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Properties

Label 15.15.9102888854...9152.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 37^{7}\cdot 701^{8}\cdot 1061^{8}$
Root discriminant $11{,}586.31$
Ramified primes $2, 37, 701, 1061$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group $(C_5^2 : C_3):C_2$ (as 15T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-50455027118296432, 394561383160313392, -509633033085546528, 262662521750738480, -62169788735286688, 6104238185833040, -9894865181840, -38266269738784, 2176728943156, -8693448072, -2154380624, 38866636, 527439, -14319, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 - 14319*x^13 + 527439*x^12 + 38866636*x^11 - 2154380624*x^10 - 8693448072*x^9 + 2176728943156*x^8 - 38266269738784*x^7 - 9894865181840*x^6 + 6104238185833040*x^5 - 62169788735286688*x^4 + 262662521750738480*x^3 - 509633033085546528*x^2 + 394561383160313392*x - 50455027118296432)
 
gp: K = bnfinit(x^15 - 7*x^14 - 14319*x^13 + 527439*x^12 + 38866636*x^11 - 2154380624*x^10 - 8693448072*x^9 + 2176728943156*x^8 - 38266269738784*x^7 - 9894865181840*x^6 + 6104238185833040*x^5 - 62169788735286688*x^4 + 262662521750738480*x^3 - 509633033085546528*x^2 + 394561383160313392*x - 50455027118296432, 1)
 

Normalized defining polynomial

\( x^{15} - 7 x^{14} - 14319 x^{13} + 527439 x^{12} + 38866636 x^{11} - 2154380624 x^{10} - 8693448072 x^{9} + 2176728943156 x^{8} - 38266269738784 x^{7} - 9894865181840 x^{6} + 6104238185833040 x^{5} - 62169788735286688 x^{4} + 262662521750738480 x^{3} - 509633033085546528 x^{2} + 394561383160313392 x - 50455027118296432 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9102888854247560835776331977265632828069564619432357343409152=2^{10}\cdot 37^{7}\cdot 701^{8}\cdot 1061^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11{,}586.31$
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:  $2, 37, 701, 1061$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{12} a^{5} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{24} a^{9} - \frac{1}{24} a^{8} + \frac{5}{24} a^{7} - \frac{1}{24} a^{6} - \frac{1}{4} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{72} a^{10} - \frac{1}{72} a^{9} + \frac{1}{72} a^{8} - \frac{1}{8} a^{7} - \frac{1}{9} a^{6} - \frac{7}{36} a^{5} + \frac{1}{9} a^{4} - \frac{7}{18} a^{3} + \frac{1}{9} a^{2} - \frac{5}{18} a + \frac{1}{9}$, $\frac{1}{144} a^{11} - \frac{1}{144} a^{10} + \frac{1}{144} a^{9} + \frac{1}{48} a^{8} + \frac{1}{9} a^{7} + \frac{5}{72} a^{6} - \frac{5}{18} a^{5} - \frac{4}{9} a^{4} + \frac{1}{18} a^{3} - \frac{17}{36} a^{2} - \frac{5}{18} a + \frac{1}{3}$, $\frac{1}{864} a^{12} - \frac{1}{864} a^{11} - \frac{1}{288} a^{10} + \frac{13}{864} a^{9} - \frac{5}{144} a^{8} + \frac{5}{108} a^{7} + \frac{11}{432} a^{6} - \frac{101}{216} a^{5} - \frac{29}{72} a^{4} + \frac{41}{216} a^{3} + \frac{11}{36} a^{2} + \frac{5}{54} a + \frac{5}{108}$, $\frac{1}{1054944} a^{13} - \frac{1}{2592} a^{12} - \frac{1409}{1054944} a^{11} - \frac{6233}{1054944} a^{10} + \frac{353}{65934} a^{9} - \frac{1240}{32967} a^{8} + \frac{20233}{175824} a^{7} - \frac{13099}{87912} a^{6} + \frac{58673}{263736} a^{5} - \frac{74779}{263736} a^{4} + \frac{12685}{65934} a^{3} - \frac{9596}{32967} a^{2} + \frac{63475}{131868} a + \frac{1192}{32967}$, $\frac{1}{4826149711750187348069561440335250564364801512921025022440280623676624909703721263353599877450738740390637120} a^{14} + \frac{497650959594068896043457946755391712399693712873665184298149286705960450904285870352767438708468802643}{1608716570583395782689853813445083521454933837640341674146760207892208303234573754451199959150246246796879040} a^{13} - \frac{30183259414924931977172481582008913533238099412985590602072119148613493869267547228835480101662368717739}{321743314116679156537970762689016704290986767528068334829352041578441660646914750890239991830049249359375808} a^{12} + \frac{2724684405439738765163481906079448885413526669430993135127941013444243535783370787516385785473174233207453}{1608716570583395782689853813445083521454933837640341674146760207892208303234573754451199959150246246796879040} a^{11} - \frac{553888800971944307581038086573085498643038315126516573336889400935232984544482310284049711337248949724165}{120653742793754683701739036008381264109120037823025625561007015591915622742593031583839996936268468509765928} a^{10} - \frac{10237292732937477233207172257903283093869378833111325481034279249674925222817698652926370885539121148399053}{603268713968773418508695180041906320545600189115128127805035077959578113712965157919199984681342342548829640} a^{9} + \frac{17467910713010468825155754604599989232539057448702021519324558863922042331077822264910488802002721944860897}{2413074855875093674034780720167625282182400756460512511220140311838312454851860631676799938725369370195318560} a^{8} + \frac{42615834153582672722729860672080670844742220621684615274353689569219058860408031320577195974325871859605}{8937314281018865459388076741361575119194076875779675967482001154956712795747631969173333106390256926649328} a^{7} - \frac{17700602400380183174110494837518600167420977282354632427631360143526041454295211908406450277694658698564171}{109685220721595167001580941825801149190109125293659659600915468719923293402357301439854542669334971372514480} a^{6} + \frac{73240451939142914586889843188469632378456772111256591442929977782735690520244670513941949642014734639131283}{402179142645848945672463453361270880363733459410085418536690051973052075808643438612799989787561561699219760} a^{5} - \frac{6596927655217290657663265877285032314633001864696096057806796846070477714460098192431574383337568359332756}{75408589246096677313586897505238290068200023639391015975629384744947264214120644739899998085167792818603705} a^{4} - \frac{67469914945535233151352128161221444436272631483320132359763054678712483833134099516968578293984573493967891}{150817178492193354627173795010476580136400047278782031951258769489894528428241289479799996170335585637207410} a^{3} + \frac{40660196619700954583102868887603734935490310579711749605408752508347556470576715072563604609181410656972407}{201089571322924472836231726680635440181866729705042709268345025986526037904321719306399994893780780849609880} a^{2} + \frac{2219783323492461431614422229349435287319440915458674496632632352754123339503698579410189957655229345408741}{5027239283073111820905793167015886004546668242626067731708625649663150947608042982659999872344519521240247} a + \frac{17365941577578527212612282852366261479223695420949206300717043214181650697156234868778880624313321793472731}{150817178492193354627173795010476580136400047278782031951258769489894528428241289479799996170335585637207410}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  $14$
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:  \( 8919670142660000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:S_3$ (as 15T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 150
The 13 conjugacy class representatives for $(C_5^2 : C_3):C_2$
Character table for $(C_5^2 : C_3):C_2$

Intermediate fields

3.3.148.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 15 sibling: data not computed
Degree 25 sibling: data not computed
Degree 30 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
37Data not computed
701Data not computed
1061Data not computed