↑

Properties

Label 20.0.46374915021...5837.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{15}\cdot 11^{9}\cdot 23^{10}$
Root discriminant $60.72$
Ramified primes $7, 11, 23$
Class number Not computed
Class group Not computed
Galois group $C_5\times C_5:D_4$ (as 20T53)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6656093149231, -2285489898392, 499850958234, -121309533205, 99672823022, -4273590944, 265747180, 17023816, 138969016, 28845701, 4957725, -327567, 471311, -24157, -8165, -5128, -291, 86, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 86*x^17 - 291*x^16 - 5128*x^15 - 8165*x^14 - 24157*x^13 + 471311*x^12 - 327567*x^11 + 4957725*x^10 + 28845701*x^9 + 138969016*x^8 + 17023816*x^7 + 265747180*x^6 - 4273590944*x^5 + 99672823022*x^4 - 121309533205*x^3 + 499850958234*x^2 - 2285489898392*x + 6656093149231)
 
gp: K = bnfinit(x^20 - 2*x^19 + 86*x^17 - 291*x^16 - 5128*x^15 - 8165*x^14 - 24157*x^13 + 471311*x^12 - 327567*x^11 + 4957725*x^10 + 28845701*x^9 + 138969016*x^8 + 17023816*x^7 + 265747180*x^6 - 4273590944*x^5 + 99672823022*x^4 - 121309533205*x^3 + 499850958234*x^2 - 2285489898392*x + 6656093149231, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 86 x^{17} - 291 x^{16} - 5128 x^{15} - 8165 x^{14} - 24157 x^{13} + 471311 x^{12} - 327567 x^{11} + 4957725 x^{10} + 28845701 x^{9} + 138969016 x^{8} + 17023816 x^{7} + 265747180 x^{6} - 4273590944 x^{5} + 99672823022 x^{4} - 121309533205 x^{3} + 499850958234 x^{2} - 2285489898392 x + 6656093149231 \)

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:  \(463749150216643050841264365840725837=7^{15}\cdot 11^{9}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.72$
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:  $7, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{16} - \frac{1}{2} a^{15} - \frac{1}{4} a^{14} - \frac{3}{8} a^{12} - \frac{3}{8} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} + \frac{1}{8} a^{8} + \frac{3}{8} a^{7} - \frac{3}{8} a^{6} - \frac{1}{8} a^{5} - \frac{3}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{1007587740153089620519771234249559774530463670060762291363613132541641239535420165021110834387013228277598742154192099824} a^{19} + \frac{13835433758307302274245895161726390238693159552312307769361443494460257314100791198294959975012354353464306907638492283}{1007587740153089620519771234249559774530463670060762291363613132541641239535420165021110834387013228277598742154192099824} a^{18} - \frac{63308460358317455161514437712232479457209267738561073711920794818156328814104904861857990604078192305282320644110610665}{1007587740153089620519771234249559774530463670060762291363613132541641239535420165021110834387013228277598742154192099824} a^{17} + \frac{15968403973740770882163643470676076449352791432036523015629146901240564586286142027453914543008024758874295858503827825}{1007587740153089620519771234249559774530463670060762291363613132541641239535420165021110834387013228277598742154192099824} a^{16} + \frac{66761090015666570063060082548381398679701791469526321058984383663121549439832226768074608688941672031088624248666302633}{503793870076544810259885617124779887265231835030381145681806566270820619767710082510555417193506614138799371077096049912} a^{15} + \frac{87904228139846053845622372437928095026566332145177986530870558465978915759321652078459412475314396965938266135621806577}{503793870076544810259885617124779887265231835030381145681806566270820619767710082510555417193506614138799371077096049912} a^{14} + \frac{117722969117838280142513476039273784097595162453800233115288018025159997240542180163515559092908900202229029244537464661}{1007587740153089620519771234249559774530463670060762291363613132541641239535420165021110834387013228277598742154192099824} a^{13} - \frac{119129258041372600832576336759924031072935448884781008724181486743377055169250797700853592118196814403588787620194148071}{251896935038272405129942808562389943632615917515190572840903283135410309883855041255277708596753307069399685538548024956} a^{12} + \frac{265249738046972148770783715837931482522476620852694643363515903953715352932029881301294175993745332023848196872111273003}{1007587740153089620519771234249559774530463670060762291363613132541641239535420165021110834387013228277598742154192099824} a^{11} - \frac{23470873630386224190136315749876331933867849090054699455100532169527682715943157954495753007447975699971922164880559597}{125948467519136202564971404281194971816307958757595286420451641567705154941927520627638854298376653534699842769274012478} a^{10} - \frac{300808325191302218512452553590836805817824253229059645301239734012899333306512791868626094524083447713231392036380152667}{1007587740153089620519771234249559774530463670060762291363613132541641239535420165021110834387013228277598742154192099824} a^{9} - \frac{152726580969295780073096390894257207092050260865616516939618554795664418345446391698189899926559218111106175594159657989}{503793870076544810259885617124779887265231835030381145681806566270820619767710082510555417193506614138799371077096049912} a^{8} + \frac{36266175038583619926121090517394521641123063513756101206035539741482570341842746917567129353218479492012540384447600151}{503793870076544810259885617124779887265231835030381145681806566270820619767710082510555417193506614138799371077096049912} a^{7} + \frac{98227085272909201600704006734728771275402101166220459391729761551973567015878178561344927449576464147772367912776576603}{503793870076544810259885617124779887265231835030381145681806566270820619767710082510555417193506614138799371077096049912} a^{6} - \frac{174985716812371236082517912454396165663803271869336042228435964607454047944224936917924753931213252969916938045471973215}{503793870076544810259885617124779887265231835030381145681806566270820619767710082510555417193506614138799371077096049912} a^{5} + \frac{167533697920779366446785288424782828969878343845826054704794041105463886992308804534998860482923226129525414871880325137}{503793870076544810259885617124779887265231835030381145681806566270820619767710082510555417193506614138799371077096049912} a^{4} - \frac{13871241019971208690308971592684451114673016792249532490487022846550481574407251931080315122835460676358320735768521973}{62974233759568101282485702140597485908153979378797643210225820783852577470963760313819427149188326767349921384637006239} a^{3} + \frac{45909970491820810545041105542340997015952202892382950986561093717474893482102569278752653166509118732478537046440693747}{1007587740153089620519771234249559774530463670060762291363613132541641239535420165021110834387013228277598742154192099824} a^{2} - \frac{8170995680742503450482104756568131900362449556771296472597386950721573580861916990977707903531479836339890832784672263}{1007587740153089620519771234249559774530463670060762291363613132541641239535420165021110834387013228277598742154192099824} a - \frac{148085736734422047516594628008090938372175466515558409931186916184245126014285153252184459323057100311285523615125754467}{1007587740153089620519771234249559774530463670060762291363613132541641239535420165021110834387013228277598742154192099824}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.1995917.1, 10.0.246071287.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ $20$ $20$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{10}$ $20$

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
7Data not computed
$11$11.10.9.5$x^{10} - 8019$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$23$23.10.5.1$x^{10} - 1058 x^{6} + 279841 x^{2} - 25745372$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
23.10.5.1$x^{10} - 1058 x^{6} + 279841 x^{2} - 25745372$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$