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Properties

Label 20.4.31901358550...8125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{11}\cdot 19^{8}\cdot 29^{8}\cdot 769$
Root discriminant $42.19$
Ramified primes $5, 19, 29, 769$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T955

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-995, 1735, 540, -1285, -334, -1399, 2918, 2027, -1789, -1629, -1359, 444, 1446, 117, 31, -312, -12, -32, 13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 13*x^18 - 32*x^17 - 12*x^16 - 312*x^15 + 31*x^14 + 117*x^13 + 1446*x^12 + 444*x^11 - 1359*x^10 - 1629*x^9 - 1789*x^8 + 2027*x^7 + 2918*x^6 - 1399*x^5 - 334*x^4 - 1285*x^3 + 540*x^2 + 1735*x - 995)
 
gp: K = bnfinit(x^20 - 2*x^19 + 13*x^18 - 32*x^17 - 12*x^16 - 312*x^15 + 31*x^14 + 117*x^13 + 1446*x^12 + 444*x^11 - 1359*x^10 - 1629*x^9 - 1789*x^8 + 2027*x^7 + 2918*x^6 - 1399*x^5 - 334*x^4 - 1285*x^3 + 540*x^2 + 1735*x - 995, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 13 x^{18} - 32 x^{17} - 12 x^{16} - 312 x^{15} + 31 x^{14} + 117 x^{13} + 1446 x^{12} + 444 x^{11} - 1359 x^{10} - 1629 x^{9} - 1789 x^{8} + 2027 x^{7} + 2918 x^{6} - 1399 x^{5} - 334 x^{4} - 1285 x^{3} + 540 x^{2} + 1735 x - 995 \)

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:  \(319013585509694102896209423828125=5^{11}\cdot 19^{8}\cdot 29^{8}\cdot 769\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.19$
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:  $5, 19, 29, 769$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} + \frac{2}{11} a^{17} - \frac{5}{11} a^{16} - \frac{5}{11} a^{15} - \frac{1}{11} a^{14} + \frac{1}{11} a^{13} - \frac{5}{11} a^{12} + \frac{5}{11} a^{11} + \frac{1}{11} a^{10} - \frac{1}{11} a^{9} - \frac{3}{11} a^{8} + \frac{2}{11} a^{7} + \frac{2}{11} a^{6} + \frac{3}{11} a^{5} - \frac{4}{11} a^{4} + \frac{3}{11} a^{3} + \frac{2}{11} a^{2} - \frac{2}{11} a - \frac{4}{11}$, $\frac{1}{2824598571846636240218165423454516868502} a^{19} - \frac{6273797402929225656661381834401135629}{256781688349694203656196856677683351682} a^{18} - \frac{311066358515333872690302075599538535011}{1412299285923318120109082711727258434251} a^{17} + \frac{242849416260903329459881519468158462978}{1412299285923318120109082711727258434251} a^{16} - \frac{426968843250007851011450514150260668001}{1412299285923318120109082711727258434251} a^{15} + \frac{602647587830028058150974907850337764496}{1412299285923318120109082711727258434251} a^{14} + \frac{703559810416105432553416657046605734571}{2824598571846636240218165423454516868502} a^{13} + \frac{334316770422510990777294473414373860534}{1412299285923318120109082711727258434251} a^{12} - \frac{524572691382732155173913067106676496640}{1412299285923318120109082711727258434251} a^{11} - \frac{689267592606468885995889487491308679890}{1412299285923318120109082711727258434251} a^{10} + \frac{1195019712366618537054894055071925303511}{2824598571846636240218165423454516868502} a^{9} - \frac{369277879093796275395549807938423896026}{1412299285923318120109082711727258434251} a^{8} - \frac{1295341443330627340501524266041482108239}{2824598571846636240218165423454516868502} a^{7} - \frac{484370873762371463515964984880668449197}{1412299285923318120109082711727258434251} a^{6} - \frac{166023597888964180733744782180119941601}{1412299285923318120109082711727258434251} a^{5} + \frac{108863863779307676321895832232206486905}{256781688349694203656196856677683351682} a^{4} - \frac{610719919468707562122291624978973742513}{2824598571846636240218165423454516868502} a^{3} - \frac{373727138639983934810084418607385699036}{1412299285923318120109082711727258434251} a^{2} - \frac{5568556014905922554407067675475740278}{128390844174847101828098428338841675841} a - \frac{783096080640488155002295929935149538539}{2824598571846636240218165423454516868502}$

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:  $11$
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:  \( 235787487.271 \) (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 $16{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ $16{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.8.0.1$x^{8} - x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
19.10.8.1$x^{10} - 209 x^{5} + 11552$$5$$2$$8$$D_5$$[\ ]_{5}^{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$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}$
769Data not computed