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Properties

Label 20.10.2056246033...4256.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{48}\cdot 31^{7}\cdot 227^{4}$
Root discriminant $51.96$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![76807, -98868, -199392, 327188, -33791, -455176, 170086, 325044, -6466, -123496, -38018, 28708, 13882, -4456, -2442, 436, 303, -20, -26, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 26*x^18 - 20*x^17 + 303*x^16 + 436*x^15 - 2442*x^14 - 4456*x^13 + 13882*x^12 + 28708*x^11 - 38018*x^10 - 123496*x^9 - 6466*x^8 + 325044*x^7 + 170086*x^6 - 455176*x^5 - 33791*x^4 + 327188*x^3 - 199392*x^2 - 98868*x + 76807)
 
gp: K = bnfinit(x^20 - 26*x^18 - 20*x^17 + 303*x^16 + 436*x^15 - 2442*x^14 - 4456*x^13 + 13882*x^12 + 28708*x^11 - 38018*x^10 - 123496*x^9 - 6466*x^8 + 325044*x^7 + 170086*x^6 - 455176*x^5 - 33791*x^4 + 327188*x^3 - 199392*x^2 - 98868*x + 76807, 1)
 

Normalized defining polynomial

\( x^{20} - 26 x^{18} - 20 x^{17} + 303 x^{16} + 436 x^{15} - 2442 x^{14} - 4456 x^{13} + 13882 x^{12} + 28708 x^{11} - 38018 x^{10} - 123496 x^{9} - 6466 x^{8} + 325044 x^{7} + 170086 x^{6} - 455176 x^{5} - 33791 x^{4} + 327188 x^{3} - 199392 x^{2} - 98868 x + 76807 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-20562460332300807986799747065184256=-\,2^{48}\cdot 31^{7}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.96$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{16} - \frac{1}{4}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{16} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{216} a^{18} + \frac{1}{24} a^{17} - \frac{1}{108} a^{15} - \frac{1}{18} a^{14} - \frac{2}{9} a^{13} + \frac{5}{54} a^{11} + \frac{11}{108} a^{10} + \frac{25}{108} a^{9} - \frac{1}{36} a^{8} - \frac{17}{36} a^{7} + \frac{5}{54} a^{6} - \frac{1}{9} a^{5} + \frac{5}{54} a^{4} + \frac{4}{27} a^{3} - \frac{43}{216} a^{2} - \frac{13}{72} a + \frac{23}{108}$, $\frac{1}{17211285059489788018811164690376130086276330669352} a^{19} - \frac{34010945982970289759326091926412993130110009447}{17211285059489788018811164690376130086276330669352} a^{18} + \frac{797711519966939630828182644197414935444655681}{956182503304988223267286927243118338126462814964} a^{17} + \frac{10956171145346641824523684587277594510342878679}{4302821264872447004702791172594032521569082667338} a^{16} - \frac{197176990552190170093993204295172019131011273059}{8605642529744894009405582345188065043138165334676} a^{15} + \frac{61382883359404357236852249081499840556252925151}{1434273754957482334900930390864677507189694222446} a^{14} + \frac{702929343264809803236468017594805949364595276613}{2868547509914964669801860781729355014379388444892} a^{13} - \frac{83813752960096035519683235032140042402326890321}{2151410632436223502351395586297016260784541333669} a^{12} + \frac{33151560856958358347497536465161416053781065853}{717136877478741167450465195432338753594847111223} a^{11} + \frac{56536106859538419191119489644804172335470778953}{318727501101662741089095642414372779375487604988} a^{10} - \frac{144047011754714648456568969903090917848905002311}{4302821264872447004702791172594032521569082667338} a^{9} + \frac{69756423228372992524901790098953400823839570463}{318727501101662741089095642414372779375487604988} a^{8} + \frac{879116883660259031261245396717462186533904312621}{8605642529744894009405582345188065043138165334676} a^{7} + \frac{1717237956138409986457990934603635663079435892497}{4302821264872447004702791172594032521569082667338} a^{6} + \frac{3537834962128425717298139583906982062136905395437}{8605642529744894009405582345188065043138165334676} a^{5} - \frac{1728194088322774882060766583048071458099318819805}{4302821264872447004702791172594032521569082667338} a^{4} + \frac{4958120584989789559027880454228066863570152483543}{17211285059489788018811164690376130086276330669352} a^{3} - \frac{3385951957914078673008610726896628369795071988435}{17211285059489788018811164690376130086276330669352} a^{2} - \frac{1368674723724307769897302674936252645140362554241}{8605642529744894009405582345188065043138165334676} a - \frac{1143303313004759963386363304541437764696541655967}{8605642529744894009405582345188065043138165334676}$

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:  $14$
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:  \( 47048090807.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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 $16{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ R $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.8.26.1$x^{8} + 4 x^{6} + 8 x^{3} + 8 x^{2} + 2$$8$$1$$26$$C_2^2:C_4$$[2, 3, 7/2, 4]$
31Data not computed
227Data not computed