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Properties

Label 20.14.4946505972...4592.1
Degree $20$
Signature $[14, 3]$
Discriminant $-\,2^{30}\cdot 7^{11}\cdot 13^{12}$
Root discriminant $38.43$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T633

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-368, 4768, -4936, -44592, 37156, 123080, -44926, -118368, 31387, 57466, -13857, -17668, 3587, 4154, -667, -762, 128, 86, -18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 18*x^18 + 86*x^17 + 128*x^16 - 762*x^15 - 667*x^14 + 4154*x^13 + 3587*x^12 - 17668*x^11 - 13857*x^10 + 57466*x^9 + 31387*x^8 - 118368*x^7 - 44926*x^6 + 123080*x^5 + 37156*x^4 - 44592*x^3 - 4936*x^2 + 4768*x - 368)
 
gp: K = bnfinit(x^20 - 4*x^19 - 18*x^18 + 86*x^17 + 128*x^16 - 762*x^15 - 667*x^14 + 4154*x^13 + 3587*x^12 - 17668*x^11 - 13857*x^10 + 57466*x^9 + 31387*x^8 - 118368*x^7 - 44926*x^6 + 123080*x^5 + 37156*x^4 - 44592*x^3 - 4936*x^2 + 4768*x - 368, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 18 x^{18} + 86 x^{17} + 128 x^{16} - 762 x^{15} - 667 x^{14} + 4154 x^{13} + 3587 x^{12} - 17668 x^{11} - 13857 x^{10} + 57466 x^{9} + 31387 x^{8} - 118368 x^{7} - 44926 x^{6} + 123080 x^{5} + 37156 x^{4} - 44592 x^{3} - 4936 x^{2} + 4768 x - 368 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-49465059721539005964938842734592=-\,2^{30}\cdot 7^{11}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.43$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{12} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{18} - \frac{1}{4} a^{15} + \frac{1}{4} a^{13} - \frac{3}{8} a^{12} + \frac{1}{4} a^{11} - \frac{3}{8} a^{10} - \frac{1}{2} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{3}{8} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4158114321233013324454025463059411416} a^{19} - \frac{158130980741711424364517298651142495}{4158114321233013324454025463059411416} a^{18} - \frac{43615160776496954189251308086525373}{519764290154126665556753182882426427} a^{17} - \frac{84338515007825531471307025747096265}{2079057160616506662227012731529705708} a^{16} - \frac{495543031992500585875303366575752749}{2079057160616506662227012731529705708} a^{15} + \frac{89497343214407392218627370368996809}{2079057160616506662227012731529705708} a^{14} + \frac{1026169165114677890388085108320750695}{4158114321233013324454025463059411416} a^{13} + \frac{1589267257002692793847674065684267231}{4158114321233013324454025463059411416} a^{12} - \frac{298604059166011300635041163340051137}{4158114321233013324454025463059411416} a^{11} + \frac{1898216901146395236012473625134899593}{4158114321233013324454025463059411416} a^{10} + \frac{1612398990511138856692475307337677629}{4158114321233013324454025463059411416} a^{9} + \frac{1297648929901572072382922374269773751}{4158114321233013324454025463059411416} a^{8} + \frac{636275782140778122468061868153830715}{4158114321233013324454025463059411416} a^{7} + \frac{1680878166558777471192485299146757797}{4158114321233013324454025463059411416} a^{6} + \frac{330707356237643771942490606642486435}{1039528580308253331113506365764852854} a^{5} + \frac{31669760806092413284802813819630809}{1039528580308253331113506365764852854} a^{4} + \frac{176140702292565955522584355180854867}{1039528580308253331113506365764852854} a^{3} - \frac{481312739513361208930938134334807959}{1039528580308253331113506365764852854} a^{2} + \frac{234309740858423248846028221962303794}{519764290154126665556753182882426427} a + \frac{218354192409054412311472662753369464}{519764290154126665556753182882426427}$

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:  $16$
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:  \( 1106160337.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.6889792.1, 10.10.379753870426112.1

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 $20$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.6.3$x^{8} - 7 x^{4} + 147$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
13Data not computed