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Properties

Label 20.0.14075038241...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{15}\cdot 31^{16}$
Root discriminant $255.52$
Ramified primes $2, 3, 5, 31$
Class number $354470464$ (GRH)
Class group $[2, 2, 2, 2, 2, 11077202]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3744560501755, -1875761820980, 1900437492720, -681170144120, 383268615741, -100036606136, 41217566898, -7805047864, 2827606183, -409081596, 153441300, -18355320, 7042586, -677260, 244530, -17204, 6083, -316, 108, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 108*x^18 - 316*x^17 + 6083*x^16 - 17204*x^15 + 244530*x^14 - 677260*x^13 + 7042586*x^12 - 18355320*x^11 + 153441300*x^10 - 409081596*x^9 + 2827606183*x^8 - 7805047864*x^7 + 41217566898*x^6 - 100036606136*x^5 + 383268615741*x^4 - 681170144120*x^3 + 1900437492720*x^2 - 1875761820980*x + 3744560501755)
 
gp: K = bnfinit(x^20 - 4*x^19 + 108*x^18 - 316*x^17 + 6083*x^16 - 17204*x^15 + 244530*x^14 - 677260*x^13 + 7042586*x^12 - 18355320*x^11 + 153441300*x^10 - 409081596*x^9 + 2827606183*x^8 - 7805047864*x^7 + 41217566898*x^6 - 100036606136*x^5 + 383268615741*x^4 - 681170144120*x^3 + 1900437492720*x^2 - 1875761820980*x + 3744560501755, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 108 x^{18} - 316 x^{17} + 6083 x^{16} - 17204 x^{15} + 244530 x^{14} - 677260 x^{13} + 7042586 x^{12} - 18355320 x^{11} + 153441300 x^{10} - 409081596 x^{9} + 2827606183 x^{8} - 7805047864 x^{7} + 41217566898 x^{6} - 100036606136 x^{5} + 383268615741 x^{4} - 681170144120 x^{3} + 1900437492720 x^{2} - 1875761820980 x + 3744560501755 \)

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:  \(1407503824193111413352299710676992000000000000000=2^{30}\cdot 3^{10}\cdot 5^{15}\cdot 31^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $255.52$
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:  $2, 3, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3720=2^{3}\cdot 3\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3720}(1,·)$, $\chi_{3720}(2209,·)$, $\chi_{3720}(2761,·)$, $\chi_{3720}(1163,·)$, $\chi_{3720}(529,·)$, $\chi_{3720}(467,·)$, $\chi_{3720}(2329,·)$, $\chi_{3720}(721,·)$, $\chi_{3720}(2267,·)$, $\chi_{3720}(481,·)$, $\chi_{3720}(2147,·)$, $\chi_{3720}(1489,·)$, $\chi_{3720}(683,·)$, $\chi_{3720}(1427,·)$, $\chi_{3720}(1403,·)$, $\chi_{3720}(1523,·)$, $\chi_{3720}(1969,·)$, $\chi_{3720}(3443,·)$, $\chi_{3720}(841,·)$, $\chi_{3720}(1907,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{6} + \frac{1}{6} a^{4} + \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} + \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{14} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{15} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{30} a^{16} - \frac{1}{15} a^{15} + \frac{1}{30} a^{12} + \frac{1}{30} a^{10} - \frac{1}{3} a^{9} - \frac{7}{15} a^{8} - \frac{1}{3} a^{6} + \frac{1}{15} a^{5} - \frac{3}{10} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{150} a^{17} + \frac{1}{25} a^{15} + \frac{1}{30} a^{14} + \frac{11}{150} a^{13} - \frac{1}{50} a^{12} + \frac{1}{25} a^{11} + \frac{2}{25} a^{10} + \frac{17}{50} a^{9} + \frac{67}{150} a^{8} + \frac{1}{3} a^{7} + \frac{17}{150} a^{6} - \frac{1}{5} a^{5} + \frac{7}{150} a^{4} - \frac{13}{30} a^{3} + \frac{7}{15} a^{2} + \frac{1}{6} a - \frac{4}{15}$, $\frac{1}{1848875100453667084624011069150} a^{18} + \frac{322096465139255656094929948}{184887510045366708462401106915} a^{17} + \frac{19929375192478054375036117201}{1848875100453667084624011069150} a^{16} - \frac{832021765711630940682392397}{123258340030244472308267404610} a^{15} - \frac{29800877958303433822927755239}{1848875100453667084624011069150} a^{14} + \frac{44379387409563844075909749009}{616291700151222361541337023050} a^{13} - \frac{15921667563609506988278677494}{308145850075611180770668511525} a^{12} - \frac{9634828083467355493270687354}{924437550226833542312005534575} a^{11} + \frac{4773790090135884682027371651}{308145850075611180770668511525} a^{10} - \frac{47947467905660914814287219151}{616291700151222361541337023050} a^{9} + \frac{91537927972708598622533036371}{369775020090733416924802213830} a^{8} - \frac{213488167029261465717428347911}{616291700151222361541337023050} a^{7} + \frac{27900183476072908684671853957}{123258340030244472308267404610} a^{6} - \frac{121262905912551217511335044851}{616291700151222361541337023050} a^{5} - \frac{21258644494095033259545976361}{184887510045366708462401106915} a^{4} + \frac{7305024767351575792376422139}{61629170015122236154133702305} a^{3} - \frac{26603472942778861243373399353}{73955004018146683384960442766} a^{2} + \frac{16440412604329293733804525847}{61629170015122236154133702305} a - \frac{72623890167473365542735769}{387198973916998342329635826}$, $\frac{1}{853814141226326895098790352986036820957266195610179393182769120150} a^{19} + \frac{76376508739108832915271545870962478}{426907070613163447549395176493018410478633097805089696591384560075} a^{18} + \frac{907871741966490620077088423700331905846202593381861797613186634}{426907070613163447549395176493018410478633097805089696591384560075} a^{17} - \frac{5706057309359198672392974735053992443646194479081201530343089079}{853814141226326895098790352986036820957266195610179393182769120150} a^{16} + \frac{14497682360540890223016542185396224407277125541010240556430895714}{426907070613163447549395176493018410478633097805089696591384560075} a^{15} + \frac{6097909002369808216035222457878217453125093674904213805843638149}{426907070613163447549395176493018410478633097805089696591384560075} a^{14} - \frac{8336626973308241247202735564319051445989166232068913861150393991}{170762828245265379019758070597207364191453239122035878636553824030} a^{13} - \frac{28075101089068673855981475905674143624102693359756879571439070879}{426907070613163447549395176493018410478633097805089696591384560075} a^{12} - \frac{4171462340210221248703170552640166047593100488917295884627914183}{85381414122632689509879035298603682095726619561017939318276912015} a^{11} + \frac{975813597440924109884445373535617202131641200529923527197316487}{142302356871054482516465058831006136826211032601696565530461520025} a^{10} - \frac{71100986915690732420710411296582631126932897885990801363191885818}{426907070613163447549395176493018410478633097805089696591384560075} a^{9} + \frac{65791927960231638874122313598070912225916152900525245415752400038}{426907070613163447549395176493018410478633097805089696591384560075} a^{8} - \frac{354287044607084543954900040096937175266728982435720501254235616053}{853814141226326895098790352986036820957266195610179393182769120150} a^{7} - \frac{90479514985230373306318970324775907266714219807983777456298919623}{284604713742108965032930117662012273652422065203393131060923040050} a^{6} - \frac{86211843445193801446564438016279915705431870323307544514680031791}{284604713742108965032930117662012273652422065203393131060923040050} a^{5} + \frac{36326605286008207045386091952336287362703612916153013433849203622}{426907070613163447549395176493018410478633097805089696591384560075} a^{4} + \frac{7211524832964554745666125217297344726815309625319042924075415324}{85381414122632689509879035298603682095726619561017939318276912015} a^{3} + \frac{9093546085290426879763383281120213779834501552376512699768690019}{34152565649053075803951614119441472838290647824407175727310764806} a^{2} + \frac{23468579631282371888169987847147069349564065497369299935073722739}{56920942748421793006586023532402454730484413040678626212184608010} a + \frac{126830451557291656828245317635451389166732271086933400417033832}{447023110589699945077900708369652785841500625973915912661135665}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{11077202}$, which has order $354470464$ (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:  \( -1 \) (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:  \( 24173706.832424585 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.72000.2, 5.5.923521.1, 10.10.2665284492003125.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
3Data not computed
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$31$31.10.8.1$x^{10} - 20491 x^{5} + 239127552$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
31.10.8.1$x^{10} - 20491 x^{5} + 239127552$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$