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Properties

Label 20.6.20562460332...4256.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{48}\cdot 31^{7}\cdot 227^{4}$
Root discriminant $51.96$
Ramified primes $2, 31, 227$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3671, 18092, 29960, -80120, 54417, 22904, -109512, 143472, -109238, 51592, -25840, 18088, -8218, 2424, -184, -640, 197, -4, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 8*x^18 - 4*x^17 + 197*x^16 - 640*x^15 - 184*x^14 + 2424*x^13 - 8218*x^12 + 18088*x^11 - 25840*x^10 + 51592*x^9 - 109238*x^8 + 143472*x^7 - 109512*x^6 + 22904*x^5 + 54417*x^4 - 80120*x^3 + 29960*x^2 + 18092*x - 3671)
 
gp: K = bnfinit(x^20 + 8*x^18 - 4*x^17 + 197*x^16 - 640*x^15 - 184*x^14 + 2424*x^13 - 8218*x^12 + 18088*x^11 - 25840*x^10 + 51592*x^9 - 109238*x^8 + 143472*x^7 - 109512*x^6 + 22904*x^5 + 54417*x^4 - 80120*x^3 + 29960*x^2 + 18092*x - 3671, 1)
 

Normalized defining polynomial

\( x^{20} + 8 x^{18} - 4 x^{17} + 197 x^{16} - 640 x^{15} - 184 x^{14} + 2424 x^{13} - 8218 x^{12} + 18088 x^{11} - 25840 x^{10} + 51592 x^{9} - 109238 x^{8} + 143472 x^{7} - 109512 x^{6} + 22904 x^{5} + 54417 x^{4} - 80120 x^{3} + 29960 x^{2} + 18092 x - 3671 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{96} a^{16} - \frac{1}{8} a^{15} - \frac{5}{48} a^{14} - \frac{1}{12} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{1}{16} a^{10} - \frac{1}{12} a^{9} - \frac{1}{24} a^{8} + \frac{1}{24} a^{7} - \frac{7}{48} a^{6} - \frac{1}{2} a^{5} + \frac{1}{48} a^{4} - \frac{1}{8} a^{3} + \frac{11}{48} a^{2} + \frac{23}{96}$, $\frac{1}{96} a^{17} - \frac{5}{48} a^{15} - \frac{1}{12} a^{14} - \frac{1}{16} a^{13} - \frac{1}{8} a^{12} - \frac{1}{16} a^{11} - \frac{1}{12} a^{10} - \frac{1}{24} a^{9} - \frac{5}{24} a^{8} - \frac{7}{48} a^{7} + \frac{1}{48} a^{5} - \frac{1}{8} a^{4} + \frac{11}{48} a^{3} - \frac{1}{2} a^{2} + \frac{23}{96} a + \frac{1}{8}$, $\frac{1}{192} a^{18} - \frac{1}{192} a^{16} + \frac{1}{48} a^{15} - \frac{1}{8} a^{14} - \frac{1}{16} a^{13} + \frac{1}{16} a^{12} + \frac{1}{48} a^{11} - \frac{17}{96} a^{10} + \frac{7}{48} a^{9} + \frac{11}{96} a^{8} - \frac{3}{16} a^{7} - \frac{1}{48} a^{6} - \frac{7}{16} a^{5} - \frac{1}{6} a^{4} - \frac{3}{16} a^{3} + \frac{5}{192} a^{2} + \frac{3}{16} a - \frac{19}{64}$, $\frac{1}{20699103224654372413053236042565684383296511040} a^{19} - \frac{57705176371736915746527272777524553101907}{215615658590149712635971208776725878992671990} a^{18} + \frac{31765209624066442228297238947764483526654429}{6899701074884790804351078680855228127765503680} a^{17} + \frac{2257730160600708901764508743744643897342821}{1724925268721197701087769670213807031941375920} a^{16} + \frac{188646512500109967195285872264628287728869061}{2587387903081796551631654505320710547912063880} a^{15} - \frac{547590833554256977446018884525293206834438919}{5174775806163593103263309010641421095824127760} a^{14} + \frac{531822745656827075044272816220541889840139727}{5174775806163593103263309010641421095824127760} a^{13} - \frac{370717448360506186675486588878492119141505623}{5174775806163593103263309010641421095824127760} a^{12} - \frac{563200556955188660618115064318521557489842577}{10349551612327186206526618021282842191648255520} a^{11} - \frac{710658021423246844019472395280729528411079741}{5174775806163593103263309010641421095824127760} a^{10} - \frac{2474860160664847817352102617939778640362893621}{10349551612327186206526618021282842191648255520} a^{9} + \frac{1099445430642420201649197966353472806117619419}{5174775806163593103263309010641421095824127760} a^{8} - \frac{197034533886726008374319565153141403813120061}{1034955161232718620652661802128284219164825552} a^{7} + \frac{516210031041334198584183057880855803672143183}{5174775806163593103263309010641421095824127760} a^{6} + \frac{135642664966641268730052216696457644923399883}{431231317180299425271942417553451757985343980} a^{5} - \frac{414135405602413531841934534967531022870978681}{5174775806163593103263309010641421095824127760} a^{4} - \frac{203881049941417179286829459271156517800892373}{1379940214976958160870215736171045625553100736} a^{3} - \frac{176381674336752382541501269554188546134065023}{1034955161232718620652661802128284219164825552} a^{2} - \frac{1182570108864394185327683005994503946189190461}{4139820644930874482610647208513136876659302208} a + \frac{1044512223259489089026508454600992802789733699}{2587387903081796551631654505320710547912063880}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  $12$
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:  \( 2962968264.51 \) (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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.12.26.64$x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{6} - 2 x^{4} + 4 x^{3} + 2$$12$$1$$26$$S_3 \times C_2^2$$[2, 3]_{3}^{2}$
31Data not computed
227Data not computed