↑

Properties

Label 20.12.7820035525...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{10}\cdot 3169^{4}\cdot 27517559^{2}$
Root discriminant $124.35$
Ramified primes $2, 5, 3169, 27517559$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1030

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![100853031438721, 0, 34720939507619, 0, 5645857496273, 0, -843346065511, 0, 5037137676, 0, 1810191294, 0, -50014572, 0, -248166, 0, 23171, 0, -275, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 275*x^18 + 23171*x^16 - 248166*x^14 - 50014572*x^12 + 1810191294*x^10 + 5037137676*x^8 - 843346065511*x^6 + 5645857496273*x^4 + 34720939507619*x^2 + 100853031438721)
 
gp: K = bnfinit(x^20 - 275*x^18 + 23171*x^16 - 248166*x^14 - 50014572*x^12 + 1810191294*x^10 + 5037137676*x^8 - 843346065511*x^6 + 5645857496273*x^4 + 34720939507619*x^2 + 100853031438721, 1)
 

Normalized defining polynomial

\( x^{20} - 275 x^{18} + 23171 x^{16} - 248166 x^{14} - 50014572 x^{12} + 1810191294 x^{10} + 5037137676 x^{8} - 843346065511 x^{6} + 5645857496273 x^{4} + 34720939507619 x^{2} + 100853031438721 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(782003552575825933249006620682240000000000=2^{20}\cdot 5^{10}\cdot 3169^{4}\cdot 27517559^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $124.35$
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:  $2, 5, 3169, 27517559$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3169} a^{14} - \frac{275}{3169} a^{12} + \frac{988}{3169} a^{10} - \frac{984}{3169} a^{8} - \frac{1414}{3169} a^{6} + \frac{1452}{3169} a^{4} - \frac{500}{3169} a^{2}$, $\frac{1}{3169} a^{15} - \frac{275}{3169} a^{13} + \frac{988}{3169} a^{11} - \frac{984}{3169} a^{9} - \frac{1414}{3169} a^{7} + \frac{1452}{3169} a^{5} - \frac{500}{3169} a^{3}$, $\frac{1}{10042561} a^{16} - \frac{275}{10042561} a^{14} + \frac{23171}{10042561} a^{12} - \frac{248166}{10042561} a^{10} + \frac{198233}{10042561} a^{8} + \frac{2530314}{10042561} a^{6} - \frac{4227946}{10042561} a^{4} - \frac{606}{3169} a^{2}$, $\frac{1}{10042561} a^{17} - \frac{275}{10042561} a^{15} + \frac{23171}{10042561} a^{13} - \frac{248166}{10042561} a^{11} + \frac{198233}{10042561} a^{9} + \frac{2530314}{10042561} a^{7} - \frac{4227946}{10042561} a^{5} - \frac{606}{3169} a^{3}$, $\frac{1}{29585832997862312006671926699406945533867827688156193174001} a^{18} + \frac{426122810514496153010222363775173116601550386348272}{29585832997862312006671926699406945533867827688156193174001} a^{16} - \frac{3188025336419381866158892278962455569935358658942734361}{29585832997862312006671926699406945533867827688156193174001} a^{14} + \frac{9675325805512267727658602851507089502180438738960883508222}{29585832997862312006671926699406945533867827688156193174001} a^{12} + \frac{4794628857714197613772919989202194308298791307773175059757}{29585832997862312006671926699406945533867827688156193174001} a^{10} + \frac{3322151752041381592951205787474859293015715321467564331963}{29585832997862312006671926699406945533867827688156193174001} a^{8} - \frac{10963117182750925438059394535692496991215370299745012488755}{29585832997862312006671926699406945533867827688156193174001} a^{6} + \frac{1946666955853750892430615769926705057248870038478406712}{9336015461616381194910674250365082213274795736243670929} a^{4} - \frac{259613571237078465311904894961603798834878275295054}{2946044639197343387475757100146759928455284233589041} a^{2} - \frac{191208238156829490578903756333824938037561072281}{929644884568426439720971000361868074615110203089}$, $\frac{1}{29585832997862312006671926699406945533867827688156193174001} a^{19} + \frac{426122810514496153010222363775173116601550386348272}{29585832997862312006671926699406945533867827688156193174001} a^{17} - \frac{3188025336419381866158892278962455569935358658942734361}{29585832997862312006671926699406945533867827688156193174001} a^{15} + \frac{9675325805512267727658602851507089502180438738960883508222}{29585832997862312006671926699406945533867827688156193174001} a^{13} + \frac{4794628857714197613772919989202194308298791307773175059757}{29585832997862312006671926699406945533867827688156193174001} a^{11} + \frac{3322151752041381592951205787474859293015715321467564331963}{29585832997862312006671926699406945533867827688156193174001} a^{9} - \frac{10963117182750925438059394535692496991215370299745012488755}{29585832997862312006671926699406945533867827688156193174001} a^{7} + \frac{1946666955853750892430615769926705057248870038478406712}{9336015461616381194910674250365082213274795736243670929} a^{5} - \frac{259613571237078465311904894961603798834878275295054}{2946044639197343387475757100146759928455284233589041} a^{3} - \frac{191208238156829490578903756333824938037561072281}{929644884568426439720971000361868074615110203089} a$

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:  $15$
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:  \( 48029980610800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1030:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1030 are not computed
Character table for t20n1030 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.8.85992371875.1

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 32 sibling: 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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3169Data not computed
27517559Data not computed