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Properties

Label 20.10.2124035093...0000.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{34}\cdot 5^{14}\cdot 1193^{4}$
Root discriminant $41.34$
Ramified primes $2, 5, 1193$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T925

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![446, -404, -5114, -5936, 6743, 22110, 16659, -7546, -20583, -12366, 1460, 7012, 3017, -1312, -1204, 50, 213, 18, -21, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 21*x^18 + 18*x^17 + 213*x^16 + 50*x^15 - 1204*x^14 - 1312*x^13 + 3017*x^12 + 7012*x^11 + 1460*x^10 - 12366*x^9 - 20583*x^8 - 7546*x^7 + 16659*x^6 + 22110*x^5 + 6743*x^4 - 5936*x^3 - 5114*x^2 - 404*x + 446)
 
gp: K = bnfinit(x^20 - 2*x^19 - 21*x^18 + 18*x^17 + 213*x^16 + 50*x^15 - 1204*x^14 - 1312*x^13 + 3017*x^12 + 7012*x^11 + 1460*x^10 - 12366*x^9 - 20583*x^8 - 7546*x^7 + 16659*x^6 + 22110*x^5 + 6743*x^4 - 5936*x^3 - 5114*x^2 - 404*x + 446, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 21 x^{18} + 18 x^{17} + 213 x^{16} + 50 x^{15} - 1204 x^{14} - 1312 x^{13} + 3017 x^{12} + 7012 x^{11} + 1460 x^{10} - 12366 x^{9} - 20583 x^{8} - 7546 x^{7} + 16659 x^{6} + 22110 x^{5} + 6743 x^{4} - 5936 x^{3} - 5114 x^{2} - 404 x + 446 \)

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:  \(-212403509369346457600000000000000=-\,2^{34}\cdot 5^{14}\cdot 1193^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.34$
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:  $2, 5, 1193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{179973618111609969807472260922250551} a^{19} + \frac{89001742077294016394330486871552513}{179973618111609969807472260922250551} a^{18} + \frac{20263894393954644978612014060810198}{179973618111609969807472260922250551} a^{17} - \frac{77632355493989565169810026439218260}{179973618111609969807472260922250551} a^{16} + \frac{81217809799392790337232017423409186}{179973618111609969807472260922250551} a^{15} + \frac{54088977027068234416381240882569398}{179973618111609969807472260922250551} a^{14} - \frac{18508567186906241699621942679849358}{179973618111609969807472260922250551} a^{13} - \frac{4993087800297991729205340851110692}{179973618111609969807472260922250551} a^{12} + \frac{37882831543304945921883015154609919}{179973618111609969807472260922250551} a^{11} - \frac{4658681820789461037401310914940114}{179973618111609969807472260922250551} a^{10} + \frac{80581443994550490501991972257829740}{179973618111609969807472260922250551} a^{9} - \frac{17366674806722238649217238380749143}{179973618111609969807472260922250551} a^{8} - \frac{20391980854054433424720906091672645}{179973618111609969807472260922250551} a^{7} + \frac{22221536550450871826987622111731495}{179973618111609969807472260922250551} a^{6} + \frac{10975764164243855938297111558671693}{179973618111609969807472260922250551} a^{5} - \frac{33123442172410458662079356583957137}{179973618111609969807472260922250551} a^{4} - \frac{79065117994853641672508883752334706}{179973618111609969807472260922250551} a^{3} - \frac{71868727911310750785865240260433851}{179973618111609969807472260922250551} a^{2} - \frac{79990693682509716746637598849224299}{179973618111609969807472260922250551} a + \frac{88743274575145161620666104250129332}{179973618111609969807472260922250551}$

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:  \( 1714924465.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T925:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 409600
The 190 conjugacy class representatives for t20n925 are not computed
Character table for t20n925 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.728703488000000.1

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 $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
5Data not computed
1193Data not computed