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Properties

Label 15.7.18866366579...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{27}\cdot 5^{6}\cdot 37^{2}\cdot 163^{5}\cdot 7557157^{2}$
Root discriminant $484.22$
Ramified primes $2, 5, 37, 163, 7557157$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T100

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![131072000, 1146880000, 3859251200, 6367436800, 5506375680, 2453729280, 499850752, 19025280, -8695296, -1130400, 24992, 10504, -432, -162, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 162*x^13 - 432*x^12 + 10504*x^11 + 24992*x^10 - 1130400*x^9 - 8695296*x^8 + 19025280*x^7 + 499850752*x^6 + 2453729280*x^5 + 5506375680*x^4 + 6367436800*x^3 + 3859251200*x^2 + 1146880000*x + 131072000)
 
gp: K = bnfinit(x^15 - 162*x^13 - 432*x^12 + 10504*x^11 + 24992*x^10 - 1130400*x^9 - 8695296*x^8 + 19025280*x^7 + 499850752*x^6 + 2453729280*x^5 + 5506375680*x^4 + 6367436800*x^3 + 3859251200*x^2 + 1146880000*x + 131072000, 1)
 

Normalized defining polynomial

\( x^{15} - 162 x^{13} - 432 x^{12} + 10504 x^{11} + 24992 x^{10} - 1130400 x^{9} - 8695296 x^{8} + 19025280 x^{7} + 499850752 x^{6} + 2453729280 x^{5} + 5506375680 x^{4} + 6367436800 x^{3} + 3859251200 x^{2} + 1146880000 x + 131072000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(18866366579450294354517385665314816000000=2^{27}\cdot 5^{6}\cdot 37^{2}\cdot 163^{5}\cdot 7557157^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $484.22$
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:  $2, 5, 37, 163, 7557157$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{5} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{6} - \frac{1}{16} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{7} - \frac{1}{32} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{256} a^{8} + \frac{1}{128} a^{6} - \frac{1}{16} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{512} a^{9} + \frac{1}{256} a^{7} - \frac{1}{32} a^{5} - \frac{1}{16} a^{3}$, $\frac{1}{2048} a^{10} - \frac{1}{1024} a^{9} + \frac{1}{1024} a^{8} - \frac{1}{512} a^{7} - \frac{1}{128} a^{6} + \frac{1}{64} a^{5} + \frac{3}{64} a^{4} + \frac{1}{32} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{40960} a^{11} - \frac{1}{4096} a^{10} - \frac{11}{20480} a^{9} - \frac{13}{10240} a^{8} + \frac{1}{640} a^{7} - \frac{9}{1280} a^{6} - \frac{5}{256} a^{5} + \frac{1}{640} a^{4} + \frac{1}{80} a^{2} + \frac{1}{4} a$, $\frac{1}{81920} a^{12} - \frac{1}{40960} a^{10} - \frac{1}{2560} a^{9} + \frac{13}{10240} a^{8} - \frac{7}{1280} a^{7} + \frac{1}{512} a^{6} + \frac{1}{80} a^{5} - \frac{5}{128} a^{4} + \frac{3}{80} a^{3} - \frac{1}{16} a^{2}$, $\frac{1}{1638400} a^{13} - \frac{1}{819200} a^{11} - \frac{1}{51200} a^{10} - \frac{27}{204800} a^{9} - \frac{27}{25600} a^{8} - \frac{51}{10240} a^{7} - \frac{23}{3200} a^{6} + \frac{27}{2560} a^{5} + \frac{83}{1600} a^{4} + \frac{29}{320} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{3300254284634640503708427878400} a^{14} + \frac{338083296622638805510273}{1650127142317320251854213939200} a^{13} - \frac{1450762753487646925964761}{1650127142317320251854213939200} a^{12} + \frac{4689095195825254973416439}{825063571158660125927106969600} a^{11} + \frac{2539627496159944302960769}{412531785579330062963553484800} a^{10} + \frac{869759715498961291936247}{1127136026173032958916812800} a^{9} - \frac{33733806337829967090119353}{34377648798277505246962790400} a^{8} + \frac{51810365445256958106853607}{17188824399138752623481395200} a^{7} + \frac{108482671797565941654786041}{8594412199569376311740697600} a^{6} - \frac{261470301303498674403152693}{12891618299354064467611046400} a^{5} - \frac{53114243807886572320192057}{3222904574838516116902761600} a^{4} - \frac{16416812178105364296591389}{322290457483851611690276160} a^{3} - \frac{220958268958290491792993}{1678596132728393810886855} a^{2} + \frac{125653196109851562622967}{2014315359274072573064226} a - \frac{299990162977805952288344}{1007157679637036286532113}$

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:  $10$
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:  \( 51017956695100000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T100:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 79 conjugacy class representatives for [1/2.S(5)^3]S(3) are not computed
Character table for [1/2.S(5)^3]S(3) is not computed

Intermediate fields

3.3.1304.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R $15$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ $15$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.4.10.6$x^{4} + 6 x^{2} + 3$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.11.11$x^{4} + 10$$4$$1$$11$$D_{4}$$[2, 3, 4]$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.5.0.1$x^{5} - x + 13$$1$$5$$0$$C_5$$[\ ]^{5}$
$163$$\Q_{163}$$x + 4$$1$$1$$0$Trivial$[\ ]$
163.2.1.2$x^{2} + 652$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
163.2.1.2$x^{2} + 652$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.1.2$x^{2} + 652$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
163.4.2.1$x^{4} + 3423 x^{2} + 3214849$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7557157Data not computed