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Properties

Label 15.15.3550599197...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{24}\cdot 5^{6}\cdot 23^{2}\cdot 37^{5}\cdot 67^{2}\cdot 39371^{4}\cdot 185021^{2}$
Root discriminant $4331.98$
Ramified primes $2, 5, 23, 37, 67, 39371, 185021$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T96

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![629936000, -2204776000, -4094584000, 9456914200, 17546867280, 3079532200, -7195456296, -4342753996, -801109368, -14432166, 8049544, 529090, -11448, -1386, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 1386*x^13 - 11448*x^12 + 529090*x^11 + 8049544*x^10 - 14432166*x^9 - 801109368*x^8 - 4342753996*x^7 - 7195456296*x^6 + 3079532200*x^5 + 17546867280*x^4 + 9456914200*x^3 - 4094584000*x^2 - 2204776000*x + 629936000)
 
gp: K = bnfinit(x^15 - 1386*x^13 - 11448*x^12 + 529090*x^11 + 8049544*x^10 - 14432166*x^9 - 801109368*x^8 - 4342753996*x^7 - 7195456296*x^6 + 3079532200*x^5 + 17546867280*x^4 + 9456914200*x^3 - 4094584000*x^2 - 2204776000*x + 629936000, 1)
 

Normalized defining polynomial

\( x^{15} - 1386 x^{13} - 11448 x^{12} + 529090 x^{11} + 8049544 x^{10} - 14432166 x^{9} - 801109368 x^{8} - 4342753996 x^{7} - 7195456296 x^{6} + 3079532200 x^{5} + 17546867280 x^{4} + 9456914200 x^{3} - 4094584000 x^{2} - 2204776000 x + 629936000 \)

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:  \(3550599197078961086196327997289859114171513438208000000=2^{24}\cdot 5^{6}\cdot 23^{2}\cdot 37^{5}\cdot 67^{2}\cdot 39371^{4}\cdot 185021^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $4331.98$
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:  $2, 5, 23, 37, 67, 39371, 185021$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{20} a^{11} - \frac{1}{20} a^{9} + \frac{1}{10} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{3}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{12} + \frac{1}{10} a^{10} + \frac{1}{20} a^{9} - \frac{1}{4} a^{8} + \frac{1}{10} a^{7} - \frac{3}{20} a^{6} + \frac{3}{10} a^{5} - \frac{2}{5} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{44400} a^{13} - \frac{19}{2220} a^{12} + \frac{59}{7400} a^{11} - \frac{817}{11100} a^{10} - \frac{57}{1480} a^{9} + \frac{614}{2775} a^{8} + \frac{4657}{22200} a^{7} + \frac{139}{925} a^{6} - \frac{1283}{3700} a^{5} - \frac{107}{925} a^{4} + \frac{121}{1110} a^{3} - \frac{232}{555} a^{2} + \frac{41}{222} a - \frac{10}{111}$, $\frac{1}{257710263600173539770582432364334639945858618573659683324000} a^{14} - \frac{259086075217370656323550536745782021723864846106078021}{25771026360017353977058243236433463994585861857365968332400} a^{13} - \frac{903818047646480671485397881158613053626500103442150576533}{128855131800086769885291216182167319972929309286829841662000} a^{12} + \frac{1346278178363628034033686211267651459530726743838783675203}{64427565900043384942645608091083659986464654643414920831000} a^{11} - \frac{3596113570746817611837273446134095886737905538097114667}{139302845189282994470585098575316021592356010039816045040} a^{10} + \frac{1443801891836830744758644508271310017124812129181552250521}{64427565900043384942645608091083659986464654643414920831000} a^{9} + \frac{7298475545713426247037496474315519129076559116891889954197}{128855131800086769885291216182167319972929309286829841662000} a^{8} - \frac{5619393263170421189713812170502097547764503391577502041407}{64427565900043384942645608091083659986464654643414920831000} a^{7} - \frac{3603545998786201635029365263514578653956239973142066116253}{21475855300014461647548536030361219995488218214471640277000} a^{6} - \frac{1841457893217060652458276362043278555902124508332454594427}{5368963825003615411887134007590304998872054553617910069250} a^{5} - \frac{1372635590181619856969290918225705794069196034461025472809}{6442756590004338494264560809108365998646465464341492083100} a^{4} + \frac{353421914136183740315325998931797659341175845954776745251}{1610689147501084623566140202277091499661616366085373020775} a^{3} - \frac{441817554926441373749753032997871436206859002025857033821}{1288551318000867698852912161821673199729293092868298416620} a^{2} - \frac{75432519206412693846840746574785435375619747121629591949}{214758553000144616475485360303612199954882182144716402770} a + \frac{13304034156354003348524505935323476609587039818385609687}{64427565900043384942645608091083659986464654643414920831}$

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:  \( 149997398390000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T96:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1296000
The 65 conjugacy class representatives for [A(5)^3]S(3)=A(5)wrS(3) are not computed
Character table for [A(5)^3]S(3)=A(5)wrS(3) is not computed

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 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ $15$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ R ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$[\ ]_{3}^{2}$
2.6.11.8$x^{6} + 4 x^{4} + 2 x^{2} + 6$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
2.6.11.8$x^{6} + 4 x^{4} + 2 x^{2} + 6$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37Data not computed
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.3.0.1$x^{3} - x + 16$$1$$3$$0$$C_3$$[\ ]^{3}$
67.3.0.1$x^{3} - x + 16$$1$$3$$0$$C_3$$[\ ]^{3}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
39371Data not computed
185021Data not computed