↑

Properties

Label 15.11.4322632523...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{16}\cdot 5^{6}\cdot 37^{5}\cdot 67\cdot 79^{2}\cdot 3557^{4}\cdot 953647^{2}$
Root discriminant $1747.36$
Ramified primes $2, 5, 37, 67, 79, 3557, 953647$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T102

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28456000, 99596000, -41261200, -366726700, -113966280, 311384340, 214275644, 5458056, -21874464, -2905263, 467900, 72493, -2448, -549, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 549*x^13 - 2448*x^12 + 72493*x^11 + 467900*x^10 - 2905263*x^9 - 21874464*x^8 + 5458056*x^7 + 214275644*x^6 + 311384340*x^5 - 113966280*x^4 - 366726700*x^3 - 41261200*x^2 + 99596000*x + 28456000)
 
gp: K = bnfinit(x^15 - 549*x^13 - 2448*x^12 + 72493*x^11 + 467900*x^10 - 2905263*x^9 - 21874464*x^8 + 5458056*x^7 + 214275644*x^6 + 311384340*x^5 - 113966280*x^4 - 366726700*x^3 - 41261200*x^2 + 99596000*x + 28456000, 1)
 

Normalized defining polynomial

\( x^{15} - 549 x^{13} - 2448 x^{12} + 72493 x^{11} + 467900 x^{10} - 2905263 x^{9} - 21874464 x^{8} + 5458056 x^{7} + 214275644 x^{6} + 311384340 x^{5} - 113966280 x^{4} - 366726700 x^{3} - 41261200 x^{2} + 99596000 x + 28456000 \)

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:  \(4322632523817045420539159292758206089739264000000=2^{16}\cdot 5^{6}\cdot 37^{5}\cdot 67\cdot 79^{2}\cdot 3557^{4}\cdot 953647^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1747.36$
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:  $2, 5, 37, 67, 79, 3557, 953647$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{40} a^{11} - \frac{9}{40} a^{9} + \frac{1}{20} a^{8} - \frac{7}{40} a^{7} + \frac{1}{4} a^{6} + \frac{17}{40} a^{5} - \frac{7}{20} a^{4} + \frac{2}{5} a^{3} + \frac{7}{20} a^{2}$, $\frac{1}{40} a^{12} + \frac{1}{40} a^{10} + \frac{1}{20} a^{9} - \frac{17}{40} a^{8} - \frac{1}{4} a^{7} - \frac{13}{40} a^{6} + \frac{3}{20} a^{5} - \frac{7}{20} a^{4} - \frac{3}{20} a^{3} - \frac{1}{2} a$, $\frac{1}{200} a^{13} + \frac{1}{200} a^{11} + \frac{1}{100} a^{10} + \frac{43}{200} a^{9} - \frac{1}{4} a^{8} + \frac{87}{200} a^{7} + \frac{43}{100} a^{6} + \frac{3}{100} a^{5} - \frac{3}{100} a^{4} - \frac{3}{10} a^{3} + \frac{1}{10} a^{2}$, $\frac{1}{53857531225910662279772270643329797833982471819833200} a^{14} - \frac{1219448947550380714182454037472545855254896119917}{13464382806477665569943067660832449458495617954958300} a^{13} - \frac{643856949532071686959446631519340698966710508885679}{53857531225910662279772270643329797833982471819833200} a^{12} + \frac{134746073557216305293082127996021360017246969773751}{13464382806477665569943067660832449458495617954958300} a^{11} + \frac{5012516397553436594375321990852019819496820415231827}{53857531225910662279772270643329797833982471819833200} a^{10} - \frac{1254781658064382574037422067244670359924311206192741}{13464382806477665569943067660832449458495617954958300} a^{9} - \frac{1316934174436633117798538980977410398033206322608213}{53857531225910662279772270643329797833982471819833200} a^{8} - \frac{430909139314804673061108804108421737512630048361413}{1346438280647766556994306766083244945849561795495830} a^{7} + \frac{3615497431395007192473499472174561995410761965661749}{26928765612955331139886135321664898916991235909916600} a^{6} + \frac{1325279470038495552847848039744933234654855538504387}{6732191403238832784971533830416224729247808977479150} a^{5} + \frac{333929294166835742130573198814582675318851074753743}{3366095701619416392485766915208112364623904488739575} a^{4} - \frac{22589313178072998658331005810952484381336881170859}{538575312259106622797722706433297978339824718198332} a^{3} + \frac{420917687225058540690250785654672044790416530757661}{2692876561295533113988613532166489891699123590991660} a^{2} - \frac{91608505608530260670717414272023802655962271429043}{269287656129553311398861353216648989169912359099166} a + \frac{21706050349718517181265326463336297619923555794239}{134643828064776655699430676608324494584956179549583}$

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:  \( 395352970925000000000 \) (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

3.3.148.1

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 $15$ $15$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.10.7$x^{6} + 2 x^{5} + 4 x^{3} + 2$$6$$1$$10$$S_4\times C_2$$[2, 8/3, 8/3]_{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$$[\ ]^{2}$
5.2.1.1$x^{2} - 5$$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}$
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$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.3.0.1$x^{3} - x + 16$$1$$3$$0$$C_3$$[\ ]^{3}$
67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
79Data not computed
3557Data not computed
953647Data not computed