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Properties

Label 20.4.23824343228...0625.2
Degree $20$
Signature $[4, 8]$
Discriminant $5^{15}\cdot 61^{7}\cdot 397^{4}$
Root discriminant $46.65$
Ramified primes $5, 61, 397$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![153355, 215710, 146950, 45950, -47929, -22055, 152131, 161485, 51809, -560, 8362, 6830, 699, -430, -136, 205, -74, -40, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^18 - 40*x^17 - 74*x^16 + 205*x^15 - 136*x^14 - 430*x^13 + 699*x^12 + 6830*x^11 + 8362*x^10 - 560*x^9 + 51809*x^8 + 161485*x^7 + 152131*x^6 - 22055*x^5 - 47929*x^4 + 45950*x^3 + 146950*x^2 + 215710*x + 153355)
 
gp: K = bnfinit(x^20 - 12*x^18 - 40*x^17 - 74*x^16 + 205*x^15 - 136*x^14 - 430*x^13 + 699*x^12 + 6830*x^11 + 8362*x^10 - 560*x^9 + 51809*x^8 + 161485*x^7 + 152131*x^6 - 22055*x^5 - 47929*x^4 + 45950*x^3 + 146950*x^2 + 215710*x + 153355, 1)
 

Normalized defining polynomial

\( x^{20} - 12 x^{18} - 40 x^{17} - 74 x^{16} + 205 x^{15} - 136 x^{14} - 430 x^{13} + 699 x^{12} + 6830 x^{11} + 8362 x^{10} - 560 x^{9} + 51809 x^{8} + 161485 x^{7} + 152131 x^{6} - 22055 x^{5} - 47929 x^{4} + 45950 x^{3} + 146950 x^{2} + 215710 x + 153355 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2382434322822520356765777587890625=5^{15}\cdot 61^{7}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.65$
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:  $5, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{17621244577460939346939992633396458934566484193399672943770841} a^{19} + \frac{1057093595665553464129811266966094346043381884671327479099544}{17621244577460939346939992633396458934566484193399672943770841} a^{18} + \frac{116047949289721948882567387126603003152688438070772042104297}{1355480352112379949764614817953573764197421861030744072597757} a^{17} + \frac{7571093578506501162512485413542610730526665822419090388741950}{17621244577460939346939992633396458934566484193399672943770841} a^{16} - \frac{3776128635397789976443249163321313826035206758984012836419071}{17621244577460939346939992633396458934566484193399672943770841} a^{15} + \frac{197359731527811691771179701659510630014892886868917383129132}{17621244577460939346939992633396458934566484193399672943770841} a^{14} - \frac{504369902316554791943019350223190962170343156431063822928304}{1355480352112379949764614817953573764197421861030744072597757} a^{13} + \frac{8772200083459193439585090847476289525861168066259756480546385}{17621244577460939346939992633396458934566484193399672943770841} a^{12} - \frac{1861497795773565213161782197798049617158003550747099433590641}{17621244577460939346939992633396458934566484193399672943770841} a^{11} - \frac{5607769806243833330211619309546921242772040219350646646881028}{17621244577460939346939992633396458934566484193399672943770841} a^{10} - \frac{6644945152440789204558915268193779451881173438993162165156834}{17621244577460939346939992633396458934566484193399672943770841} a^{9} - \frac{4852569124090613581045921029369650750043453954799041904430614}{17621244577460939346939992633396458934566484193399672943770841} a^{8} - \frac{2451333922365147152429782565038429451626287430749807340444522}{17621244577460939346939992633396458934566484193399672943770841} a^{7} - \frac{5522047807488174390715154980699450311722544357789041786512479}{17621244577460939346939992633396458934566484193399672943770841} a^{6} - \frac{8767285524539362410527059758037810085239363284209956236530663}{17621244577460939346939992633396458934566484193399672943770841} a^{5} - \frac{3240723015773494768468935429991439338296172241972558977200403}{17621244577460939346939992633396458934566484193399672943770841} a^{4} - \frac{160331966967243134948687917727230486657940485837263429179971}{17621244577460939346939992633396458934566484193399672943770841} a^{3} + \frac{6418726313434752425182540571415923867068270891400126263040764}{17621244577460939346939992633396458934566484193399672943770841} a^{2} - \frac{4529351396385993506460199961861111694854422681541596507821643}{17621244577460939346939992633396458934566484193399672943770841} a - \frac{6369602015573335251515815382187544641454263287328411245035303}{17621244577460939346939992633396458934566484193399672943770841}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 134531227.411 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.24217.1, 10.10.1832697153125.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
61Data not computed
397Data not computed