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Properties

Label 20.0.33289080246...3461.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{15}\cdot 1567^{7}$
Root discriminant $29.93$
Ramified primes $3, 1567$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4:S_5$ (as 20T120)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![656181, 309501, 2473983, 184761, 1865295, -266436, 632475, -213489, 180504, -78273, 45207, -18018, 8985, -3417, 1452, -486, 175, -49, 15, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 15*x^18 - 49*x^17 + 175*x^16 - 486*x^15 + 1452*x^14 - 3417*x^13 + 8985*x^12 - 18018*x^11 + 45207*x^10 - 78273*x^9 + 180504*x^8 - 213489*x^7 + 632475*x^6 - 266436*x^5 + 1865295*x^4 + 184761*x^3 + 2473983*x^2 + 309501*x + 656181)
 
gp: K = bnfinit(x^20 - 4*x^19 + 15*x^18 - 49*x^17 + 175*x^16 - 486*x^15 + 1452*x^14 - 3417*x^13 + 8985*x^12 - 18018*x^11 + 45207*x^10 - 78273*x^9 + 180504*x^8 - 213489*x^7 + 632475*x^6 - 266436*x^5 + 1865295*x^4 + 184761*x^3 + 2473983*x^2 + 309501*x + 656181, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 15 x^{18} - 49 x^{17} + 175 x^{16} - 486 x^{15} + 1452 x^{14} - 3417 x^{13} + 8985 x^{12} - 18018 x^{11} + 45207 x^{10} - 78273 x^{9} + 180504 x^{8} - 213489 x^{7} + 632475 x^{6} - 266436 x^{5} + 1865295 x^{4} + 184761 x^{3} + 2473983 x^{2} + 309501 x + 656181 \)

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:  \(332890802464111776751903153461=3^{15}\cdot 1567^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.93$
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:  $3, 1567$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{11} + \frac{1}{9} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{11} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4}$, $\frac{1}{27} a^{16} - \frac{1}{27} a^{15} - \frac{1}{27} a^{13} + \frac{1}{27} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{27} a^{17} - \frac{1}{27} a^{15} - \frac{1}{27} a^{14} + \frac{1}{27} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{27} a^{18} + \frac{1}{27} a^{15} + \frac{1}{27} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5}$, $\frac{1}{21061959151770517334557206098143884433057955318205} a^{19} - \frac{203357541172588780516935965428732572547932148}{369508055294219602360652738563927797071192198565} a^{18} + \frac{78165599959450717466099718128196825409914075094}{7020653050590172444852402032714628144352651772735} a^{17} - \frac{26084210055455627158673921592047693480944746178}{21061959151770517334557206098143884433057955318205} a^{16} + \frac{423376346502412555465531567688685682548317174336}{21061959151770517334557206098143884433057955318205} a^{15} - \frac{34708942472777808803482802104391237871359188579}{780072561176685827205822448079403127150294641415} a^{14} + \frac{333944332620921591032756933389444970373346689781}{7020653050590172444852402032714628144352651772735} a^{13} - \frac{996360807740690487107775897904479638412833139188}{21061959151770517334557206098143884433057955318205} a^{12} - \frac{884570229329072256642326485218988523250135390473}{7020653050590172444852402032714628144352651772735} a^{11} - \frac{30960482324530704692951673040001281346913445522}{1404130610118034488970480406542925628870530354547} a^{10} - \frac{371887101600694805915789412430403488307389139522}{2340217683530057481617467344238209381450883924245} a^{9} + \frac{625247442333151255203097352529038737153158250436}{7020653050590172444852402032714628144352651772735} a^{8} + \frac{232345663068210993972853725565876720442806982927}{2340217683530057481617467344238209381450883924245} a^{7} - \frac{70857903487947579311235178669515860179774077337}{156014512235337165441164489615880625430058928283} a^{6} - \frac{36887418375130362765983385938140937433922750491}{156014512235337165441164489615880625430058928283} a^{5} - \frac{102220355236462352993573467104101314826967830938}{780072561176685827205822448079403127150294641415} a^{4} - \frac{335101557595853979683788481064867731833260547414}{780072561176685827205822448079403127150294641415} a^{3} - \frac{63355524170713043403300304139052291258910288109}{780072561176685827205822448079403127150294641415} a^{2} - \frac{348288232282506194437298325403595921283190007713}{780072561176685827205822448079403127150294641415} a - \frac{282700831300553172076872280440876434805356360201}{780072561176685827205822448079403127150294641415}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{3429109882286431309626812}{20607080989937621100431303643615} a^{19} + \frac{10002436430664925846179113}{20607080989937621100431303643615} a^{18} - \frac{83456511482534514882390211}{20607080989937621100431303643615} a^{17} + \frac{383043254049478450582116484}{20607080989937621100431303643615} a^{16} - \frac{1143485102502070245902065658}{20607080989937621100431303643615} a^{15} + \frac{3617457786876888166414349194}{20607080989937621100431303643615} a^{14} - \frac{965715942158472628077517006}{2289675665548624566714589293735} a^{13} + \frac{7869818418729298681888438658}{6869026996645873700143767881205} a^{12} - \frac{4913195764530665612806061732}{2289675665548624566714589293735} a^{11} + \frac{6712446117889654848052465795}{1373805399329174740028753576241} a^{10} - \frac{11080340231179783394016582739}{2289675665548624566714589293735} a^{9} + \frac{21217614652094159038635619399}{2289675665548624566714589293735} a^{8} + \frac{20471481098319224236994846024}{2289675665548624566714589293735} a^{7} + \frac{870806390554974965646682106}{457935133109724913342917858747} a^{6} + \frac{25287592053952962080212495841}{152645044369908304447639286249} a^{5} - \frac{23300693751479940126464195431}{763225221849541522238196431245} a^{4} + \frac{642090148171851598823968306322}{763225221849541522238196431245} a^{3} + \frac{54913962426964093121635344732}{763225221849541522238196431245} a^{2} + \frac{1335968939303500127699958703494}{763225221849541522238196431245} a + \frac{463632580821208773819803659478}{763225221849541522238196431245} \) (order $6$)
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:  \( 11798986.2811 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:S_5$ (as 20T120):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 19 conjugacy class representatives for $C_4:S_5$
Character table for $C_4:S_5$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.42309.2, 5.3.14103.1, 10.0.596683827.1

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 $20$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
3Data not computed
1567Data not computed