↑

Properties

Label 20.8.51714682171...4672.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{16}\cdot 13^{16}\cdot 17^{9}$
Root discriminant $48.49$
Ramified primes $2, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-391, -8466, 17272, 150562, 270099, 210906, 92334, 43966, 21516, -2786, -11190, -4614, -628, -274, 210, 106, 91, -20, -2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 2*x^18 - 20*x^17 + 91*x^16 + 106*x^15 + 210*x^14 - 274*x^13 - 628*x^12 - 4614*x^11 - 11190*x^10 - 2786*x^9 + 21516*x^8 + 43966*x^7 + 92334*x^6 + 210906*x^5 + 270099*x^4 + 150562*x^3 + 17272*x^2 - 8466*x - 391)
 
gp: K = bnfinit(x^20 - 4*x^19 - 2*x^18 - 20*x^17 + 91*x^16 + 106*x^15 + 210*x^14 - 274*x^13 - 628*x^12 - 4614*x^11 - 11190*x^10 - 2786*x^9 + 21516*x^8 + 43966*x^7 + 92334*x^6 + 210906*x^5 + 270099*x^4 + 150562*x^3 + 17272*x^2 - 8466*x - 391, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 2 x^{18} - 20 x^{17} + 91 x^{16} + 106 x^{15} + 210 x^{14} - 274 x^{13} - 628 x^{12} - 4614 x^{11} - 11190 x^{10} - 2786 x^{9} + 21516 x^{8} + 43966 x^{7} + 92334 x^{6} + 210906 x^{5} + 270099 x^{4} + 150562 x^{3} + 17272 x^{2} - 8466 x - 391 \)

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:  \(5171468217146897008836483283484672=2^{16}\cdot 13^{16}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.49$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{5} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{3}{8} a^{4} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{14} - \frac{1}{16} a^{13} + \frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{3}{16} a^{9} - \frac{3}{16} a^{8} + \frac{3}{16} a^{7} + \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{5}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{16} a^{2} + \frac{3}{16} a + \frac{7}{16}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{3}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{8} a^{2} + \frac{3}{8} a + \frac{5}{16}$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{8} a^{2} + \frac{7}{16} a + \frac{3}{8}$, $\frac{1}{25760} a^{18} + \frac{11}{6440} a^{17} + \frac{177}{25760} a^{16} + \frac{22}{805} a^{15} + \frac{431}{12880} a^{14} + \frac{85}{2576} a^{13} - \frac{589}{6440} a^{12} + \frac{61}{920} a^{11} - \frac{81}{1610} a^{10} - \frac{19}{1840} a^{9} - \frac{463}{2576} a^{8} - \frac{81}{3220} a^{7} + \frac{741}{12880} a^{6} - \frac{2017}{12880} a^{5} - \frac{901}{6440} a^{4} - \frac{461}{6440} a^{3} + \frac{12599}{25760} a^{2} - \frac{737}{12880} a + \frac{11}{1120}$, $\frac{1}{927545978918639847112745406697760} a^{19} - \frac{3954761266163273796205711877}{463772989459319923556372703348880} a^{18} + \frac{4321208201544106353479212079365}{185509195783727969422549081339552} a^{17} + \frac{252908080351234747797361522294}{28985811841207495222273293959305} a^{16} + \frac{188899424852318771222924087189}{11594324736482998088909317583722} a^{15} + \frac{8474847141238890095479174845891}{231886494729659961778186351674440} a^{14} - \frac{9019145106957722063086558723583}{463772989459319923556372703348880} a^{13} - \frac{19164051684297757883022148356017}{463772989459319923556372703348880} a^{12} - \frac{1243382073192573425713057352419}{92754597891863984711274540669776} a^{11} + \frac{6951579700795022762687507673127}{57971623682414990444546587918610} a^{10} + \frac{5125199387856024100879216385841}{115943247364829980889093175837220} a^{9} - \frac{4104762553253356536292646119969}{463772989459319923556372703348880} a^{8} + \frac{656175521133283550781923818951}{16563321052118568698441882262460} a^{7} - \frac{10330467840863786371114660597155}{46377298945931992355637270334888} a^{6} + \frac{78551318083749162381039915715559}{463772989459319923556372703348880} a^{5} + \frac{6996214015400989188238273512287}{66253284208474274793767529049840} a^{4} + \frac{113527756838311080952646647453861}{927545978918639847112745406697760} a^{3} - \frac{11097646972651124365738419664509}{66253284208474274793767529049840} a^{2} + \frac{40669420028969287623581458608099}{185509195783727969422549081339552} a + \frac{5343220675143920284295506944751}{20164043019970431458972726232560}$

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:  \( 4161383152.19 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.10158928.1, 10.10.1341649635419392.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.12.11.6$x^{12} - 13312$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.6.5.2$x^{6} + 51$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$