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Properties

Label 15.15.3454864189...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{2}\cdot 229^{5}\cdot 56437^{4}\cdot 164142457^{2}$
Root discriminant $7990.28$
Ramified primes $2, 5, 7, 229, 56437, 164142457$
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![14447872000, 75851328000, 98787324800, -37835364800, -101017715040, -14869135600, 25585029672, 7850882532, -677187108, -230491062, 3664210, 1702691, -7884, -2619, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2619*x^13 - 7884*x^12 + 1702691*x^11 + 3664210*x^10 - 230491062*x^9 - 677187108*x^8 + 7850882532*x^7 + 25585029672*x^6 - 14869135600*x^5 - 101017715040*x^4 - 37835364800*x^3 + 98787324800*x^2 + 75851328000*x + 14447872000)
 
gp: K = bnfinit(x^15 - 2619*x^13 - 7884*x^12 + 1702691*x^11 + 3664210*x^10 - 230491062*x^9 - 677187108*x^8 + 7850882532*x^7 + 25585029672*x^6 - 14869135600*x^5 - 101017715040*x^4 - 37835364800*x^3 + 98787324800*x^2 + 75851328000*x + 14447872000, 1)
 

Normalized defining polynomial

\( x^{15} - 2619 x^{13} - 7884 x^{12} + 1702691 x^{11} + 3664210 x^{10} - 230491062 x^{9} - 677187108 x^{8} + 7850882532 x^{7} + 25585029672 x^{6} - 14869135600 x^{5} - 101017715040 x^{4} - 37835364800 x^{3} + 98787324800 x^{2} + 75851328000 x + 14447872000 \)

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:  \(34548641890596001947725147253590025498737248434540544000000=2^{18}\cdot 5^{6}\cdot 7^{2}\cdot 229^{5}\cdot 56437^{4}\cdot 164142457^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $7990.28$
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:  $2, 5, 7, 229, 56437, 164142457$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{10} + \frac{1}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{16} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{160} a^{11} + \frac{1}{160} a^{9} - \frac{1}{40} a^{8} + \frac{31}{160} a^{7} + \frac{1}{16} a^{6} - \frac{1}{80} a^{5} + \frac{3}{40} a^{4} - \frac{17}{40} a^{3} - \frac{1}{20} a^{2}$, $\frac{1}{320} a^{12} + \frac{1}{320} a^{10} - \frac{1}{80} a^{9} + \frac{31}{320} a^{8} + \frac{1}{32} a^{7} + \frac{39}{160} a^{6} + \frac{3}{80} a^{5} - \frac{17}{80} a^{4} - \frac{11}{40} a^{3} - \frac{1}{2} a$, $\frac{1}{3200} a^{13} + \frac{1}{3200} a^{11} - \frac{21}{800} a^{10} + \frac{111}{3200} a^{9} - \frac{7}{320} a^{8} - \frac{321}{1600} a^{7} + \frac{23}{800} a^{6} - \frac{77}{800} a^{5} + \frac{89}{400} a^{4} + \frac{9}{20} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{741279891489036352654232056700376502962708747707304650830233228800} a^{14} + \frac{25905423886861947836243446081753688704892707485425328444138291}{185319972872259088163558014175094125740677186926826162707558307200} a^{13} - \frac{323311198517539024752116942888237509104232040777958091498305979}{741279891489036352654232056700376502962708747707304650830233228800} a^{12} - \frac{24459650048917882168160663507332679789962788984515558907442519}{9265998643612954408177900708754706287033859346341308135377915360} a^{11} + \frac{2477565394721608044891189287075895505401131289225722296301706231}{148255978297807270530846411340075300592541749541460930166046645760} a^{10} - \frac{14074811769519305888069414512289175030171852447775698711298193973}{370639945744518176327116028350188251481354373853652325415116614400} a^{9} + \frac{45352611513032877076385275156109972632324481137854100345344544049}{370639945744518176327116028350188251481354373853652325415116614400} a^{8} - \frac{38869116245198151093445787422837179090419474045029542921680775299}{185319972872259088163558014175094125740677186926826162707558307200} a^{7} + \frac{7088334655163000774125435246904753458277398155374564746024086857}{37063994574451817632711602835018825148135437385365232541511661440} a^{6} + \frac{2621775982403510039622081928160150864130844504769181796298730351}{18531997287225908816355801417509412574067718692682616270755830720} a^{5} + \frac{1189188803571209568368359252667358085764304407484902535996252503}{46329993218064772040889503543773531435169296731706540676889576800} a^{4} + \frac{307372799486417339625299806707184027140465497433892686328169099}{926599864361295440817790070875470628703385934634130813537791536} a^{3} + \frac{174056006371211195737863670705718990248475204059250722645452719}{463299932180647720408895035437735314351692967317065406768895768} a^{2} + \frac{31626544660773286983948879496942185409677363215609287958732571}{231649966090323860204447517718867657175846483658532703384447884} a + \frac{28353623316033888217286145800611823527737949230865748313035930}{57912491522580965051111879429716914293961620914633175846111971}$

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:  \( 15924172782800000000000000 \) (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

3.3.229.1

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 R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ $15$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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}$
2.4.6.6$x^{4} - 20$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.5$x^{4} + 2 x^{2} - 4$$2$$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$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
229Data not computed
56437Data not computed
164142457Data not computed