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Properties

Label 20.12.2650043292...0625.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{10}\cdot 97^{10}\cdot 60662149^{2}$
Root discriminant $132.18$
Ramified primes $5, 97, 60662149$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T656

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![44742407803561, 108814306471, -21240654931404, -16737690057, 4431198685244, 140062232, -535688310011, 94277553, 41603791746, -5916096, -2170492588, 129241, 77072033, -679, -1840209, -6, 28290, 0, -253, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 253*x^18 + 28290*x^16 - 6*x^15 - 1840209*x^14 - 679*x^13 + 77072033*x^12 + 129241*x^11 - 2170492588*x^10 - 5916096*x^9 + 41603791746*x^8 + 94277553*x^7 - 535688310011*x^6 + 140062232*x^5 + 4431198685244*x^4 - 16737690057*x^3 - 21240654931404*x^2 + 108814306471*x + 44742407803561)
 
gp: K = bnfinit(x^20 - 253*x^18 + 28290*x^16 - 6*x^15 - 1840209*x^14 - 679*x^13 + 77072033*x^12 + 129241*x^11 - 2170492588*x^10 - 5916096*x^9 + 41603791746*x^8 + 94277553*x^7 - 535688310011*x^6 + 140062232*x^5 + 4431198685244*x^4 - 16737690057*x^3 - 21240654931404*x^2 + 108814306471*x + 44742407803561, 1)
 

Normalized defining polynomial

\( x^{20} - 253 x^{18} + 28290 x^{16} - 6 x^{15} - 1840209 x^{14} - 679 x^{13} + 77072033 x^{12} + 129241 x^{11} - 2170492588 x^{10} - 5916096 x^{9} + 41603791746 x^{8} + 94277553 x^{7} - 535688310011 x^{6} + 140062232 x^{5} + 4431198685244 x^{4} - 16737690057 x^{3} - 21240654931404 x^{2} + 108814306471 x + 44742407803561 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2650043292770687765039331210445682119140625=5^{10}\cdot 97^{10}\cdot 60662149^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $132.18$
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:  $5, 97, 60662149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{13} a^{15} - \frac{3}{13} a^{14} + \frac{1}{13} a^{13} + \frac{5}{13} a^{12} + \frac{6}{13} a^{11} + \frac{4}{13} a^{10} + \frac{6}{13} a^{9} + \frac{1}{13} a^{8} - \frac{2}{13} a^{7} + \frac{2}{13} a^{6} + \frac{2}{13} a^{5} - \frac{4}{13} a^{4} - \frac{6}{13} a^{3} - \frac{1}{13} a^{2} - \frac{1}{13} a$, $\frac{1}{13} a^{16} + \frac{5}{13} a^{14} - \frac{5}{13} a^{13} - \frac{5}{13} a^{12} - \frac{4}{13} a^{11} + \frac{5}{13} a^{10} + \frac{6}{13} a^{9} + \frac{1}{13} a^{8} - \frac{4}{13} a^{7} - \frac{5}{13} a^{6} + \frac{2}{13} a^{5} - \frac{5}{13} a^{4} - \frac{6}{13} a^{3} - \frac{4}{13} a^{2} - \frac{3}{13} a$, $\frac{1}{13} a^{17} - \frac{3}{13} a^{14} + \frac{3}{13} a^{13} - \frac{3}{13} a^{12} + \frac{1}{13} a^{11} - \frac{1}{13} a^{10} - \frac{3}{13} a^{9} + \frac{4}{13} a^{8} + \frac{5}{13} a^{7} + \frac{5}{13} a^{6} - \frac{2}{13} a^{5} + \frac{1}{13} a^{4} + \frac{2}{13} a^{2} + \frac{5}{13} a$, $\frac{1}{338} a^{18} + \frac{5}{338} a^{17} + \frac{11}{338} a^{16} + \frac{1}{169} a^{15} + \frac{15}{338} a^{14} + \frac{1}{338} a^{13} - \frac{57}{338} a^{12} + \frac{159}{338} a^{11} + \frac{41}{338} a^{10} - \frac{45}{338} a^{9} - \frac{77}{169} a^{8} + \frac{93}{338} a^{7} - \frac{35}{338} a^{6} - \frac{159}{338} a^{5} - \frac{109}{338} a^{4} + \frac{83}{169} a^{3} + \frac{83}{338} a^{2} - \frac{1}{2}$, $\frac{1}{18997343114057587199133872324020049810603760871790001387843083314834179726867048268658} a^{19} + \frac{16656700381534912403151504385946515672569465649417889483153894450959190861141914141}{18997343114057587199133872324020049810603760871790001387843083314834179726867048268658} a^{18} + \frac{423299995953519917802610013097782631206761397532084232862516027316536836828830287093}{18997343114057587199133872324020049810603760871790001387843083314834179726867048268658} a^{17} + \frac{207033989460912173340497523592867287234182196618362720107896807824575176465442422240}{9498671557028793599566936162010024905301880435895000693921541657417089863433524134329} a^{16} - \frac{261604693816216780882810042215204342372027216896937392612509080939902522148917474785}{18997343114057587199133872324020049810603760871790001387843083314834179726867048268658} a^{15} - \frac{791478091724803936083068609017964408024566203960330638059405789171520178348343259021}{2713906159151083885590553189145721401514822981684285912549011902119168532409578324094} a^{14} + \frac{519220695367394317496040852500977815265517233277474316946323937788536098339967602921}{2713906159151083885590553189145721401514822981684285912549011902119168532409578324094} a^{13} - \frac{1170992656543014312570873001173822929603287371633266214754577852935159430079954747505}{2713906159151083885590553189145721401514822981684285912549011902119168532409578324094} a^{12} - \frac{1079238677020289720714478044681701181081260008218913253212253740412086518112741538399}{18997343114057587199133872324020049810603760871790001387843083314834179726867048268658} a^{11} - \frac{4880864415753628387260105680122647868514986915117329945128946972962867649631177282433}{18997343114057587199133872324020049810603760871790001387843083314834179726867048268658} a^{10} + \frac{220576048234232719004342037779102096496450477509942014192330685874171510669568860607}{730667042848368738428225858616155761946298495068846207224733973647468451033348010333} a^{9} + \frac{790627026695637746989974379335301853621284056808959765403092573070988611799907279359}{2713906159151083885590553189145721401514822981684285912549011902119168532409578324094} a^{8} + \frac{5965433853034228796747684686097564382512436992152233474829010767712236920590689033639}{18997343114057587199133872324020049810603760871790001387843083314834179726867048268658} a^{7} - \frac{8744760424123981544466362227105744430383040467675571020235460199705957052312234939511}{18997343114057587199133872324020049810603760871790001387843083314834179726867048268658} a^{6} + \frac{833224288486936067080291517591126332296164907069263615260866936477101308908808015935}{2713906159151083885590553189145721401514822981684285912549011902119168532409578324094} a^{5} - \frac{547465798471441962190894930644629700867866721549120148338005090779556530333096109819}{9498671557028793599566936162010024905301880435895000693921541657417089863433524134329} a^{4} - \frac{2622440737830505346799320625574054825773146822133991463688700036373241715476619033171}{18997343114057587199133872324020049810603760871790001387843083314834179726867048268658} a^{3} - \frac{4249231965516333409549517585320224096227705955242022556872045943009775112052678122259}{9498671557028793599566936162010024905301880435895000693921541657417089863433524134329} a^{2} + \frac{495707930533519962573637420089851201878429240969309928609799522900316121985394261269}{1461334085696737476856451717232311523892596990137692414449467947294936902066696020666} a + \frac{4381147522021944318122611056073686516091992918819749117764317642738446989510801515}{56205157142182210648325066047396597072792191928372785171133382588266803925642154641}$

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:  $15$
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:  \( 140481468190000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T656:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 70 conjugacy class representatives for t20n656 are not computed
Character table for t20n656 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{97}) \), \(\Q(\sqrt{485}) \), \(\Q(\sqrt{5}, \sqrt{97})\), 10.6.189569215625.1

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 24 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed
60662149Data not computed