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Properties

Label 20.12.4747646778...3125.2
Degree $20$
Signature $[12, 4]$
Discriminant $5^{15}\cdot 11^{5}\cdot 9931^{5}$
Root discriminant $60.79$
Ramified primes $5, 11, 9931$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23291, 188709, 189952, -793637, -364886, 1140214, 101226, -652342, 7640, 199639, -8747, -39219, 4413, 6014, -1602, -804, 305, 75, -28, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 28*x^18 + 75*x^17 + 305*x^16 - 804*x^15 - 1602*x^14 + 6014*x^13 + 4413*x^12 - 39219*x^11 - 8747*x^10 + 199639*x^9 + 7640*x^8 - 652342*x^7 + 101226*x^6 + 1140214*x^5 - 364886*x^4 - 793637*x^3 + 189952*x^2 + 188709*x + 23291)
 
gp: K = bnfinit(x^20 - 3*x^19 - 28*x^18 + 75*x^17 + 305*x^16 - 804*x^15 - 1602*x^14 + 6014*x^13 + 4413*x^12 - 39219*x^11 - 8747*x^10 + 199639*x^9 + 7640*x^8 - 652342*x^7 + 101226*x^6 + 1140214*x^5 - 364886*x^4 - 793637*x^3 + 189952*x^2 + 188709*x + 23291, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 28 x^{18} + 75 x^{17} + 305 x^{16} - 804 x^{15} - 1602 x^{14} + 6014 x^{13} + 4413 x^{12} - 39219 x^{11} - 8747 x^{10} + 199639 x^{9} + 7640 x^{8} - 652342 x^{7} + 101226 x^{6} + 1140214 x^{5} - 364886 x^{4} - 793637 x^{3} + 189952 x^{2} + 188709 x + 23291 \)

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:  \(474764677846779705681219512939453125=5^{15}\cdot 11^{5}\cdot 9931^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.79$
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:  $5, 11, 9931$
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}{3801340598129900983364994029040963693285507077016289} a^{19} - \frac{123288673109621043498548052110828263378105111690672}{3801340598129900983364994029040963693285507077016289} a^{18} - \frac{1859963877586146376730574560331776242219869721962331}{3801340598129900983364994029040963693285507077016289} a^{17} + \frac{802419713026088593570674384656834215593484798688460}{3801340598129900983364994029040963693285507077016289} a^{16} - \frac{621561362251076729839636445995550352849193839148233}{3801340598129900983364994029040963693285507077016289} a^{15} - \frac{930581545436218981087846051558864417194114121969730}{3801340598129900983364994029040963693285507077016289} a^{14} + \frac{1518349744551879795811250008599211033639745886298304}{3801340598129900983364994029040963693285507077016289} a^{13} - \frac{1130908991012194507612602647922771111544748465694037}{3801340598129900983364994029040963693285507077016289} a^{12} + \frac{478188848101215439646532670126437575154463653220209}{3801340598129900983364994029040963693285507077016289} a^{11} + \frac{330661172731028889820785090669251531496484562956915}{3801340598129900983364994029040963693285507077016289} a^{10} + \frac{1166450148818882689571210290918846553388837351064150}{3801340598129900983364994029040963693285507077016289} a^{9} - \frac{687781786758112944059483288310917909364389967885627}{3801340598129900983364994029040963693285507077016289} a^{8} - \frac{1591603923977084825890099870814089839154247552002216}{3801340598129900983364994029040963693285507077016289} a^{7} + \frac{717788717819075757120924311248867131912012342529236}{3801340598129900983364994029040963693285507077016289} a^{6} + \frac{897467804670231720525210605732558419481369896211804}{3801340598129900983364994029040963693285507077016289} a^{5} + \frac{501262302873261274751029184418971056346839304372514}{3801340598129900983364994029040963693285507077016289} a^{4} - \frac{1756783389767295588215443119741682847967653747853404}{3801340598129900983364994029040963693285507077016289} a^{3} + \frac{1335629625017957367263819154416407407583094338351693}{3801340598129900983364994029040963693285507077016289} a^{2} + \frac{355647224063105230831209276653577185130029576102490}{3801340598129900983364994029040963693285507077016289} a - \frac{1013828674122302393246316475894506829000390119263328}{3801340598129900983364994029040963693285507077016289}$

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:  \( 48335741865.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.932312193828125.1

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 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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
9931Data not computed