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Properties

Label 20.16.4759732575...0625.1
Degree $20$
Signature $[16, 2]$
Discriminant $3^{6}\cdot 5^{10}\cdot 401^{8}$
Root discriminant $34.19$
Ramified primes $3, 5, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 10, -31, -242, 311, 1548, -1811, -3047, 3191, 2409, -1181, -1191, -1076, 663, 728, -233, -95, 34, -7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 7*x^18 + 34*x^17 - 95*x^16 - 233*x^15 + 728*x^14 + 663*x^13 - 1076*x^12 - 1191*x^11 - 1181*x^10 + 2409*x^9 + 3191*x^8 - 3047*x^7 - 1811*x^6 + 1548*x^5 + 311*x^4 - 242*x^3 - 31*x^2 + 10*x + 1)
 
gp: K = bnfinit(x^20 - x^19 - 7*x^18 + 34*x^17 - 95*x^16 - 233*x^15 + 728*x^14 + 663*x^13 - 1076*x^12 - 1191*x^11 - 1181*x^10 + 2409*x^9 + 3191*x^8 - 3047*x^7 - 1811*x^6 + 1548*x^5 + 311*x^4 - 242*x^3 - 31*x^2 + 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 7 x^{18} + 34 x^{17} - 95 x^{16} - 233 x^{15} + 728 x^{14} + 663 x^{13} - 1076 x^{12} - 1191 x^{11} - 1181 x^{10} + 2409 x^{9} + 3191 x^{8} - 3047 x^{7} - 1811 x^{6} + 1548 x^{5} + 311 x^{4} - 242 x^{3} - 31 x^{2} + 10 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4759732575183046916538369140625=3^{6}\cdot 5^{10}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.19$
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:  $3, 5, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{69} a^{17} - \frac{7}{69} a^{16} - \frac{4}{69} a^{15} - \frac{8}{23} a^{14} - \frac{1}{69} a^{13} - \frac{10}{69} a^{12} + \frac{32}{69} a^{11} - \frac{26}{69} a^{10} + \frac{2}{69} a^{9} - \frac{26}{69} a^{8} - \frac{5}{23} a^{7} - \frac{26}{69} a^{6} - \frac{17}{69} a^{5} + \frac{13}{69} a^{4} - \frac{31}{69} a^{2} + \frac{22}{69} a + \frac{7}{69}$, $\frac{1}{207} a^{18} + \frac{16}{207} a^{16} + \frac{17}{207} a^{15} - \frac{100}{207} a^{14} - \frac{86}{207} a^{13} + \frac{31}{207} a^{12} - \frac{26}{69} a^{11} + \frac{3}{23} a^{10} + \frac{19}{69} a^{9} + \frac{79}{207} a^{8} - \frac{62}{207} a^{7} + \frac{8}{207} a^{6} + \frac{101}{207} a^{5} + \frac{91}{207} a^{4} + \frac{38}{207} a^{3} - \frac{19}{69} a^{2} + \frac{1}{9} a + \frac{49}{207}$, $\frac{1}{356181416366940438489} a^{19} + \frac{504211672315034818}{356181416366940438489} a^{18} + \frac{1933631491644684589}{356181416366940438489} a^{17} - \frac{11541044763313440050}{118727138788980146163} a^{16} + \frac{17188522294177136890}{356181416366940438489} a^{15} + \frac{246649466183387066}{1720683170854784727} a^{14} - \frac{17871037367653805347}{356181416366940438489} a^{13} + \frac{172896083910218681539}{356181416366940438489} a^{12} - \frac{22608609549222748885}{118727138788980146163} a^{11} - \frac{13174214258401656098}{39575712929660048721} a^{10} - \frac{64851325691309099399}{356181416366940438489} a^{9} - \frac{149010761721674217745}{356181416366940438489} a^{8} - \frac{55954602133413075607}{118727138788980146163} a^{7} - \frac{109257025177371180998}{356181416366940438489} a^{6} + \frac{8989013226864778}{746711564710566957} a^{5} + \frac{4242805253929393741}{13191904309886682907} a^{4} + \frac{4504850368285539677}{356181416366940438489} a^{3} - \frac{102905451985516872394}{356181416366940438489} a^{2} - \frac{18759142828361159225}{39575712929660048721} a + \frac{119246135693650025923}{356181416366940438489}$

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:  $17$
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:  \( 384531814.135 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 5.5.160801.1, 10.10.80803005003125.1, 10.8.2181681135084375.1, 10.8.698137963227.1

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 R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5Data not computed
401Data not computed