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Properties

Label 20.16.1465621300...3125.1
Degree $20$
Signature $[16, 2]$
Discriminant $5^{15}\cdot 6029^{6}$
Root discriminant $45.53$
Ramified primes $5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T802

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![47569, -112287, -143919, 432233, 55887, -598148, 208344, 332408, -262419, -28380, 113953, -39697, -18624, 15540, -219, -2324, 394, 148, -38, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 38*x^18 + 148*x^17 + 394*x^16 - 2324*x^15 - 219*x^14 + 15540*x^13 - 18624*x^12 - 39697*x^11 + 113953*x^10 - 28380*x^9 - 262419*x^8 + 332408*x^7 + 208344*x^6 - 598148*x^5 + 55887*x^4 + 432233*x^3 - 143919*x^2 - 112287*x + 47569)
 
gp: K = bnfinit(x^20 - 3*x^19 - 38*x^18 + 148*x^17 + 394*x^16 - 2324*x^15 - 219*x^14 + 15540*x^13 - 18624*x^12 - 39697*x^11 + 113953*x^10 - 28380*x^9 - 262419*x^8 + 332408*x^7 + 208344*x^6 - 598148*x^5 + 55887*x^4 + 432233*x^3 - 143919*x^2 - 112287*x + 47569, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 38 x^{18} + 148 x^{17} + 394 x^{16} - 2324 x^{15} - 219 x^{14} + 15540 x^{13} - 18624 x^{12} - 39697 x^{11} + 113953 x^{10} - 28380 x^{9} - 262419 x^{8} + 332408 x^{7} + 208344 x^{6} - 598148 x^{5} + 55887 x^{4} + 432233 x^{3} - 143919 x^{2} - 112287 x + 47569 \)

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:  \(1465621300755332525247833251953125=5^{15}\cdot 6029^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.53$
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:  $5, 6029$
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}{691333053630997285596160257737558316150211897} a^{19} - \frac{123070106344231911620815606009340727984402459}{691333053630997285596160257737558316150211897} a^{18} - \frac{98026159404204401485349432829531345690944780}{691333053630997285596160257737558316150211897} a^{17} + \frac{135054588849121403584192059791280791200098882}{691333053630997285596160257737558316150211897} a^{16} - \frac{77128511416198130556578856636224436915220653}{691333053630997285596160257737558316150211897} a^{15} + \frac{146447385484657643907047617242302391872208196}{691333053630997285596160257737558316150211897} a^{14} - \frac{340273887573182575822021812057408720125994277}{691333053630997285596160257737558316150211897} a^{13} + \frac{332516987947408163705171460199044669473981117}{691333053630997285596160257737558316150211897} a^{12} + \frac{303394470982841103473624775794772098259563080}{691333053630997285596160257737558316150211897} a^{11} - \frac{79504409069146362586550917785405848501525574}{691333053630997285596160257737558316150211897} a^{10} + \frac{22993850331767289540985073302148817144093168}{691333053630997285596160257737558316150211897} a^{9} - \frac{225188088126140091720439933468455615458780636}{691333053630997285596160257737558316150211897} a^{8} + \frac{337639032596262484863988208066711099835390453}{691333053630997285596160257737558316150211897} a^{7} - \frac{256373813263751922564197204215699413206982715}{691333053630997285596160257737558316150211897} a^{6} + \frac{298046963216151741621551684579781016440572417}{691333053630997285596160257737558316150211897} a^{5} - \frac{130525864544276508999579214681829175746612681}{691333053630997285596160257737558316150211897} a^{4} + \frac{194085233314225047614686514790638017285492405}{691333053630997285596160257737558316150211897} a^{3} - \frac{294897365051622122092745223000264539380040349}{691333053630997285596160257737558316150211897} a^{2} + \frac{57526986802460250660012283501721709278357083}{691333053630997285596160257737558316150211897} a - \frac{75579125254605365650900194669019163224914786}{691333053630997285596160257737558316150211897}$

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:  \( 5551914929.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T802:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n802 are not computed
Character table for t20n802 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
6029Data not computed