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Properties

Label 20.20.8121996692...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 5^{13}\cdot 6029^{10}$
Root discriminant $442.06$
Ramified primes $2, 5, 6029$
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![25878125, 0, -983215875, 0, 9682058280, 0, -13873345570, 0, 4239133720, 0, -571963368, 0, 40234241, 0, -1524616, 0, 30002, 0, -283, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 283*x^18 + 30002*x^16 - 1524616*x^14 + 40234241*x^12 - 571963368*x^10 + 4239133720*x^8 - 13873345570*x^6 + 9682058280*x^4 - 983215875*x^2 + 25878125)
 
gp: K = bnfinit(x^20 - 283*x^18 + 30002*x^16 - 1524616*x^14 + 40234241*x^12 - 571963368*x^10 + 4239133720*x^8 - 13873345570*x^6 + 9682058280*x^4 - 983215875*x^2 + 25878125, 1)
 

Normalized defining polynomial

\( x^{20} - 283 x^{18} + 30002 x^{16} - 1524616 x^{14} + 40234241 x^{12} - 571963368 x^{10} + 4239133720 x^{8} - 13873345570 x^{6} + 9682058280 x^{4} - 983215875 x^{2} + 25878125 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(81219966925743319869894819510534963457280000000000000=2^{20}\cdot 5^{13}\cdot 6029^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $442.06$
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:  $2, 5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{14} + \frac{2}{5} a^{12} - \frac{1}{5} a^{10} + \frac{1}{5} a^{8} + \frac{2}{5} a^{6}$, $\frac{1}{325} a^{17} - \frac{133}{325} a^{15} - \frac{23}{325} a^{13} + \frac{84}{325} a^{11} + \frac{141}{325} a^{9} - \frac{68}{325} a^{7} - \frac{1}{65} a^{5} - \frac{29}{65} a^{3} - \frac{24}{65} a$, $\frac{1}{72205180584885355796690638053204021625073375} a^{18} - \frac{2203360007988010278527241566742655619972233}{72205180584885355796690638053204021625073375} a^{16} + \frac{12757622738425502897508974626842838984333602}{72205180584885355796690638053204021625073375} a^{14} - \frac{994614083112382591993005075054329809776141}{72205180584885355796690638053204021625073375} a^{12} + \frac{8109291633205731040590822040339244863557191}{72205180584885355796690638053204021625073375} a^{10} + \frac{19350921590446018009401746979354437128242807}{72205180584885355796690638053204021625073375} a^{8} - \frac{6839436166505369319141423655956465088599086}{14441036116977071159338127610640804325014675} a^{6} - \frac{2378870321698165104161231300785522994985139}{14441036116977071159338127610640804325014675} a^{4} - \frac{3656395447837024332273116892722971397919494}{14441036116977071159338127610640804325014675} a^{2} - \frac{373242452421772388808726665253271282386}{3417996714077413292151036120861728834323}$, $\frac{1}{2527181320470987452884172331862140756877568125} a^{19} + \frac{18337856162308361370931911817468122337717}{2527181320470987452884172331862140756877568125} a^{17} + \frac{145280722476338504018783340990859824358347286}{361025902924426778983453190266020108125366875} a^{15} + \frac{886573682644939914047327299609715105243048884}{2527181320470987452884172331862140756877568125} a^{13} + \frac{50321551052061785198656118132981595967446241}{2527181320470987452884172331862140756877568125} a^{11} - \frac{894877749507410102308696639448136482832301618}{2527181320470987452884172331862140756877568125} a^{9} + \frac{136237906284775151090300948363315503916161694}{505436264094197490576834466372428151375513625} a^{7} - \frac{27476291100935772687922151959668014708926552}{72205180584885355796690638053204021625073375} a^{5} + \frac{206294052714368087138104276831208722250370781}{505436264094197490576834466372428151375513625} a^{3} + \frac{364291493238877594497798454404774187435866}{1555188504905223047928721434992086619616965} a$

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:  $19$
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:  \( 403285850329000000000 \) (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

\(\Q(\sqrt{6029}) \), 4.4.2907907280.1, 5.5.753625.1, 10.10.124464771269983455453125.1

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 $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.5.2$x^{6} + 10$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
6029Data not computed