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Properties

Label 20.0.31241493709...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 17^{10}$
Root discriminant $59.53$
Ramified primes $2, 3, 5, 17$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![39062500, 0, -43750000, 22500000, 24562500, -15243750, -5542500, 4014000, 910200, -436230, -188889, 24135, 31653, -5334, -657, -99, -33, 84, -13, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 13*x^18 + 84*x^17 - 33*x^16 - 99*x^15 - 657*x^14 - 5334*x^13 + 31653*x^12 + 24135*x^11 - 188889*x^10 - 436230*x^9 + 910200*x^8 + 4014000*x^7 - 5542500*x^6 - 15243750*x^5 + 24562500*x^4 + 22500000*x^3 - 43750000*x^2 + 39062500)
 
gp: K = bnfinit(x^20 - 3*x^19 - 13*x^18 + 84*x^17 - 33*x^16 - 99*x^15 - 657*x^14 - 5334*x^13 + 31653*x^12 + 24135*x^11 - 188889*x^10 - 436230*x^9 + 910200*x^8 + 4014000*x^7 - 5542500*x^6 - 15243750*x^5 + 24562500*x^4 + 22500000*x^3 - 43750000*x^2 + 39062500, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 13 x^{18} + 84 x^{17} - 33 x^{16} - 99 x^{15} - 657 x^{14} - 5334 x^{13} + 31653 x^{12} + 24135 x^{11} - 188889 x^{10} - 436230 x^{9} + 910200 x^{8} + 4014000 x^{7} - 5542500 x^{6} - 15243750 x^{5} + 24562500 x^{4} + 22500000 x^{3} - 43750000 x^{2} + 39062500 \)

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:  \(312414937093187559824400000000000000=2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.53$
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:  $2, 3, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{25} a^{12} + \frac{2}{25} a^{11} - \frac{3}{25} a^{10} - \frac{6}{25} a^{9} + \frac{12}{25} a^{8} + \frac{11}{25} a^{7} - \frac{2}{25} a^{6} + \frac{6}{25} a^{5} + \frac{8}{25} a^{4} + \frac{11}{25} a^{2}$, $\frac{1}{125} a^{13} + \frac{2}{125} a^{12} - \frac{3}{125} a^{11} - \frac{56}{125} a^{10} + \frac{62}{125} a^{9} - \frac{39}{125} a^{8} + \frac{23}{125} a^{7} + \frac{31}{125} a^{6} + \frac{58}{125} a^{5} + \frac{2}{5} a^{4} - \frac{14}{125} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{625} a^{14} + \frac{2}{625} a^{13} - \frac{3}{625} a^{12} - \frac{56}{625} a^{11} + \frac{62}{625} a^{10} - \frac{39}{625} a^{9} - \frac{102}{625} a^{8} + \frac{156}{625} a^{7} - \frac{192}{625} a^{6} + \frac{12}{25} a^{5} - \frac{14}{625} a^{4} + \frac{8}{25} a^{3} - \frac{2}{25} a^{2} - \frac{1}{5} a$, $\frac{1}{6250} a^{15} - \frac{3}{6250} a^{14} - \frac{13}{6250} a^{13} + \frac{42}{3125} a^{12} - \frac{33}{6250} a^{11} - \frac{99}{6250} a^{10} - \frac{657}{6250} a^{9} + \frac{458}{3125} a^{8} + \frac{403}{6250} a^{7} - \frac{173}{1250} a^{6} - \frac{1389}{6250} a^{5} + \frac{127}{625} a^{4} - \frac{46}{125} a^{3} + \frac{6}{25} a^{2} + \frac{1}{5} a$, $\frac{1}{31250} a^{16} + \frac{1}{15625} a^{15} + \frac{11}{15625} a^{14} + \frac{119}{31250} a^{13} + \frac{237}{31250} a^{12} - \frac{1532}{15625} a^{11} + \frac{974}{15625} a^{10} - \frac{10569}{31250} a^{9} - \frac{117}{31250} a^{8} + \frac{54}{625} a^{7} - \frac{7657}{15625} a^{6} + \frac{373}{1250} a^{5} + \frac{192}{625} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{156250} a^{17} + \frac{1}{78125} a^{16} - \frac{3}{156250} a^{15} - \frac{28}{78125} a^{14} + \frac{31}{78125} a^{13} + \frac{918}{78125} a^{12} - \frac{1977}{156250} a^{11} - \frac{5547}{78125} a^{10} - \frac{5721}{78125} a^{9} + \frac{981}{3125} a^{8} - \frac{58139}{156250} a^{7} + \frac{79}{3125} a^{6} - \frac{2227}{6250} a^{5} + \frac{12}{625} a^{4} - \frac{49}{125} a^{3} + \frac{8}{25} a^{2} - \frac{2}{5} a$, $\frac{1}{913439843750} a^{18} - \frac{1033994}{456719921875} a^{17} + \frac{6709567}{913439843750} a^{16} + \frac{62910639}{913439843750} a^{15} - \frac{19716023}{913439843750} a^{14} - \frac{3596916219}{913439843750} a^{13} + \frac{16724937483}{913439843750} a^{12} + \frac{38907140561}{913439843750} a^{11} + \frac{328396496643}{913439843750} a^{10} + \frac{20111794411}{182687968750} a^{9} + \frac{69765673011}{913439843750} a^{8} + \frac{89501109287}{182687968750} a^{7} + \frac{294983467}{3653759375} a^{6} - \frac{64253467}{292300750} a^{5} - \frac{6121944}{730751875} a^{4} - \frac{19149079}{146150375} a^{3} + \frac{13400167}{29230075} a^{2} + \frac{1086098}{5846015} a - \frac{465450}{1169203}$, $\frac{1}{83318930409739793084890644695901864136718750} a^{19} - \frac{12438646795395668428626926563544}{41659465204869896542445322347950932068359375} a^{18} - \frac{824180799177857594821495638225067556}{382196928485044922407755250898632404296875} a^{17} - \frac{1009545728728695841186831670440289836711}{83318930409739793084890644695901864136718750} a^{16} + \frac{2610332117757781718277647854884523213401}{41659465204869896542445322347950932068359375} a^{15} - \frac{5601935379120380221963669909481214456817}{11902704344248541869270092099414552019531250} a^{14} - \frac{183299382159434913833966559980225473748017}{83318930409739793084890644695901864136718750} a^{13} + \frac{1649210630054149053278101051417029934087561}{83318930409739793084890644695901864136718750} a^{12} + \frac{2589559383287198559103160931333999553311784}{41659465204869896542445322347950932068359375} a^{11} - \frac{7053676629473172849700081169228432257861439}{16663786081947958616978128939180372827343750} a^{10} - \frac{10814350698511931455775827645752582899776839}{83318930409739793084890644695901864136718750} a^{9} + \frac{5510854655463738611819391824947788763777017}{16663786081947958616978128939180372827343750} a^{8} - \frac{1383135673656290011249340996446869568694553}{3332757216389591723395625787836074565468750} a^{7} + \frac{65236357259858039943408565390214466042359}{133310288655583668935825031513442982618750} a^{6} + \frac{42996009906670329940338138574219618266307}{133310288655583668935825031513442982618750} a^{5} + \frac{5001029252929903404899384063849728083629}{13331028865558366893582503151344298261875} a^{4} - \frac{6445836496017711460159521501791826574}{380886539015953339816642947181265664625} a^{3} + \frac{173800078557078551515973478595265729453}{533241154622334675743300126053771930475} a^{2} - \frac{8540705490657565129133201967416322912}{106648230924466935148660025210754386095} a - \frac{8078004389844261664400585005155331032}{21329646184893387029732005042150877219}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 4271304143.474904 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{-51}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{-15}, \sqrt{-51})\), 5.1.162000.1, 10.0.393660000000.1, 10.0.111788181324000000.1, 10.2.186313635540000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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}$
$3$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$