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Properties

Label 15.15.1920888206...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{18}\cdot 5^{6}\cdot 101^{4}\cdot 107^{4}\cdot 193\cdot 229^{5}\cdot 9337^{2}\cdot 18013^{2}$
Root discriminant $5652.41$
Ramified primes $2, 5, 101, 107, 193, 229, 9337, 18013$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T102

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2766592000, 14524608000, 18916572800, -7245012800, -19343665440, -2890123600, 4909046392, 1541051052, -129545388, -49569282, 1241310, 494981, -5364, -1449, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 1449*x^13 - 5364*x^12 + 494981*x^11 + 1241310*x^10 - 49569282*x^9 - 129545388*x^8 + 1541051052*x^7 + 4909046392*x^6 - 2890123600*x^5 - 19343665440*x^4 - 7245012800*x^3 + 18916572800*x^2 + 14524608000*x + 2766592000)
 
gp: K = bnfinit(x^15 - 1449*x^13 - 5364*x^12 + 494981*x^11 + 1241310*x^10 - 49569282*x^9 - 129545388*x^8 + 1541051052*x^7 + 4909046392*x^6 - 2890123600*x^5 - 19343665440*x^4 - 7245012800*x^3 + 18916572800*x^2 + 14524608000*x + 2766592000, 1)
 

Normalized defining polynomial

\( x^{15} - 1449 x^{13} - 5364 x^{12} + 494981 x^{11} + 1241310 x^{10} - 49569282 x^{9} - 129545388 x^{8} + 1541051052 x^{7} + 4909046392 x^{6} - 2890123600 x^{5} - 19343665440 x^{4} - 7245012800 x^{3} + 18916572800 x^{2} + 14524608000 x + 2766592000 \)

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:  \(192088820668994844315798629752765566154819292680192000000=2^{18}\cdot 5^{6}\cdot 101^{4}\cdot 107^{4}\cdot 193\cdot 229^{5}\cdot 9337^{2}\cdot 18013^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $5652.41$
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:  $2, 5, 101, 107, 193, 229, 9337, 18013$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{4} a^{7} - \frac{3}{16} a^{6} - \frac{1}{8} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{160} a^{11} - \frac{9}{160} a^{9} - \frac{1}{40} a^{8} + \frac{21}{160} a^{7} + \frac{3}{16} a^{6} + \frac{39}{80} a^{5} + \frac{13}{40} a^{4} - \frac{17}{40} a^{3} + \frac{9}{20} a^{2} - \frac{1}{2} a$, $\frac{1}{320} a^{12} - \frac{9}{320} a^{10} - \frac{1}{80} a^{9} + \frac{21}{320} a^{8} - \frac{5}{32} a^{7} + \frac{39}{160} a^{6} - \frac{7}{80} a^{5} + \frac{23}{80} a^{4} + \frac{19}{40} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{3200} a^{13} - \frac{9}{3200} a^{11} + \frac{9}{800} a^{10} + \frac{21}{3200} a^{9} - \frac{5}{64} a^{8} - \frac{121}{1600} a^{7} + \frac{3}{800} a^{6} + \frac{43}{800} a^{5} - \frac{91}{400} a^{4} - \frac{9}{40} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{53837884186915303054625033076466812642446957435239502067200} a^{14} + \frac{226547651294910546366419947407544395355725974616094071}{2691894209345765152731251653823340632122347871761975103360} a^{13} - \frac{9076395762419135249591964503004504504923451349686423849}{53837884186915303054625033076466812642446957435239502067200} a^{12} + \frac{2776240975434314267045583574162126396579444847118411611}{3364867761682206440914064567279175790152934839702468879200} a^{11} - \frac{159979562702327433080105324279565982361494159513596618557}{7691126312416471864946433296638116091778136776462786009600} a^{10} + \frac{5290540693087428462226615119975653340974091119263970411}{153822526248329437298928665932762321835562735529255720192} a^{9} - \frac{388553081844822392324310652006715020687195671545175574283}{3845563156208235932473216648319058045889068388231393004800} a^{8} - \frac{277576736685164992468792500518986567511684256628335154811}{1922781578104117966236608324159529022944534194115696502400} a^{7} - \frac{80937169888338572585024553055956344976079973608717953021}{363769487749427723342061034300451436773290252940807446400} a^{6} - \frac{2834827702263511774047897587096570816654118528585005220651}{6729735523364412881828129134558351580305869679404937758400} a^{5} - \frac{304766629583196717512473958950748942822675200425201106219}{672973552336441288182812913455835158030586967940493775840} a^{4} - \frac{13404543411584138281577349885151395245760030481885451901}{67297355233644128818281291345583515803058696794049377584} a^{3} - \frac{34629742526207761654636911925380537232290774628310996073}{168243388084110322045703228363958789507646741985123443960} a^{2} - \frac{234475543733381638824813711738791509665103513214442141}{16824338808411032204570322836395878950764674198512344396} a - \frac{1464321201217947446604082171819457568500114019096268276}{4206084702102758051142580709098969737691168549628086099}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 1226973821300000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T102:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10368000
The 140 conjugacy class representatives for [S(5)^3]S(3)=S(5)wrS(3) are not computed
Character table for [S(5)^3]S(3)=S(5)wrS(3) is not computed

Intermediate fields

3.3.229.1

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 $15$ R ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.5$x^{4} + 2 x^{2} - 4$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.6$x^{4} - 20$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.5.4.1$x^{5} - 101$$5$$1$$4$$C_5$$[\ ]_{5}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
107Data not computed
$193$$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 5$$1$$1$$0$Trivial$[\ ]$
193.2.1.2$x^{2} + 965$$2$$1$$1$$C_2$$[\ ]_{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
193.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
229Data not computed
9337Data not computed
18013Data not computed