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Properties

Label 20.4.31514842292...0625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{10}\cdot 13^{6}\cdot 401^{8}$
Root discriminant $53.08$
Ramified primes $5, 13, 401$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-21951, 40581, -66339, 102969, 27869, -132550, -11621, 89618, 56360, -31355, -35089, 12570, 8622, -4116, -488, 360, -193, 14, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 15*x^18 + 14*x^17 - 193*x^16 + 360*x^15 - 488*x^14 - 4116*x^13 + 8622*x^12 + 12570*x^11 - 35089*x^10 - 31355*x^9 + 56360*x^8 + 89618*x^7 - 11621*x^6 - 132550*x^5 + 27869*x^4 + 102969*x^3 - 66339*x^2 + 40581*x - 21951)
 
gp: K = bnfinit(x^20 - 6*x^19 + 15*x^18 + 14*x^17 - 193*x^16 + 360*x^15 - 488*x^14 - 4116*x^13 + 8622*x^12 + 12570*x^11 - 35089*x^10 - 31355*x^9 + 56360*x^8 + 89618*x^7 - 11621*x^6 - 132550*x^5 + 27869*x^4 + 102969*x^3 - 66339*x^2 + 40581*x - 21951, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 15 x^{18} + 14 x^{17} - 193 x^{16} + 360 x^{15} - 488 x^{14} - 4116 x^{13} + 8622 x^{12} + 12570 x^{11} - 35089 x^{10} - 31355 x^{9} + 56360 x^{8} + 89618 x^{7} - 11621 x^{6} - 132550 x^{5} + 27869 x^{4} + 102969 x^{3} - 66339 x^{2} + 40581 x - 21951 \)

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:  \(31514842292848707138778668056640625=5^{10}\cdot 13^{6}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.08$
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:  $5, 13, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{1219044177} a^{18} - \frac{26969809}{406348059} a^{17} - \frac{3118450}{406348059} a^{16} + \frac{84357359}{1219044177} a^{15} - \frac{122102311}{1219044177} a^{14} + \frac{144756767}{406348059} a^{13} - \frac{49282340}{1219044177} a^{12} - \frac{15459676}{406348059} a^{11} - \frac{66195723}{135449353} a^{10} - \frac{60190001}{135449353} a^{9} + \frac{449658659}{1219044177} a^{8} + \frac{479518024}{1219044177} a^{7} - \frac{5656276}{93772629} a^{6} + \frac{577301879}{1219044177} a^{5} - \frac{291201647}{1219044177} a^{4} - \frac{479729113}{1219044177} a^{3} + \frac{591452825}{1219044177} a^{2} - \frac{12751189}{406348059} a - \frac{7507001}{135449353}$, $\frac{1}{311160905928609611207666805609593135363546652621} a^{19} - \frac{25896381530009760566977030973500948097}{103720301976203203735888935203197711787848884207} a^{18} + \frac{10604188221549178295696699281931643772808237096}{103720301976203203735888935203197711787848884207} a^{17} - \frac{2259910287329818440720852524315115748608620512}{311160905928609611207666805609593135363546652621} a^{16} - \frac{22913113594875628531351182504003044644791529291}{311160905928609611207666805609593135363546652621} a^{15} + \frac{360598429930908314963439264274303885734501853}{34573433992067734578629645067732570595949628069} a^{14} - \frac{145639149686339029871824564853497958060955431516}{311160905928609611207666805609593135363546652621} a^{13} + \frac{33134486181479025174628596673067914768566643241}{103720301976203203735888935203197711787848884207} a^{12} - \frac{3859724926482795983687278916820424200611594485}{34573433992067734578629645067732570595949628069} a^{11} + \frac{12126781830529526849602610560105667770377790349}{103720301976203203735888935203197711787848884207} a^{10} + \frac{91959546185146574885899234837930220316096747041}{311160905928609611207666805609593135363546652621} a^{9} - \frac{91054613054773140617288478184253521600572781109}{311160905928609611207666805609593135363546652621} a^{8} + \frac{8210428594954404002879939017749973199496435757}{18303582701682918306333341506446655021385097213} a^{7} - \frac{54464927896041366140826308805746779225243720240}{311160905928609611207666805609593135363546652621} a^{6} - \frac{3764172945251827859101640802930451478793096620}{311160905928609611207666805609593135363546652621} a^{5} - \frac{88129948839585969393751365987938949234796438201}{311160905928609611207666805609593135363546652621} a^{4} - \frac{58511753938410056970679181175035255057767023643}{311160905928609611207666805609593135363546652621} a^{3} - \frac{10143950010371765974812432218252122378681823995}{34573433992067734578629645067732570595949628069} a^{2} + \frac{5520090455674456529240941359000983699045889752}{34573433992067734578629645067732570595949628069} a - \frac{3572635975454925460912452212379466136662938678}{11524477997355911526209881689244190198649876023}$

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:  \( 654238305.599 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T141:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n141
Character table for t20n141 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.160801.1, 10.2.177524201991865625.1, 10.10.80803005003125.1, 10.2.56807744637397.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed