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Properties

Label 20.0.19901427351...4809.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 71^{2}\cdot 401^{8}$
Root discriminant $29.17$
Ramified primes $3, 71, 401$
Class number $8$
Class group $[2, 2, 2]$
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -30, 127, -174, 542, -677, 1529, -1316, 2016, -1557, 1819, -1174, 1044, -553, 405, -165, 95, -22, 11, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 11*x^18 - 22*x^17 + 95*x^16 - 165*x^15 + 405*x^14 - 553*x^13 + 1044*x^12 - 1174*x^11 + 1819*x^10 - 1557*x^9 + 2016*x^8 - 1316*x^7 + 1529*x^6 - 677*x^5 + 542*x^4 - 174*x^3 + 127*x^2 - 30*x + 9)
 
gp: K = bnfinit(x^20 - x^19 + 11*x^18 - 22*x^17 + 95*x^16 - 165*x^15 + 405*x^14 - 553*x^13 + 1044*x^12 - 1174*x^11 + 1819*x^10 - 1557*x^9 + 2016*x^8 - 1316*x^7 + 1529*x^6 - 677*x^5 + 542*x^4 - 174*x^3 + 127*x^2 - 30*x + 9, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 11 x^{18} - 22 x^{17} + 95 x^{16} - 165 x^{15} + 405 x^{14} - 553 x^{13} + 1044 x^{12} - 1174 x^{11} + 1819 x^{10} - 1557 x^{9} + 2016 x^{8} - 1316 x^{7} + 1529 x^{6} - 677 x^{5} + 542 x^{4} - 174 x^{3} + 127 x^{2} - 30 x + 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(199014273518726850560805214809=3^{10}\cdot 71^{2}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.17$
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:  $3, 71, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} - \frac{7}{23} a^{17} - \frac{6}{23} a^{16} - \frac{10}{23} a^{15} + \frac{3}{23} a^{14} - \frac{7}{23} a^{13} - \frac{6}{23} a^{12} + \frac{11}{23} a^{11} - \frac{2}{23} a^{10} + \frac{6}{23} a^{9} - \frac{8}{23} a^{8} + \frac{4}{23} a^{6} - \frac{6}{23} a^{5} - \frac{5}{23} a^{4} + \frac{6}{23} a^{3} - \frac{4}{23} a^{2} + \frac{2}{23} a + \frac{6}{23}$, $\frac{1}{107470261264958617741809} a^{19} + \frac{587029044963445945841}{107470261264958617741809} a^{18} + \frac{25041145502476871584073}{107470261264958617741809} a^{17} - \frac{26193971584545531509920}{107470261264958617741809} a^{16} + \frac{6457521636875662824836}{107470261264958617741809} a^{15} - \frac{9794591655875073993743}{35823420421652872580603} a^{14} - \frac{9648690839424698853542}{35823420421652872580603} a^{13} - \frac{14194420546654935960385}{107470261264958617741809} a^{12} - \frac{12731703745072915511278}{35823420421652872580603} a^{11} - \frac{50659538824109079387847}{107470261264958617741809} a^{10} - \frac{49150272495197901567443}{107470261264958617741809} a^{9} + \frac{11454713088843497838705}{35823420421652872580603} a^{8} - \frac{15421097153455932693081}{35823420421652872580603} a^{7} - \frac{7806783855479031389780}{107470261264958617741809} a^{6} + \frac{21858980096745338219588}{107470261264958617741809} a^{5} - \frac{756075397400028109927}{2027740778584124863053} a^{4} - \frac{38601380126266399883503}{107470261264958617741809} a^{3} - \frac{7835534230396191706002}{35823420421652872580603} a^{2} - \frac{12373960431823705586774}{107470261264958617741809} a - \frac{1583052395719880223028}{35823420421652872580603}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{194010850685649379930}{4672620054998200771383} a^{19} + \frac{202315093618640485249}{4672620054998200771383} a^{18} - \frac{2125206241409576554835}{4672620054998200771383} a^{17} + \frac{4322573973795906668698}{4672620054998200771383} a^{16} - \frac{18409650957476633795156}{4672620054998200771383} a^{15} + \frac{10739743876181693974285}{1557540018332733590461} a^{14} - \frac{25980339304791689521110}{1557540018332733590461} a^{13} + \frac{106142922676065595735645}{4672620054998200771383} a^{12} - \frac{65868479691289797152855}{1557540018332733590461} a^{11} + \frac{220611320056258759096096}{4672620054998200771383} a^{10} - \frac{337324842195392342887960}{4672620054998200771383} a^{9} + \frac{94140057663763241029281}{1557540018332733590461} a^{8} - \frac{119699311053142648684185}{1557540018332733590461} a^{7} + \frac{222538174715123208419918}{4672620054998200771383} a^{6} - \frac{259299222504684198706346}{4672620054998200771383} a^{5} + \frac{1893395069407948512022}{88162642547135863611} a^{4} - \frac{76629441060102054522335}{4672620054998200771383} a^{3} + \frac{3429144480775289108668}{1557540018332733590461} a^{2} - \frac{16198306560933001769017}{4672620054998200771383} a + \frac{1226111350597296424030}{1557540018332733590461} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1072932.19861 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T141:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n141
Character table for t20n141 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.160801.1, 10.8.1835844273671.1, 10.0.6283241669043.1, 10.2.446110158502053.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed