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Properties

Label 20.0.41115038262...2272.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{42}\cdot 2657^{7}$
Root discriminant $67.72$
Ramified primes $2, 2657$
Class number $24$ (GRH)
Class group $[2, 12]$ (GRH)
Galois group 20T547

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![177738116, -239357408, 317189688, -190676344, 156661796, -46839408, 41595536, -9065852, 9654213, -1503972, 1626322, -232900, 215087, -21488, 18668, -1900, 1199, -116, 50, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 50*x^18 - 116*x^17 + 1199*x^16 - 1900*x^15 + 18668*x^14 - 21488*x^13 + 215087*x^12 - 232900*x^11 + 1626322*x^10 - 1503972*x^9 + 9654213*x^8 - 9065852*x^7 + 41595536*x^6 - 46839408*x^5 + 156661796*x^4 - 190676344*x^3 + 317189688*x^2 - 239357408*x + 177738116)
 
gp: K = bnfinit(x^20 - 4*x^19 + 50*x^18 - 116*x^17 + 1199*x^16 - 1900*x^15 + 18668*x^14 - 21488*x^13 + 215087*x^12 - 232900*x^11 + 1626322*x^10 - 1503972*x^9 + 9654213*x^8 - 9065852*x^7 + 41595536*x^6 - 46839408*x^5 + 156661796*x^4 - 190676344*x^3 + 317189688*x^2 - 239357408*x + 177738116, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 50 x^{18} - 116 x^{17} + 1199 x^{16} - 1900 x^{15} + 18668 x^{14} - 21488 x^{13} + 215087 x^{12} - 232900 x^{11} + 1626322 x^{10} - 1503972 x^{9} + 9654213 x^{8} - 9065852 x^{7} + 41595536 x^{6} - 46839408 x^{5} + 156661796 x^{4} - 190676344 x^{3} + 317189688 x^{2} - 239357408 x + 177738116 \)

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:  \(4111503826253885814528363168920502272=2^{42}\cdot 2657^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.72$
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:  $2, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{16} - \frac{1}{8} a^{15} - \frac{1}{8} a^{14} + \frac{3}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{3}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{14} - \frac{1}{8} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{8} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{917643307179727241729417398654483988874341868759792416337078640006810385805224} a^{19} - \frac{32450335951912534420003575494226348098654610674597949572293005127687763708035}{917643307179727241729417398654483988874341868759792416337078640006810385805224} a^{18} - \frac{4466135138329519433748073920887989650820777563339837242901634108347121838397}{917643307179727241729417398654483988874341868759792416337078640006810385805224} a^{17} - \frac{78698687903563817871674660622390838510179404845477943313178352528394725870511}{917643307179727241729417398654483988874341868759792416337078640006810385805224} a^{16} - \frac{21469082743733841875513457991987048016056641672642765303558453667828044235011}{458821653589863620864708699327241994437170934379896208168539320003405192902612} a^{15} - \frac{2040384609823668665448712948350495249521757990163486393833344405810836186459}{229410826794931810432354349663620997218585467189948104084269660001702596451306} a^{14} - \frac{25936783829783927951499772330781133286611351292118270286938960533996985353123}{229410826794931810432354349663620997218585467189948104084269660001702596451306} a^{13} - \frac{10285226453447917010575803895752378328979516573859182892887504555916847330787}{458821653589863620864708699327241994437170934379896208168539320003405192902612} a^{12} - \frac{54024790963195148408100790915584616540472437843133813079915223230285338631629}{917643307179727241729417398654483988874341868759792416337078640006810385805224} a^{11} + \frac{35926112577332121400239194984463510978641212120463725208100797895172882659633}{917643307179727241729417398654483988874341868759792416337078640006810385805224} a^{10} - \frac{7874539037767468117302180766765973999944993728839534952740053164909177335597}{917643307179727241729417398654483988874341868759792416337078640006810385805224} a^{9} + \frac{389845560621045306841688368814846060503322282665570524050557862618878787050111}{917643307179727241729417398654483988874341868759792416337078640006810385805224} a^{8} - \frac{215300351361386836012004549408699142911741124758337237723943181168703540748197}{458821653589863620864708699327241994437170934379896208168539320003405192902612} a^{7} - \frac{110425319489841458210350976105944453852035370469103732412264500175050846857541}{458821653589863620864708699327241994437170934379896208168539320003405192902612} a^{6} - \frac{96816180308935541454126200362399650744535406302147125832710420616055414128373}{458821653589863620864708699327241994437170934379896208168539320003405192902612} a^{5} + \frac{2354571071106011333669640163148421264815389153811684526129047420925738460491}{458821653589863620864708699327241994437170934379896208168539320003405192902612} a^{4} + \frac{65530727683809668988517702175088416627556918748246821016005059896747726126979}{229410826794931810432354349663620997218585467189948104084269660001702596451306} a^{3} - \frac{38709012505534012229404089333858333553942038465651057492900486926255787038549}{229410826794931810432354349663620997218585467189948104084269660001702596451306} a^{2} + \frac{3699813428790511376501300742267592902089541249477500448155054429716504812885}{114705413397465905216177174831810498609292733594974052042134830000851298225653} a - \frac{8601812242397225005366460017330142349556066978944690121310665534616581287383}{114705413397465905216177174831810498609292733594974052042134830000851298225653}$

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

Class group and class number

$C_{2}\times C_{12}$, which has order $24$ (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:  \( 955634612.02 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T547:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 28800
The 41 conjugacy class representatives for t20n547
Character table for t20n547 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.680192.2, 10.6.925322313728.1

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
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 R $20$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.12.26.64$x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{6} - 2 x^{4} + 4 x^{3} + 2$$12$$1$$26$$S_3 \times C_2^2$$[2, 3]_{3}^{2}$
2657Data not computed