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Properties

Label 20.10.5092151360...0000.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{8}\cdot 5^{10}\cdot 1093^{8}$
Root discriminant $48.46$
Ramified primes $2, 5, 1093$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2284, 8566, -12243, -41964, 54262, 79301, -136209, -14161, 96743, -14527, -22883, -9144, 10339, 2953, -1757, -887, 275, 107, -28, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 28*x^18 + 107*x^17 + 275*x^16 - 887*x^15 - 1757*x^14 + 2953*x^13 + 10339*x^12 - 9144*x^11 - 22883*x^10 - 14527*x^9 + 96743*x^8 - 14161*x^7 - 136209*x^6 + 79301*x^5 + 54262*x^4 - 41964*x^3 - 12243*x^2 + 8566*x + 2284)
 
gp: K = bnfinit(x^20 - 4*x^19 - 28*x^18 + 107*x^17 + 275*x^16 - 887*x^15 - 1757*x^14 + 2953*x^13 + 10339*x^12 - 9144*x^11 - 22883*x^10 - 14527*x^9 + 96743*x^8 - 14161*x^7 - 136209*x^6 + 79301*x^5 + 54262*x^4 - 41964*x^3 - 12243*x^2 + 8566*x + 2284, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 28 x^{18} + 107 x^{17} + 275 x^{16} - 887 x^{15} - 1757 x^{14} + 2953 x^{13} + 10339 x^{12} - 9144 x^{11} - 22883 x^{10} - 14527 x^{9} + 96743 x^{8} - 14161 x^{7} - 136209 x^{6} + 79301 x^{5} + 54262 x^{4} - 41964 x^{3} - 12243 x^{2} + 8566 x + 2284 \)

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:  \(-5092151360596147886766002500000000=-\,2^{8}\cdot 5^{10}\cdot 1093^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.46$
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:  $2, 5, 1093$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{8} - \frac{1}{10} a^{7} - \frac{1}{10} a^{6} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a + \frac{2}{5}$, $\frac{1}{10} a^{13} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{10} a^{8} + \frac{1}{10} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{10} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{10} a^{14} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{10} a^{9} + \frac{1}{10} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{10} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{20} a^{15} - \frac{1}{20} a^{14} - \frac{1}{20} a^{13} - \frac{1}{10} a^{11} + \frac{7}{20} a^{10} - \frac{1}{5} a^{9} + \frac{1}{10} a^{8} - \frac{1}{20} a^{7} + \frac{1}{5} a^{6} - \frac{1}{2} a^{5} + \frac{3}{20} a^{4} - \frac{9}{20} a^{3} + \frac{3}{20} a^{2} + \frac{1}{5} a$, $\frac{1}{100} a^{16} - \frac{1}{100} a^{15} + \frac{1}{100} a^{14} - \frac{1}{25} a^{13} - \frac{1}{25} a^{12} - \frac{3}{100} a^{11} - \frac{11}{50} a^{10} + \frac{9}{50} a^{9} - \frac{41}{100} a^{8} - \frac{11}{25} a^{7} + \frac{1}{25} a^{6} - \frac{19}{100} a^{5} - \frac{7}{100} a^{4} - \frac{29}{100} a^{3} + \frac{7}{50} a^{2} + \frac{7}{25} a - \frac{9}{25}$, $\frac{1}{200} a^{17} - \frac{1}{200} a^{16} - \frac{1}{50} a^{15} - \frac{9}{200} a^{14} + \frac{1}{200} a^{13} - \frac{3}{200} a^{12} + \frac{9}{100} a^{11} - \frac{7}{200} a^{10} + \frac{19}{200} a^{9} - \frac{7}{100} a^{8} + \frac{39}{200} a^{7} + \frac{61}{200} a^{6} - \frac{47}{200} a^{5} - \frac{17}{100} a^{4} + \frac{99}{200} a^{3} - \frac{77}{200} a^{2} + \frac{37}{100} a - \frac{1}{2}$, $\frac{1}{65600} a^{18} + \frac{17}{16400} a^{17} - \frac{67}{65600} a^{16} + \frac{1479}{65600} a^{15} - \frac{9}{4100} a^{14} - \frac{121}{8200} a^{13} - \frac{413}{65600} a^{12} - \frac{15543}{65600} a^{11} + \frac{12387}{32800} a^{10} - \frac{99}{320} a^{9} - \frac{18933}{65600} a^{8} + \frac{2429}{32800} a^{7} + \frac{4293}{32800} a^{6} + \frac{18169}{65600} a^{5} + \frac{21121}{65600} a^{4} + \frac{203}{6560} a^{3} - \frac{969}{2624} a^{2} - \frac{6963}{32800} a + \frac{7101}{16400}$, $\frac{1}{6807677046753088956554717449600} a^{19} - \frac{6238631574188889734925551}{1361535409350617791310943489920} a^{18} - \frac{2438086395723783762685924527}{6807677046753088956554717449600} a^{17} + \frac{7739114062916651643263497901}{1701919261688272239138679362400} a^{16} + \frac{8430347126226485840191806331}{1361535409350617791310943489920} a^{15} - \frac{26746470731300972364425129673}{850959630844136119569339681200} a^{14} + \frac{311694165872504746279383651323}{6807677046753088956554717449600} a^{13} - \frac{58047101215465548944170912117}{1701919261688272239138679362400} a^{12} - \frac{1240246891542485704399686294609}{6807677046753088956554717449600} a^{11} - \frac{3027091160022226888486108877121}{6807677046753088956554717449600} a^{10} - \frac{833691798391882603336018098967}{1701919261688272239138679362400} a^{9} + \frac{3327528121698902811225311035333}{6807677046753088956554717449600} a^{8} + \frac{388406148495847680657028354849}{1701919261688272239138679362400} a^{7} + \frac{700033385158464832977371865907}{6807677046753088956554717449600} a^{6} + \frac{1455049674291085100710159674917}{3403838523376544478277358724800} a^{5} - \frac{74893312077670321255401984809}{272307081870123558262188697984} a^{4} - \frac{1009723139170994521847632249091}{6807677046753088956554717449600} a^{3} - \frac{768465027358080470990973715287}{6807677046753088956554717449600} a^{2} - \frac{580804641883947673572125552809}{3403838523376544478277358724800} a - \frac{160606120349246208424543558071}{340383852337654447827735872480}$

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:  \( 13569875145.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.1194649.1, 10.10.4459956978753125.1

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 $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1093Data not computed