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Properties

Label 20.0.37884228190...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{16}\cdot 5^{15}\cdot 7^{8}\cdot 29^{8}$
Root discriminant $67.44$
Ramified primes $3, 5, 7, 29$
Class number $95$ (GRH)
Class group $[95]$ (GRH)
Galois group $C_2\times C_5:F_5$ (as 20T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![92416, 1184384, 6042880, 4720352, 4160432, 2705304, 3262784, -464666, 1250475, -298521, 238424, -62610, 36318, -7039, 4619, -1170, 467, -107, 37, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 37*x^18 - 107*x^17 + 467*x^16 - 1170*x^15 + 4619*x^14 - 7039*x^13 + 36318*x^12 - 62610*x^11 + 238424*x^10 - 298521*x^9 + 1250475*x^8 - 464666*x^7 + 3262784*x^6 + 2705304*x^5 + 4160432*x^4 + 4720352*x^3 + 6042880*x^2 + 1184384*x + 92416)
 
gp: K = bnfinit(x^20 - 5*x^19 + 37*x^18 - 107*x^17 + 467*x^16 - 1170*x^15 + 4619*x^14 - 7039*x^13 + 36318*x^12 - 62610*x^11 + 238424*x^10 - 298521*x^9 + 1250475*x^8 - 464666*x^7 + 3262784*x^6 + 2705304*x^5 + 4160432*x^4 + 4720352*x^3 + 6042880*x^2 + 1184384*x + 92416, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 37 x^{18} - 107 x^{17} + 467 x^{16} - 1170 x^{15} + 4619 x^{14} - 7039 x^{13} + 36318 x^{12} - 62610 x^{11} + 238424 x^{10} - 298521 x^{9} + 1250475 x^{8} - 464666 x^{7} + 3262784 x^{6} + 2705304 x^{5} + 4160432 x^{4} + 4720352 x^{3} + 6042880 x^{2} + 1184384 x + 92416 \)

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:  \(3788422819041915442557973663330078125=3^{16}\cdot 5^{15}\cdot 7^{8}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.44$
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:  $3, 5, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{11} + \frac{1}{10} a^{8} - \frac{1}{2} a^{7} - \frac{1}{10} a^{6} - \frac{1}{2} a^{5} + \frac{1}{10} a^{4} - \frac{1}{2} a^{2} + \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{11} + \frac{1}{10} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2} - \frac{1}{2} a + \frac{2}{5}$, $\frac{1}{20} a^{14} - \frac{1}{20} a^{13} - \frac{1}{20} a^{12} + \frac{1}{20} a^{11} + \frac{1}{20} a^{10} - \frac{1}{10} a^{9} - \frac{1}{4} a^{8} + \frac{7}{20} a^{7} - \frac{1}{10} a^{5} + \frac{3}{10} a^{4} + \frac{1}{20} a^{3} + \frac{9}{20} a^{2} - \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{40} a^{15} - \frac{1}{40} a^{14} + \frac{1}{40} a^{13} + \frac{1}{40} a^{12} - \frac{1}{40} a^{11} - \frac{1}{20} a^{10} + \frac{7}{40} a^{9} - \frac{3}{8} a^{8} + \frac{9}{20} a^{7} + \frac{1}{4} a^{6} + \frac{1}{5} a^{5} - \frac{1}{40} a^{4} - \frac{1}{40} a^{3} + \frac{3}{20} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{80} a^{16} - \frac{1}{80} a^{15} + \frac{1}{80} a^{14} + \frac{1}{80} a^{13} - \frac{1}{80} a^{12} + \frac{9}{40} a^{11} - \frac{13}{80} a^{10} + \frac{1}{16} a^{9} + \frac{9}{40} a^{8} - \frac{1}{8} a^{7} - \frac{2}{5} a^{6} - \frac{1}{80} a^{5} + \frac{39}{80} a^{4} - \frac{7}{40} a^{3} - \frac{3}{10} a^{2} + \frac{1}{10} a$, $\frac{1}{160} a^{17} - \frac{1}{160} a^{16} + \frac{1}{160} a^{15} + \frac{1}{160} a^{14} - \frac{1}{160} a^{13} + \frac{1}{80} a^{12} + \frac{11}{160} a^{11} + \frac{1}{32} a^{10} + \frac{9}{80} a^{9} + \frac{7}{80} a^{8} - \frac{9}{20} a^{7} + \frac{11}{32} a^{6} + \frac{39}{160} a^{5} - \frac{3}{16} a^{4} - \frac{2}{5} a^{3} + \frac{3}{10} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{320} a^{18} - \frac{1}{320} a^{17} + \frac{1}{320} a^{16} + \frac{1}{320} a^{15} - \frac{1}{320} a^{14} + \frac{1}{160} a^{13} + \frac{11}{320} a^{12} - \frac{15}{64} a^{11} + \frac{9}{160} a^{10} + \frac{7}{160} a^{9} - \frac{19}{40} a^{8} - \frac{5}{64} a^{7} - \frac{41}{320} a^{6} + \frac{13}{32} a^{5} + \frac{3}{10} a^{4} + \frac{2}{5} a^{3} - \frac{1}{20} a^{2} + \frac{1}{5} a$, $\frac{1}{23812501798560489581411054813871927802886059779799083935360} a^{19} - \frac{15495952456616827358416600340016638045133500310404993659}{23812501798560489581411054813871927802886059779799083935360} a^{18} - \frac{54184817759006483668099948872557624830414702289872479581}{23812501798560489581411054813871927802886059779799083935360} a^{17} - \frac{88338830838773137490654604708256259885166611866978935361}{23812501798560489581411054813871927802886059779799083935360} a^{16} + \frac{87913471136963172725254413786624044515791539702464885869}{23812501798560489581411054813871927802886059779799083935360} a^{15} - \frac{21289805982148460993103919798105169180001815135151773499}{1190625089928024479070552740693596390144302988989954196768} a^{14} + \frac{1056985515740198942163286028134805760533634112465946679471}{23812501798560489581411054813871927802886059779799083935360} a^{13} + \frac{101312413124793764316018895689958694403432876262595923419}{4762500359712097916282210962774385560577211955959816787072} a^{12} + \frac{213099536955919379136769533132897468519581555128639500693}{2976562724820061197676381851733990975360757472474885491920} a^{11} + \frac{260812391020639514451370243798688928632749989796267892241}{11906250899280244790705527406935963901443029889899541967680} a^{10} + \frac{46326485130090994180271917788482462357322879218213666803}{5953125449640122395352763703467981950721514944949770983840} a^{9} - \frac{10743156524270858810504961304590799527705709307681993424537}{23812501798560489581411054813871927802886059779799083935360} a^{8} + \frac{5300290051932086279196945699700320235752201667124855627393}{23812501798560489581411054813871927802886059779799083935360} a^{7} - \frac{1954886479354090304761167278680870427802889358886622131127}{5953125449640122395352763703467981950721514944949770983840} a^{6} + \frac{54525802875654429678351933212074397613716136678394703621}{1190625089928024479070552740693596390144302988989954196768} a^{5} - \frac{572182622312931402739332975199093948822069704455260558483}{1488281362410030598838190925866995487680378736237442745960} a^{4} + \frac{2022316332622138475290805715952187265765959017080809289}{7259909084926978530918004516424368232587213347499720712} a^{3} - \frac{22343071405218390598302642371512771287719312112435456961}{744140681205015299419095462933497743840189368118721372980} a^{2} - \frac{6093688059961808083510355176350195872303440177780305202}{37207034060250764970954773146674887192009468405936068649} a - \frac{2039439975699405506060800235733781823640827918495395893}{9791324752697569729198624512282865050528807475246333855}$

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

Class group and class number

$C_{95}$, which has order $95$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{11606485575895960989022814520861931613595524629752705}{4762500359712097916282210962774385560577211955959816787072} a^{19} + \frac{292708611027886675708114660899769655047376235267964673}{23812501798560489581411054813871927802886059779799083935360} a^{18} - \frac{2163058933750098874493593475209425477641169489452931657}{23812501798560489581411054813871927802886059779799083935360} a^{17} + \frac{6325376829192463317318237197341778138169472778901542367}{23812501798560489581411054813871927802886059779799083935360} a^{16} - \frac{5504737612291202103150857685769749367609432841225441019}{4762500359712097916282210962774385560577211955959816787072} a^{15} + \frac{34831563084562899804935002285139968210601027886038076297}{11906250899280244790705527406935963901443029889899541967680} a^{14} - \frac{54650981460612686378425820120077555388549189529664856619}{4762500359712097916282210962774385560577211955959816787072} a^{13} + \frac{426976528836001066901790840887349521135716219599482981403}{23812501798560489581411054813871927802886059779799083935360} a^{12} - \frac{214801239219769809311714508997558488621849965048965617735}{2381250179856048958141105481387192780288605977979908393536} a^{11} + \frac{1888910664970247122296365685907851225285566991352255345557}{11906250899280244790705527406935963901443029889899541967680} a^{10} - \frac{885039090165536058690377433766989666075915852475954197831}{1488281362410030598838190925866995487680378736237442745960} a^{9} + \frac{18348286564658516900008009922021160353839395575848083488301}{23812501798560489581411054813871927802886059779799083935360} a^{8} - \frac{74519897705585996112455560028830860507305650027306233660111}{23812501798560489581411054813871927802886059779799083935360} a^{7} + \frac{16323706800424116933809860554183339095016247672829016004977}{11906250899280244790705527406935963901443029889899541967680} a^{6} - \frac{1535692225980452920247190285392631654343247368725557871252}{186035170301253824854773865733374435960047342029680343245} a^{5} - \frac{8721981240906707115607049968414626843431231425123337359863}{1488281362410030598838190925866995487680378736237442745960} a^{4} - \frac{379585793447716265508626013295307028705485425526804691639}{36299545424634892654590022582121841162936066737498603560} a^{3} - \frac{396087049218914127212388729129837630540872297066280751545}{37207034060250764970954773146674887192009468405936068649} a^{2} - \frac{5238442865565211510336239781680971259001373181193080427419}{372070340602507649709547731466748871920094684059360686490} a - \frac{17345816191882407086737657456871134698281533067385138189}{9791324752697569729198624512282865050528807475246333855} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 631632012.968 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_5:F_5$ (as 20T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 20 conjugacy class representatives for $C_2\times C_5:F_5$
Character table for $C_2\times C_5:F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.10.870450781956328125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.10.8.1$x^{10} - 899 x^{5} + 204363$$5$$2$$8$$D_5$$[\ ]_{5}^{2}$