a^{13} - \frac{149}{5720} a^{12} + \frac{53}{1716} a^{11} - \frac{1}{104} a^{10} - \frac{7}{572} a^{9} - \frac{1953}{5720} a^{8} - \frac{85}{572} a^{7} - \frac{267}{572} a^{6} + \frac{141}{572} a^{5} + \frac{259}{1144} a^{4} - \frac{81}{286} a^{3} - \frac{485}{1144} a^{2} - \frac{229}{572} a + \frac{15}{286}$, $\frac{1}{147127874105097376167897963600} a^{19} - \frac{21580010829951202453271}{6130328087712390673662415150} a^{18} + \frac{317703034855195445501294213}{29425574821019475233579592720} a^{17} + \frac{1457319711827149742156001}{55730255342839915215112865} a^{16} + \frac{428983434632562695633706899}{14712787410509737616789796360} a^{15} + \frac{17705329945120613192029478}{613032808771239067366241515} a^{14} - \frac{292894537518144272493649661}{9808524940339825077859864240} a^{13} + \frac{507298104857607025290488}{167190766028519745645338595} a^{12} + \frac{316290097397655934309581381}{9808524940339825077859864240} a^{11} - \frac{61803360133751482186966549}{3678196852627434404197449090} a^{10} + \frac{3175056774134619987587478467}{9808524940339825077859864240} a^{9} - \frac{18779249717759313000196268}{55730255342839915215112865} a^{8} - \frac{27353940252426316341307797}{75450191848767885214306648} a^{7} + \frac{4942765633611215346844055}{980852494033982507785986424} a^{6} - \frac{717558652260491158121218393}{1961704988067965015571972848} a^{5} + \frac{456089136101740565686248455}{980852494033982507785986424} a^{4} - \frac{619778835005437642454612477}{1961704988067965015571972848} a^{3} + \frac{43134324948985822360903987}{122606561754247813473248303} a^{2} + \frac{46332371512547206819621377}{122606561754247813473248303} a + \frac{53894859271763444637497079}{245213123508495626946496606}$

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:  \( 128509830940000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_5:F_5$ (as 20T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 20 conjugacy class representatives for $C_2\times C_5:F_5$
Character table for $C_2\times C_5:F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 10.10.32036132812500000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
3Data not computed
5Data not computed