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Properties

Label 20.8.12357827945...5625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{10}\cdot 19^{7}\cdot 1699^{5}$
Root discriminant $40.23$
Ramified primes $5, 19, 1699$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T756

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-199, 3164, -8412, -16238, 31942, 20133, -25303, -27485, 11305, 8065, 3900, 2779, 2913, 760, -247, 263, -145, 1, 10, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 10*x^18 + x^17 - 145*x^16 + 263*x^15 - 247*x^14 + 760*x^13 + 2913*x^12 + 2779*x^11 + 3900*x^10 + 8065*x^9 + 11305*x^8 - 27485*x^7 - 25303*x^6 + 20133*x^5 + 31942*x^4 - 16238*x^3 - 8412*x^2 + 3164*x - 199)
 
gp: K = bnfinit(x^20 - 6*x^19 + 10*x^18 + x^17 - 145*x^16 + 263*x^15 - 247*x^14 + 760*x^13 + 2913*x^12 + 2779*x^11 + 3900*x^10 + 8065*x^9 + 11305*x^8 - 27485*x^7 - 25303*x^6 + 20133*x^5 + 31942*x^4 - 16238*x^3 - 8412*x^2 + 3164*x - 199, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 10 x^{18} + x^{17} - 145 x^{16} + 263 x^{15} - 247 x^{14} + 760 x^{13} + 2913 x^{12} + 2779 x^{11} + 3900 x^{10} + 8065 x^{9} + 11305 x^{8} - 27485 x^{7} - 25303 x^{6} + 20133 x^{5} + 31942 x^{4} - 16238 x^{3} - 8412 x^{2} + 3164 x - 199 \)

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:  \(123578279458495716101657822265625=5^{10}\cdot 19^{7}\cdot 1699^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.23$
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:  $5, 19, 1699$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18} + \frac{5}{19} a^{17} - \frac{8}{19} a^{16} + \frac{4}{19} a^{15} + \frac{8}{19} a^{14} + \frac{2}{19} a^{13} + \frac{8}{19} a^{12} - \frac{1}{19} a^{11} + \frac{2}{19} a^{9} + \frac{8}{19} a^{8} + \frac{8}{19} a^{7} - \frac{2}{19} a^{6} - \frac{9}{19} a^{5} - \frac{5}{19} a^{4} + \frac{6}{19} a^{3} - \frac{3}{19} a^{2} - \frac{8}{19} a + \frac{3}{19}$, $\frac{1}{359820799856421128063305725768571597872056845945663} a^{19} - \frac{3335304764099172960825637892835109850576920117371}{359820799856421128063305725768571597872056845945663} a^{18} + \frac{76734198368010796819079963960646537632758184319869}{359820799856421128063305725768571597872056845945663} a^{17} - \frac{51890739855957214770675374290400254362369089309297}{359820799856421128063305725768571597872056845945663} a^{16} + \frac{157509945157964302660720228258312497521168821972395}{359820799856421128063305725768571597872056845945663} a^{15} + \frac{73977790274007680008752362569930136632358171057695}{359820799856421128063305725768571597872056845945663} a^{14} + \frac{95023093076746044051735790830769227238267937135072}{359820799856421128063305725768571597872056845945663} a^{13} - \frac{39750069991019554886956255217232228646977447307954}{359820799856421128063305725768571597872056845945663} a^{12} + \frac{151977516135874885876212048137055045852495311515082}{359820799856421128063305725768571597872056845945663} a^{11} - \frac{121343669224803729255431882559727842925702594253572}{359820799856421128063305725768571597872056845945663} a^{10} + \frac{104426551085332117870666020311886615275188695275199}{359820799856421128063305725768571597872056845945663} a^{9} + \frac{473207788991373086720016890161403907874903791805}{18937936834548480424384511882556399888002991891877} a^{8} - \frac{88603602484184810800251253832575649317719714442749}{359820799856421128063305725768571597872056845945663} a^{7} + \frac{32972419090694617257805119342357890940358854331049}{359820799856421128063305725768571597872056845945663} a^{6} - \frac{4779436853394437655720689713513769947660683409724}{359820799856421128063305725768571597872056845945663} a^{5} - \frac{133506639557893300183268346710382018869472525267064}{359820799856421128063305725768571597872056845945663} a^{4} - \frac{142131506351574707072421748911144560653180513894430}{359820799856421128063305725768571597872056845945663} a^{3} - \frac{127999502725514152354736634281648437611963138308481}{359820799856421128063305725768571597872056845945663} a^{2} + \frac{33250526601265207120517908432086213084505695215857}{359820799856421128063305725768571597872056845945663} a - \frac{40024815759636042741684973044972199287141110857038}{359820799856421128063305725768571597872056845945663}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 109638614.529 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T756:

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 t20n756 are not computed
Character table for t20n756 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.3256446753125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
1699Data not computed