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Properties

Label 20.0.30210858588...0625.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 31^{2}\cdot 409^{5}\cdot 167711^{2}$
Root discriminant $47.21$
Ramified primes $5, 31, 409, 167711$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group 20T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53486371, -43814976, 89094183, -53623372, 62910983, -30201330, 26190577, -10416274, 7236290, -2439278, 1398203, -404725, 192547, -47783, 18822, -3900, 1266, -202, 53, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 53*x^18 - 202*x^17 + 1266*x^16 - 3900*x^15 + 18822*x^14 - 47783*x^13 + 192547*x^12 - 404725*x^11 + 1398203*x^10 - 2439278*x^9 + 7236290*x^8 - 10416274*x^7 + 26190577*x^6 - 30201330*x^5 + 62910983*x^4 - 53623372*x^3 + 89094183*x^2 - 43814976*x + 53486371)
 
gp: K = bnfinit(x^20 - 5*x^19 + 53*x^18 - 202*x^17 + 1266*x^16 - 3900*x^15 + 18822*x^14 - 47783*x^13 + 192547*x^12 - 404725*x^11 + 1398203*x^10 - 2439278*x^9 + 7236290*x^8 - 10416274*x^7 + 26190577*x^6 - 30201330*x^5 + 62910983*x^4 - 53623372*x^3 + 89094183*x^2 - 43814976*x + 53486371, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 53 x^{18} - 202 x^{17} + 1266 x^{16} - 3900 x^{15} + 18822 x^{14} - 47783 x^{13} + 192547 x^{12} - 404725 x^{11} + 1398203 x^{10} - 2439278 x^{9} + 7236290 x^{8} - 10416274 x^{7} + 26190577 x^{6} - 30201330 x^{5} + 62910983 x^{4} - 53623372 x^{3} + 89094183 x^{2} - 43814976 x + 53486371 \)

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:  \(3021085858887094365354739931640625=5^{10}\cdot 31^{2}\cdot 409^{5}\cdot 167711^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.21$
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:  $5, 31, 409, 167711$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{41} a^{18} + \frac{5}{41} a^{17} - \frac{14}{41} a^{16} + \frac{16}{41} a^{15} - \frac{11}{41} a^{14} - \frac{19}{41} a^{13} - \frac{7}{41} a^{12} + \frac{3}{41} a^{11} - \frac{1}{41} a^{10} - \frac{6}{41} a^{9} - \frac{4}{41} a^{8} - \frac{18}{41} a^{7} - \frac{4}{41} a^{6} - \frac{3}{41} a^{5} + \frac{10}{41} a^{4} + \frac{8}{41} a^{3} - \frac{15}{41} a^{2} + \frac{16}{41} a - \frac{18}{41}$, $\frac{1}{26996323760847331975389487777489634792486445036523629657835087} a^{19} + \frac{262139386490205934017541507529434973644168992459434049825648}{26996323760847331975389487777489634792486445036523629657835087} a^{18} + \frac{7172391831022495875190711763296798801595501460036741653575786}{26996323760847331975389487777489634792486445036523629657835087} a^{17} + \frac{875921527414276240645336841568816520954384665186424944227235}{26996323760847331975389487777489634792486445036523629657835087} a^{16} + \frac{4306693209519538507907525508808815049975380304242557961451463}{26996323760847331975389487777489634792486445036523629657835087} a^{15} - \frac{2903235529238526513900668910098710148361619177148422994201852}{26996323760847331975389487777489634792486445036523629657835087} a^{14} - \frac{4229967142257445625162394983119459179248217121052733889410406}{26996323760847331975389487777489634792486445036523629657835087} a^{13} - \frac{11271641147180991899370063642376244463603523202236357362848742}{26996323760847331975389487777489634792486445036523629657835087} a^{12} - \frac{11060136663130380132968217975148097583874813034822827952402280}{26996323760847331975389487777489634792486445036523629657835087} a^{11} - \frac{10916587767379030116639905523372281753900606662917723804184531}{26996323760847331975389487777489634792486445036523629657835087} a^{10} - \frac{7699364369437746830143825245401794104435288001985953618478936}{26996323760847331975389487777489634792486445036523629657835087} a^{9} + \frac{380004357341638619679326616522059561678789552147559096729655}{26996323760847331975389487777489634792486445036523629657835087} a^{8} + \frac{11850695486691037612940596742980055589966814371661513427712411}{26996323760847331975389487777489634792486445036523629657835087} a^{7} + \frac{6731314642723473269751161340664448890134240644783847523782697}{26996323760847331975389487777489634792486445036523629657835087} a^{6} - \frac{7798234782067999452066147721255569135778970610470061514995160}{26996323760847331975389487777489634792486445036523629657835087} a^{5} + \frac{10429190673192720197804243171820765016989355460070260354673659}{26996323760847331975389487777489634792486445036523629657835087} a^{4} + \frac{9026929270804138215307178230734252908098622786783826275405983}{26996323760847331975389487777489634792486445036523629657835087} a^{3} + \frac{4904196353807481916989904543873534903733988298344505797789763}{26996323760847331975389487777489634792486445036523629657835087} a^{2} + \frac{651209477423917242615845649309019424487401817372802826698536}{26996323760847331975389487777489634792486445036523629657835087} a + \frac{1684822465717946904309325267823774684823338329717110891411108}{26996323760847331975389487777489634792486445036523629657835087}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 39336445.4795 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 115200
The 119 conjugacy class representatives for t20n781 are not computed
Character table for t20n781 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.10225.1, 10.6.16247003125.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ $20$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
31Data not computed
409Data not computed
167711Data not computed