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Properties

Label 20.0.53092940800...9361.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 111847^{2}\cdot 847789^{2}$
Root discriminant $21.69$
Ramified primes $3, 111847, 847789$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1021

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 16, -22, 48, -38, 97, -47, 102, 6, 74, 12, 20, 33, 23, -6, 19, -2, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 5*x^18 - 2*x^17 + 19*x^16 - 6*x^15 + 23*x^14 + 33*x^13 + 20*x^12 + 12*x^11 + 74*x^10 + 6*x^9 + 102*x^8 - 47*x^7 + 97*x^6 - 38*x^5 + 48*x^4 - 22*x^3 + 16*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 5*x^18 - 2*x^17 + 19*x^16 - 6*x^15 + 23*x^14 + 33*x^13 + 20*x^12 + 12*x^11 + 74*x^10 + 6*x^9 + 102*x^8 - 47*x^7 + 97*x^6 - 38*x^5 + 48*x^4 - 22*x^3 + 16*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 5 x^{18} - 2 x^{17} + 19 x^{16} - 6 x^{15} + 23 x^{14} + 33 x^{13} + 20 x^{12} + 12 x^{11} + 74 x^{10} + 6 x^{9} + 102 x^{8} - 47 x^{7} + 97 x^{6} - 38 x^{5} + 48 x^{4} - 22 x^{3} + 16 x^{2} - 4 x + 1 \)

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:  \(530929408000357237198679361=3^{10}\cdot 111847^{2}\cdot 847789^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.69$
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:  $3, 111847, 847789$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{12576498905395} a^{19} - \frac{3698766099172}{12576498905395} a^{18} + \frac{5521915809767}{12576498905395} a^{17} + \frac{4380626021141}{12576498905395} a^{16} + \frac{2831346423623}{12576498905395} a^{15} + \frac{1309311225696}{12576498905395} a^{14} - \frac{1921274941523}{12576498905395} a^{13} + \frac{7500397166}{12576498905395} a^{12} + \frac{1499019750609}{12576498905395} a^{11} + \frac{2833152197328}{12576498905395} a^{10} - \frac{2095580846569}{12576498905395} a^{9} + \frac{558245242776}{2515299781079} a^{8} - \frac{299134635058}{12576498905395} a^{7} - \frac{3490014912829}{12576498905395} a^{6} + \frac{2695744129621}{12576498905395} a^{5} - \frac{5789210271804}{12576498905395} a^{4} + \frac{5838682742342}{12576498905395} a^{3} - \frac{3310170426974}{12576498905395} a^{2} - \frac{426043966390}{2515299781079} a + \frac{519507451956}{12576498905395}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{3361633261626}{12576498905395} a^{19} - \frac{3623347259102}{12576498905395} a^{18} + \frac{17114844342352}{12576498905395} a^{17} - \frac{7763233506459}{12576498905395} a^{16} + \frac{64212095893223}{12576498905395} a^{15} - \frac{23608941050929}{12576498905395} a^{14} + \frac{78858766968322}{12576498905395} a^{13} + \frac{110612692903771}{12576498905395} a^{12} + \frac{56775544615164}{12576498905395} a^{11} + \frac{44129580534528}{12576498905395} a^{10} + \frac{258084852658526}{12576498905395} a^{9} + \frac{1104763728065}{2515299781079} a^{8} + \frac{348735687556197}{12576498905395} a^{7} - \frac{155733737429844}{12576498905395} a^{6} + \frac{346459276204521}{12576498905395} a^{5} - \frac{126694173253874}{12576498905395} a^{4} + \frac{160929947023112}{12576498905395} a^{3} - \frac{49403472273574}{12576498905395} a^{2} + \frac{10667919130648}{2515299781079} a - \frac{729562165914}{12576498905395} \) (order $6$)
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:  \( 249242.742274 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1021:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7257600
The 84 conjugacy class representatives for t20n1021 are not computed
Character table for t20n1021 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.8.94822656283.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
3Data not computed
111847Data not computed
847789Data not computed