↑

Properties

Label 20.0.10800753337...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 31^{4}\cdot 71^{4}\cdot 2161^{5}$
Root discriminant $71.07$
Ramified primes $5, 31, 71, 2161$
Class number $44$ (GRH)
Class group $[2, 22]$ (GRH)
Galois group 20T168

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![48517907351, -16688451949, 43219440899, -12094971499, 17011183013, -3897482918, 3910053860, -732444355, 583819605, -88813431, 59444659, -7251477, 4194686, -400895, 202822, -14503, 6424, -311, 120, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 120*x^18 - 311*x^17 + 6424*x^16 - 14503*x^15 + 202822*x^14 - 400895*x^13 + 4194686*x^12 - 7251477*x^11 + 59444659*x^10 - 88813431*x^9 + 583819605*x^8 - 732444355*x^7 + 3910053860*x^6 - 3897482918*x^5 + 17011183013*x^4 - 12094971499*x^3 + 43219440899*x^2 - 16688451949*x + 48517907351)
 
gp: K = bnfinit(x^20 - 3*x^19 + 120*x^18 - 311*x^17 + 6424*x^16 - 14503*x^15 + 202822*x^14 - 400895*x^13 + 4194686*x^12 - 7251477*x^11 + 59444659*x^10 - 88813431*x^9 + 583819605*x^8 - 732444355*x^7 + 3910053860*x^6 - 3897482918*x^5 + 17011183013*x^4 - 12094971499*x^3 + 43219440899*x^2 - 16688451949*x + 48517907351, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 120 x^{18} - 311 x^{17} + 6424 x^{16} - 14503 x^{15} + 202822 x^{14} - 400895 x^{13} + 4194686 x^{12} - 7251477 x^{11} + 59444659 x^{10} - 88813431 x^{9} + 583819605 x^{8} - 732444355 x^{7} + 3910053860 x^{6} - 3897482918 x^{5} + 17011183013 x^{4} - 12094971499 x^{3} + 43219440899 x^{2} - 16688451949 x + 48517907351 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10800753337779736826690692535166015625=5^{10}\cdot 31^{4}\cdot 71^{4}\cdot 2161^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.07$
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:  $5, 31, 71, 2161$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1899443827966768408465218204536349178804737575768501922292838174604737787891} a^{19} + \frac{687430357437447435426467403417593248459102113085814708693452179611334480455}{1899443827966768408465218204536349178804737575768501922292838174604737787891} a^{18} - \frac{99443771186541422400130627968155739387318856403007483752703192556115416616}{1899443827966768408465218204536349178804737575768501922292838174604737787891} a^{17} - \frac{315565059816576677755195670880096998620668364927280600108428304081567483234}{633147942655589469488406068178783059601579191922833974097612724868245929297} a^{16} - \frac{44245549457140925154513338109880094341749075196675107584569161163731160421}{1899443827966768408465218204536349178804737575768501922292838174604737787891} a^{15} + \frac{95667459272274601493300525021476506899689749161742662034491266725004538092}{1899443827966768408465218204536349178804737575768501922292838174604737787891} a^{14} + \frac{270186275704017413760546472219183611704629269377757666335404636977483593743}{633147942655589469488406068178783059601579191922833974097612724868245929297} a^{13} + \frac{76660682243419803894536957833417495624115501413808670382576127991491865751}{1899443827966768408465218204536349178804737575768501922292838174604737787891} a^{12} + \frac{449836278365690696728392102481096748406728204173334491968066529365758596309}{1899443827966768408465218204536349178804737575768501922292838174604737787891} a^{11} + \frac{671800158754282245484727152287287563281220706396588914342011455180922142004}{1899443827966768408465218204536349178804737575768501922292838174604737787891} a^{10} - \frac{535628494842749739334191748400507950584441964874762274905299373859655290565}{1899443827966768408465218204536349178804737575768501922292838174604737787891} a^{9} + \frac{442767871424873990789481831500323951427188396301899122276790621354011129442}{1899443827966768408465218204536349178804737575768501922292838174604737787891} a^{8} - \frac{750930892014182085533636451104167116728510447761524606772004402838857767714}{1899443827966768408465218204536349178804737575768501922292838174604737787891} a^{7} - \frac{59590074615494535596102566105877307145047113090474239512183090305790784757}{633147942655589469488406068178783059601579191922833974097612724868245929297} a^{6} - \frac{102457885373271865052707472467368668418202013824802275074885949905470202532}{1899443827966768408465218204536349178804737575768501922292838174604737787891} a^{5} - \frac{7096684068147033899031369357222814448737410087182173614309832758860882021}{21342065482772678746800204545352237964098175008634853059470091849491435819} a^{4} + \frac{313232143607356230665205232580157883679789765145303423360651888958738057074}{633147942655589469488406068178783059601579191922833974097612724868245929297} a^{3} + \frac{779807272336458409883741269132440379168296213836124357879729113136257234684}{1899443827966768408465218204536349178804737575768501922292838174604737787891} a^{2} + \frac{183299945595106096752003824620381472071481285629802025451629028995152142638}{633147942655589469488406068178783059601579191922833974097612724868245929297} a - \frac{175067257971016611300779554870679265332942888452566470614121917613218450897}{1899443827966768408465218204536349178804737575768501922292838174604737787891}$

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

Class group and class number

$C_{2}\times C_{22}$, which has order $44$ (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:  $9$
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:  \( 289210628.932 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T168:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 800
The 44 conjugacy class representatives for t20n168
Character table for t20n168 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.54025.1, 10.2.15138753125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.10.0.1$x^{10} - x + 22$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2161Data not computed