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Properties

Label 15.11.3805507799...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 337\cdot 53201^{4}\cdot 1103777641^{2}$
Root discriminant $6897.54$
Ramified primes $2, 5, 7, 337, 53201, 1103777641$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T101

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-111570583552000, -292872781824000, -331573827993600, -211243210342400, -82841158287360, -20553073039360, -3198404365312, -301773467776, -16826161536, -655037640, -29003136, -909588, -864, -54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 54*x^13 - 864*x^12 - 909588*x^11 - 29003136*x^10 - 655037640*x^9 - 16826161536*x^8 - 301773467776*x^7 - 3198404365312*x^6 - 20553073039360*x^5 - 82841158287360*x^4 - 211243210342400*x^3 - 331573827993600*x^2 - 292872781824000*x - 111570583552000)
 
gp: K = bnfinit(x^15 - 54*x^13 - 864*x^12 - 909588*x^11 - 29003136*x^10 - 655037640*x^9 - 16826161536*x^8 - 301773467776*x^7 - 3198404365312*x^6 - 20553073039360*x^5 - 82841158287360*x^4 - 211243210342400*x^3 - 331573827993600*x^2 - 292872781824000*x - 111570583552000, 1)
 

Normalized defining polynomial

\( x^{15} - 54 x^{13} - 864 x^{12} - 909588 x^{11} - 29003136 x^{10} - 655037640 x^{9} - 16826161536 x^{8} - 301773467776 x^{7} - 3198404365312 x^{6} - 20553073039360 x^{5} - 82841158287360 x^{4} - 211243210342400 x^{3} - 331573827993600 x^{2} - 292872781824000 x - 111570583552000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3805507799533955316962266792656268375035273432076288000000=2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 337\cdot 53201^{4}\cdot 1103777641^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $6897.54$
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:  $2, 5, 7, 337, 53201, 1103777641$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{8} a^{3} - \frac{1}{4} a$, $\frac{1}{16} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{80} a^{5} + \frac{1}{40} a^{3} - \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{320} a^{6} - \frac{1}{40} a^{4} - \frac{1}{40} a^{3} - \frac{1}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{640} a^{7} - \frac{1}{160} a^{5} - \frac{1}{80} a^{4} - \frac{3}{160} a^{3} - \frac{7}{40} a^{2} + \frac{1}{4} a$, $\frac{1}{6400} a^{8} + \frac{1}{1600} a^{6} - \frac{1}{400} a^{5} + \frac{21}{1600} a^{4} + \frac{9}{200} a^{3} - \frac{9}{100} a^{2} - \frac{2}{5}$, $\frac{1}{409600} a^{9} - \frac{1}{12800} a^{8} + \frac{117}{204800} a^{7} + \frac{3}{12800} a^{6} + \frac{139}{102400} a^{5} + \frac{3}{320} a^{4} - \frac{2089}{51200} a^{3} + \frac{519}{3200} a^{2} + \frac{41}{640} a + \frac{31}{80}$, $\frac{1}{1638400} a^{10} + \frac{53}{819200} a^{8} - \frac{3}{51200} a^{7} + \frac{523}{409600} a^{6} + \frac{17}{12800} a^{5} + \frac{3479}{204800} a^{4} - \frac{363}{12800} a^{3} - \frac{659}{12800} a^{2} + \frac{7}{64} a - \frac{3}{10}$, $\frac{1}{1048576000} a^{11} - \frac{1}{26214400} a^{10} - \frac{467}{524288000} a^{9} - \frac{3599}{65536000} a^{8} + \frac{183843}{262144000} a^{7} + \frac{13597}{32768000} a^{6} - \frac{135949}{26214400} a^{5} - \frac{224809}{16384000} a^{4} + \frac{66279}{4096000} a^{3} + \frac{14487}{64000} a^{2} - \frac{1973}{102400} a + \frac{6209}{12800}$, $\frac{1}{20971520000} a^{12} + \frac{13}{10485760000} a^{10} - \frac{727}{655360000} a^{9} - \frac{406077}{5242880000} a^{8} + \frac{29523}{163840000} a^{7} + \frac{146323}{524288000} a^{6} + \frac{857353}{163840000} a^{5} + \frac{1455229}{81920000} a^{4} - \frac{565629}{10240000} a^{3} + \frac{39979}{2048000} a^{2} - \frac{3287}{32000} a - \frac{2479}{6400}$, $\frac{1}{10485760000000} a^{13} + \frac{1}{524288000000} a^{12} - \frac{1507}{5242880000000} a^{11} + \frac{133957}{1310720000000} a^{10} + \frac{1262643}{2621440000000} a^{9} + \frac{24602759}{655360000000} a^{8} + \frac{45158083}{262144000000} a^{7} - \frac{405850593}{327680000000} a^{6} - \frac{110411333}{20480000000} a^{5} + \frac{187247797}{10240000000} a^{4} - \frac{40501501}{1024000000} a^{3} - \frac{16434169}{256000000} a^{2} + \frac{2274313}{6400000} a + \frac{103897}{400000}$, $\frac{1}{42949672960000000} a^{14} + \frac{213}{5368709120000000} a^{13} + \frac{67333}{21474836480000000} a^{12} - \frac{179049}{536870912000000} a^{11} - \frac{2950894181}{10737418240000000} a^{10} + \frac{702094731}{1342177280000000} a^{9} + \frac{204325638727}{5368709120000000} a^{8} - \frac{485239993939}{671088640000000} a^{7} - \frac{78226061497}{67108864000000} a^{6} - \frac{19179866241}{10485760000000} a^{5} - \frac{61181382931}{20971520000000} a^{4} + \frac{1210340761}{104857600000} a^{3} + \frac{1025477481}{262144000000} a^{2} - \frac{1275724407}{3276800000} a - \frac{127132351}{819200000}$

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:  $12$
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:  \( 473856133750000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T101:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 133 conjugacy class representatives for [S(5)^3]3=S(5)wr3 are not computed
Character table for [S(5)^3]3=S(5)wr3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ $15$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$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}$
7Data not computed
337Data not computed
53201Data not computed
1103777641Data not computed