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Properties

Label 15.15.6919859191...3125.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{12}\cdot 5^{5}\cdot 79^{4}\cdot 401^{7}\cdot 50329^{4}$
Root discriminant $3884.54$
Ramified primes $3, 5, 79, 401, 50329$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T43

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2402311520555338669, -202156325407652112, 113854569421380357, 6977459265924253, -1901998615798914, -102907697118067, 14664282257229, 781942057449, -54750997444, -3060924423, 91964349, 5784843, -50296, -4452, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 4452*x^13 - 50296*x^12 + 5784843*x^11 + 91964349*x^10 - 3060924423*x^9 - 54750997444*x^8 + 781942057449*x^7 + 14664282257229*x^6 - 102907697118067*x^5 - 1901998615798914*x^4 + 6977459265924253*x^3 + 113854569421380357*x^2 - 202156325407652112*x - 2402311520555338669)
 
gp: K = bnfinit(x^15 - x^14 - 4452*x^13 - 50296*x^12 + 5784843*x^11 + 91964349*x^10 - 3060924423*x^9 - 54750997444*x^8 + 781942057449*x^7 + 14664282257229*x^6 - 102907697118067*x^5 - 1901998615798914*x^4 + 6977459265924253*x^3 + 113854569421380357*x^2 - 202156325407652112*x - 2402311520555338669, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 4452 x^{13} - 50296 x^{12} + 5784843 x^{11} + 91964349 x^{10} - 3060924423 x^{9} - 54750997444 x^{8} + 781942057449 x^{7} + 14664282257229 x^{6} - 102907697118067 x^{5} - 1901998615798914 x^{4} + 6977459265924253 x^{3} + 113854569421380357 x^{2} - 202156325407652112 x - 2402311520555338669 \)

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:  \(691985919132121715398802656928813567369454726910003125=3^{12}\cdot 5^{5}\cdot 79^{4}\cdot 401^{7}\cdot 50329^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3884.54$
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:  $3, 5, 79, 401, 50329$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13} a^{13} + \frac{3}{13} a^{12} + \frac{4}{13} a^{11} - \frac{2}{13} a^{10} - \frac{4}{13} a^{9} - \frac{3}{13} a^{8} - \frac{2}{13} a^{7} - \frac{2}{13} a^{6} + \frac{3}{13} a^{5} - \frac{3}{13} a^{4} - \frac{2}{13} a^{3} + \frac{2}{13} a^{2} + \frac{1}{13} a - \frac{2}{13}$, $\frac{1}{244615765644746150397084267039354503279840847889527934655247591389995011232412123000667514791047033751306281} a^{14} + \frac{1305817030947032281631062247714297725788307283052630708049630563413385044118977507133544899050900170467305}{244615765644746150397084267039354503279840847889527934655247591389995011232412123000667514791047033751306281} a^{13} - \frac{97326438573114727184593444306231380281233052443825837626814246940556829970026422050802323953007614664055719}{244615765644746150397084267039354503279840847889527934655247591389995011232412123000667514791047033751306281} a^{12} - \frac{46316798045210573458670657749208263728480188219715336306601370626762405887858804360873376553328961938346000}{244615765644746150397084267039354503279840847889527934655247591389995011232412123000667514791047033751306281} a^{11} + \frac{1316937548584560242424627455063432783130593407126656104845352711210871585761828410575041016584459191174993}{244615765644746150397084267039354503279840847889527934655247591389995011232412123000667514791047033751306281} a^{10} + \frac{114394093790680939013824031571713703049321137937661782786818430094948655177487335729855454763338083683171612}{244615765644746150397084267039354503279840847889527934655247591389995011232412123000667514791047033751306281} a^{9} - \frac{7386638000346716482719419977244087634696716295572033306902110164723413459157992027494835290572460704570872}{244615765644746150397084267039354503279840847889527934655247591389995011232412123000667514791047033751306281} a^{8} - \frac{101269878911345593782252626659329641017598857175519791186617559613526404257146068868091978660397240524160549}{244615765644746150397084267039354503279840847889527934655247591389995011232412123000667514791047033751306281} a^{7} - \frac{88465884176074308657672868991947587972337432038707362779952466231191907967031676844562551550362191004102602}{244615765644746150397084267039354503279840847889527934655247591389995011232412123000667514791047033751306281} a^{6} - \frac{2288265965415419157444211094383868291186376128198840261733270495355824339782557792637424255614943738807382}{6611236909317463524245520730793364953509212105122376612303988956486351654930057378396419318676946858143413} a^{5} - \frac{5838763057613039068927294111294566300019840399990700496037509398440935108772602648987339722086709523740853}{244615765644746150397084267039354503279840847889527934655247591389995011232412123000667514791047033751306281} a^{4} - \frac{8103509127262343196102964497883563578034813305874487479012070794589146671424081486913854309103928345653374}{244615765644746150397084267039354503279840847889527934655247591389995011232412123000667514791047033751306281} a^{3} + \frac{42617334900520123264993994771784246272970947522838139943620484969945928788012764874813784163257082064896708}{244615765644746150397084267039354503279840847889527934655247591389995011232412123000667514791047033751306281} a^{2} + \frac{74376092483605598211106466266898430635619767253258933671558056668266869115428845763805763506870064085790536}{244615765644746150397084267039354503279840847889527934655247591389995011232412123000667514791047033751306281} a + \frac{1320677758634341573527979248923303096382973829758464156987860929792975349613621714870137896052849914875374}{18816597357288165415160328233796500252295449837655994973480583953076539325570163307743654983926694903946637}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 5820768023810000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T43:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1620
The 24 conjugacy class representatives for [3^4:2]D(5)
Character table for [3^4:2]D(5) is not computed

Intermediate fields

5.5.160801.1

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

Sibling fields

Degree 15 siblings: data not computed
Degree 30 siblings: data not computed
Degree 45 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.6.8.10$x^{6} + 6 x^{5} + 36$$3$$2$$8$$S_3\times C_3$$[2, 2]^{2}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$5$5.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.6.4.2$x^{6} - 79 x^{3} + 18723$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
401Data not computed
50329Data not computed