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Properties

Label 20.6.11715220217...1875.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,5^{14}\cdot 19^{9}\cdot 29^{6}$
Root discriminant $31.87$
Ramified primes $5, 19, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T955

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -10, 33, -85, -23, 260, 47, 2115, 4441, 3878, 2186, 1042, 174, -121, -74, -118, -62, 5, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + x^18 + 5*x^17 - 62*x^16 - 118*x^15 - 74*x^14 - 121*x^13 + 174*x^12 + 1042*x^11 + 2186*x^10 + 3878*x^9 + 4441*x^8 + 2115*x^7 + 47*x^6 + 260*x^5 - 23*x^4 - 85*x^3 + 33*x^2 - 10*x - 4)
 
gp: K = bnfinit(x^20 - 2*x^19 + x^18 + 5*x^17 - 62*x^16 - 118*x^15 - 74*x^14 - 121*x^13 + 174*x^12 + 1042*x^11 + 2186*x^10 + 3878*x^9 + 4441*x^8 + 2115*x^7 + 47*x^6 + 260*x^5 - 23*x^4 - 85*x^3 + 33*x^2 - 10*x - 4, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + x^{18} + 5 x^{17} - 62 x^{16} - 118 x^{15} - 74 x^{14} - 121 x^{13} + 174 x^{12} + 1042 x^{11} + 2186 x^{10} + 3878 x^{9} + 4441 x^{8} + 2115 x^{7} + 47 x^{6} + 260 x^{5} - 23 x^{4} - 85 x^{3} + 33 x^{2} - 10 x - 4 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1171522021720880762078857421875=-\,5^{14}\cdot 19^{9}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.87$
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:  $5, 19, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{6} a^{18} - \frac{1}{2} a^{17} + \frac{1}{3} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{6} a^{13} - \frac{1}{6} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{2} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{595587001386916319495251794315438} a^{19} - \frac{21200001463411969527652233359665}{297793500693458159747625897157719} a^{18} - \frac{14699108382752011064121007993555}{595587001386916319495251794315438} a^{17} + \frac{226586047450811780052127998511079}{595587001386916319495251794315438} a^{16} - \frac{24329895135874324590755620797274}{99264500231152719915875299052573} a^{15} - \frac{43128513915114313274868297756920}{297793500693458159747625897157719} a^{14} - \frac{30389183954645148908610420221980}{297793500693458159747625897157719} a^{13} + \frac{224175598577413012568600790037255}{595587001386916319495251794315438} a^{12} - \frac{128243015863167848840052721702982}{297793500693458159747625897157719} a^{11} + \frac{36942465966629011670442638035520}{99264500231152719915875299052573} a^{10} - \frac{87431069587168876532531490506672}{297793500693458159747625897157719} a^{9} + \frac{74235236678990857152206952919372}{297793500693458159747625897157719} a^{8} - \frac{88159994489224282595816618717945}{198529000462305439831750598105146} a^{7} - \frac{63917121801696316603203828055465}{198529000462305439831750598105146} a^{6} - \frac{221243345847763994010043430651593}{595587001386916319495251794315438} a^{5} - \frac{78513985632269638357575145690052}{297793500693458159747625897157719} a^{4} + \frac{8142774407154142615192578054509}{198529000462305439831750598105146} a^{3} - \frac{24355813905656738686503958772683}{595587001386916319495251794315438} a^{2} + \frac{87851491488275770018053892307149}{198529000462305439831750598105146} a + \frac{83625647155943437320247808262635}{297793500693458159747625897157719}$

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:  $12$
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:  \( 42996373.5338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T955:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 819200
The 275 conjugacy class representatives for t20n955 are not computed
Character table for t20n955 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.9932496465625.1

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 $16{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.8.0.1$x^{8} - x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$