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Properties

Label 20.0.15970981195...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{27}\cdot 11^{8}$
Root discriminant $22.92$
Ramified primes $5, 11$
Class number $1$
Class group Trivial
Galois group $C_5\times C_5:C_4$ (as 20T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 60, -255, 900, -2544, 6125, -11360, 15660, -15605, 11436, -5995, 2515, -1155, 735, -344, 155, -65, 25, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 25*x^18 - 65*x^17 + 155*x^16 - 344*x^15 + 735*x^14 - 1155*x^13 + 2515*x^12 - 5995*x^11 + 11436*x^10 - 15605*x^9 + 15660*x^8 - 11360*x^7 + 6125*x^6 - 2544*x^5 + 900*x^4 - 255*x^3 + 60*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 25*x^18 - 65*x^17 + 155*x^16 - 344*x^15 + 735*x^14 - 1155*x^13 + 2515*x^12 - 5995*x^11 + 11436*x^10 - 15605*x^9 + 15660*x^8 - 11360*x^7 + 6125*x^6 - 2544*x^5 + 900*x^4 - 255*x^3 + 60*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 25 x^{18} - 65 x^{17} + 155 x^{16} - 344 x^{15} + 735 x^{14} - 1155 x^{13} + 2515 x^{12} - 5995 x^{11} + 11436 x^{10} - 15605 x^{9} + 15660 x^{8} - 11360 x^{7} + 6125 x^{6} - 2544 x^{5} + 900 x^{4} - 255 x^{3} + 60 x^{2} - 10 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1597098119556903839111328125=5^{27}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.92$
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:  $5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{7} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{8} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{9} + \frac{1}{5} a^{4}$, $\frac{1}{25} a^{15} + \frac{2}{25} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{25} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{1}{25}$, $\frac{1}{25} a^{16} + \frac{2}{25} a^{11} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{25} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{25} a - \frac{2}{5}$, $\frac{1}{25} a^{17} + \frac{2}{25} a^{12} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{25} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{25} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{25} a^{18} + \frac{2}{25} a^{13} + \frac{2}{5} a^{9} - \frac{2}{25} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{25} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{10874483360528688625775} a^{19} - \frac{123201842104494154927}{10874483360528688625775} a^{18} + \frac{14393713747355083242}{2174896672105737725155} a^{17} + \frac{13427054673237857966}{10874483360528688625775} a^{16} - \frac{75827261419039949294}{10874483360528688625775} a^{15} - \frac{699513503899424501858}{10874483360528688625775} a^{14} - \frac{554672855056765595649}{10874483360528688625775} a^{13} - \frac{4621585385359142102}{434979334421147545031} a^{12} + \frac{896181033829738349802}{10874483360528688625775} a^{11} - \frac{1035138415159644372033}{10874483360528688625775} a^{10} + \frac{4590220439621173559543}{10874483360528688625775} a^{9} + \frac{4331673662744647942049}{10874483360528688625775} a^{8} + \frac{976972916965341822516}{2174896672105737725155} a^{7} - \frac{4765671204769444806672}{10874483360528688625775} a^{6} - \frac{3415144953845430653347}{10874483360528688625775} a^{5} - \frac{5211743493249187214011}{10874483360528688625775} a^{4} + \frac{1580994856551151781052}{10874483360528688625775} a^{3} + \frac{459886186989755463272}{2174896672105737725155} a^{2} - \frac{3206349459523612152596}{10874483360528688625775} a - \frac{5102995081050238522026}{10874483360528688625775}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{656024931138028197448}{2174896672105737725155} a^{19} + \frac{14730770632829111758664}{10874483360528688625775} a^{18} - \frac{74287156883596743223086}{10874483360528688625775} a^{17} + \frac{34901363778058400904157}{2174896672105737725155} a^{16} - \frac{415087110073216617435313}{10874483360528688625775} a^{15} + \frac{181513992811300578249671}{2174896672105737725155} a^{14} - \frac{1926292992665803650291397}{10874483360528688625775} a^{13} + \frac{2759046933808621380731623}{10874483360528688625775} a^{12} - \frac{1348403476684773525529383}{2174896672105737725155} a^{11} + \frac{16101178407526436503391289}{10874483360528688625775} a^{10} - \frac{5788263370689989466946993}{2174896672105737725155} a^{9} + \frac{35562852317639642792680807}{10874483360528688625775} a^{8} - \frac{31786645704627253008916283}{10874483360528688625775} a^{7} + \frac{780162713744233201053457}{434979334421147545031} a^{6} - \frac{8955738548207175256492399}{10874483360528688625775} a^{5} + \frac{127421099859264825545081}{434979334421147545031} a^{4} - \frac{1041402946192566740845299}{10874483360528688625775} a^{3} + \frac{135027380103406984134321}{10874483360528688625775} a^{2} - \frac{9818228766618912066279}{2174896672105737725155} a + \frac{3817665940115336676048}{10874483360528688625775} \) (order $10$)
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:  \( 1031961.46725 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:C_4$ (as 20T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 40 conjugacy class representatives for $C_5\times C_5:C_4$
Character table for $C_5\times C_5:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.10.17872314453125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: 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 $20$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$