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Properties

Label 20.8.29658361313...0000.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{24}\cdot 5^{13}\cdot 3469^{4}$
Root discriminant $33.39$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T755

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, 454, -35, -3358, 3473, 3616, -9302, 6362, -463, -232, -1203, 1012, -43, -164, 76, -124, 67, -18, 11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 11*x^18 - 18*x^17 + 67*x^16 - 124*x^15 + 76*x^14 - 164*x^13 - 43*x^12 + 1012*x^11 - 1203*x^10 - 232*x^9 - 463*x^8 + 6362*x^7 - 9302*x^6 + 3616*x^5 + 3473*x^4 - 3358*x^3 - 35*x^2 + 454*x + 31)
 
gp: K = bnfinit(x^20 - 6*x^19 + 11*x^18 - 18*x^17 + 67*x^16 - 124*x^15 + 76*x^14 - 164*x^13 - 43*x^12 + 1012*x^11 - 1203*x^10 - 232*x^9 - 463*x^8 + 6362*x^7 - 9302*x^6 + 3616*x^5 + 3473*x^4 - 3358*x^3 - 35*x^2 + 454*x + 31, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 11 x^{18} - 18 x^{17} + 67 x^{16} - 124 x^{15} + 76 x^{14} - 164 x^{13} - 43 x^{12} + 1012 x^{11} - 1203 x^{10} - 232 x^{9} - 463 x^{8} + 6362 x^{7} - 9302 x^{6} + 3616 x^{5} + 3473 x^{4} - 3358 x^{3} - 35 x^{2} + 454 x + 31 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2965836131318190080000000000000=2^{24}\cdot 5^{13}\cdot 3469^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.39$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{41} a^{18} - \frac{4}{41} a^{17} + \frac{13}{41} a^{16} + \frac{9}{41} a^{15} + \frac{10}{41} a^{14} - \frac{14}{41} a^{13} - \frac{16}{41} a^{12} - \frac{8}{41} a^{11} - \frac{14}{41} a^{10} + \frac{2}{41} a^{9} + \frac{14}{41} a^{8} - \frac{20}{41} a^{7} + \frac{6}{41} a^{6} - \frac{17}{41} a^{5} - \frac{10}{41} a^{4} - \frac{18}{41} a^{3} + \frac{16}{41} a^{2} + \frac{20}{41} a + \frac{1}{41}$, $\frac{1}{1450656333086796388827377566746944081} a^{19} + \frac{11962022649014231921213374165820383}{1450656333086796388827377566746944081} a^{18} - \frac{10072374735316063345376831515742134}{35381861782604789971399452847486441} a^{17} + \frac{17409396686661236419689831482333586}{35381861782604789971399452847486441} a^{16} + \frac{269213247621818722597535358631718115}{1450656333086796388827377566746944081} a^{15} - \frac{55543072770136889136567033438144234}{1450656333086796388827377566746944081} a^{14} + \frac{9620552685695818429841084480164558}{35381861782604789971399452847486441} a^{13} - \frac{413392676038619354679536035499810134}{1450656333086796388827377566746944081} a^{12} - \frac{326776157905702401050569323421045645}{1450656333086796388827377566746944081} a^{11} - \frac{522586584080405189303490548253963118}{1450656333086796388827377566746944081} a^{10} + \frac{2428666494231942432737626490312015}{35381861782604789971399452847486441} a^{9} + \frac{366919509471981360595940521679008308}{1450656333086796388827377566746944081} a^{8} - \frac{575502052915430581671708226424511683}{1450656333086796388827377566746944081} a^{7} - \frac{258508649648654432349856929838169309}{1450656333086796388827377566746944081} a^{6} + \frac{543431914707654771607203122312731950}{1450656333086796388827377566746944081} a^{5} + \frac{8061350216494127924922547427393507}{46795365583445044800883147314417551} a^{4} + \frac{187197436681140701767421634412438119}{1450656333086796388827377566746944081} a^{3} - \frac{408357718644455286714107467233380221}{1450656333086796388827377566746944081} a^{2} - \frac{43138820131286691307174111998683148}{1450656333086796388827377566746944081} a - \frac{2351359289878899706114737120819029}{46795365583445044800883147314417551}$

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:  $13$
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:  \( 56848953.9954 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T755:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n755 are not computed
Character table for t20n755 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 R $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
2Data not computed
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
3469Data not computed