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Properties

Label 20.4.33984561499...0000.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{38}\cdot 5^{14}\cdot 1193^{4}$
Root discriminant $47.49$
Ramified primes $2, 5, 1193$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-200, 280, 618, -320, -577, -360, 1190, -40, 543, -800, 452, -720, 144, -420, 134, -120, 50, -20, 14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 14*x^18 - 20*x^17 + 50*x^16 - 120*x^15 + 134*x^14 - 420*x^13 + 144*x^12 - 720*x^11 + 452*x^10 - 800*x^9 + 543*x^8 - 40*x^7 + 1190*x^6 - 360*x^5 - 577*x^4 - 320*x^3 + 618*x^2 + 280*x - 200)
 
gp: K = bnfinit(x^20 + 14*x^18 - 20*x^17 + 50*x^16 - 120*x^15 + 134*x^14 - 420*x^13 + 144*x^12 - 720*x^11 + 452*x^10 - 800*x^9 + 543*x^8 - 40*x^7 + 1190*x^6 - 360*x^5 - 577*x^4 - 320*x^3 + 618*x^2 + 280*x - 200, 1)
 

Normalized defining polynomial

\( x^{20} + 14 x^{18} - 20 x^{17} + 50 x^{16} - 120 x^{15} + 134 x^{14} - 420 x^{13} + 144 x^{12} - 720 x^{11} + 452 x^{10} - 800 x^{9} + 543 x^{8} - 40 x^{7} + 1190 x^{6} - 360 x^{5} - 577 x^{4} - 320 x^{3} + 618 x^{2} + 280 x - 200 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3398456149909543321600000000000000=2^{38}\cdot 5^{14}\cdot 1193^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.49$
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:  $2, 5, 1193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{411235077805018779588592578080} a^{19} - \frac{678995062835029179073640327}{41123507780501877958859257808} a^{18} + \frac{87140270907824351758534949097}{205617538902509389794296289040} a^{17} + \frac{1271822189946893786118432793}{2570219236281367372428703613} a^{16} + \frac{10310859172429397255567140069}{41123507780501877958859257808} a^{15} + \frac{7912558797158797085394463483}{20561753890250938979429628904} a^{14} - \frac{409786895726399036494153793}{205617538902509389794296289040} a^{13} - \frac{4144331824625228853336298087}{10280876945125469489714814452} a^{12} - \frac{3410061181839293367692956693}{12851096181406836862143518065} a^{11} + \frac{344732851544184952139843793}{734348353223247820693915318} a^{10} - \frac{2955739970802985588594241339}{7908366880865745761319088040} a^{9} + \frac{4317310400846374448567009537}{10280876945125469489714814452} a^{8} - \frac{169291914102851802098901578897}{411235077805018779588592578080} a^{7} - \frac{16972691953960168311400903757}{41123507780501877958859257808} a^{6} - \frac{18413127705714900949310859419}{41123507780501877958859257808} a^{5} + \frac{366053926374050851531867809}{2937393412892991282775661272} a^{4} - \frac{107338497122742621253549792617}{411235077805018779588592578080} a^{3} + \frac{16502918782323226412722317903}{41123507780501877958859257808} a^{2} + \frac{73035548540271300826353271219}{205617538902509389794296289040} a + \frac{9779727952383555125375154849}{20561753890250938979429628904}$

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:  $11$
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:  \( 2457878590.06 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 116 conjugacy class representatives for t20n872 are not computed
Character table for t20n872 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.728703488000000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
5Data not computed
1193Data not computed