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Properties

Label 15.7.69334557885...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{16}\cdot 5^{5}\cdot 37^{5}\cdot 67^{2}\cdot 167^{4}\cdot 1182491^{2}$
Root discriminant $528.12$
Ramified primes $2, 5, 37, 67, 167, 1182491$
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![-1336000, 4676000, 1937200, -17217700, 5350680, 14540540, -10195764, 375236, 1097784, -177453, -35100, 7403, -72, -189, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 189*x^13 - 72*x^12 + 7403*x^11 - 35100*x^10 - 177453*x^9 + 1097784*x^8 + 375236*x^7 - 10195764*x^6 + 14540540*x^5 + 5350680*x^4 - 17217700*x^3 + 1937200*x^2 + 4676000*x - 1336000)
 
gp: K = bnfinit(x^15 - 189*x^13 - 72*x^12 + 7403*x^11 - 35100*x^10 - 177453*x^9 + 1097784*x^8 + 375236*x^7 - 10195764*x^6 + 14540540*x^5 + 5350680*x^4 - 17217700*x^3 + 1937200*x^2 + 4676000*x - 1336000, 1)
 

Normalized defining polynomial

\( x^{15} - 189 x^{13} - 72 x^{12} + 7403 x^{11} - 35100 x^{10} - 177453 x^{9} + 1097784 x^{8} + 375236 x^{7} - 10195764 x^{6} + 14540540 x^{5} + 5350680 x^{4} - 17217700 x^{3} + 1937200 x^{2} + 4676000 x - 1336000 \)

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:  \(69334557885208174528044608269101670400000=2^{16}\cdot 5^{5}\cdot 37^{5}\cdot 67^{2}\cdot 167^{4}\cdot 1182491^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $528.12$
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:  $2, 5, 37, 67, 167, 1182491$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{10} - \frac{1}{10} a^{9} - \frac{1}{4} a^{8} + \frac{2}{5} a^{7} + \frac{7}{20} a^{6} - \frac{1}{5} a^{5} - \frac{1}{4} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{40} a^{11} - \frac{9}{40} a^{9} - \frac{1}{20} a^{8} + \frac{3}{40} a^{7} - \frac{1}{4} a^{6} + \frac{7}{40} a^{5} - \frac{3}{20} a^{4} + \frac{2}{5} a^{3} + \frac{3}{20} a^{2} - \frac{1}{2} a$, $\frac{1}{40} a^{12} - \frac{1}{40} a^{10} + \frac{1}{20} a^{9} + \frac{3}{40} a^{8} - \frac{3}{20} a^{7} - \frac{17}{40} a^{6} - \frac{9}{20} a^{5} + \frac{2}{5} a^{4} + \frac{9}{20} a^{3} + \frac{1}{10} a^{2}$, $\frac{1}{200} a^{13} + \frac{1}{200} a^{11} - \frac{1}{100} a^{10} - \frac{47}{200} a^{9} - \frac{3}{20} a^{8} + \frac{77}{200} a^{7} - \frac{13}{100} a^{6} + \frac{43}{100} a^{5} - \frac{27}{100} a^{4} + \frac{3}{10} a^{3} - \frac{1}{10} a^{2}$, $\frac{1}{2630340581387841221853132038187999891305200} a^{14} + \frac{2124745158767420362064549079630790170873}{1315170290693920610926566019093999945652600} a^{13} + \frac{8823477342785745632098115719934747777201}{2630340581387841221853132038187999891305200} a^{12} + \frac{1554688699208529551763591395275631344341}{657585145346960305463283009546999972826300} a^{11} - \frac{2936194282134497740598238713286080862819}{2630340581387841221853132038187999891305200} a^{10} + \frac{16030333937941699079764594508707548206311}{328792572673480152731641504773499986413150} a^{9} - \frac{924683074094609505241364767864161426418343}{2630340581387841221853132038187999891305200} a^{8} - \frac{128377423487663216045536852768876410740491}{657585145346960305463283009546999972826300} a^{7} - \frac{21448498146251393984807604212577959636287}{263034058138784122185313203818799989130520} a^{6} - \frac{175090036477675299058125681677786796153219}{1315170290693920610926566019093999945652600} a^{5} + \frac{62669478777951852938474338861950651818516}{164396286336740076365820752386749993206575} a^{4} + \frac{25169352462621407232567523368059374044577}{131517029069392061092656601909399994565260} a^{3} + \frac{15862136458403400871772043310276073041489}{65758514534696030546328300954699997282630} a^{2} + \frac{61598943171546178300095584306246260417}{6575851453469603054632830095469999728263} a + \frac{1704153196570747643149215591450372417073}{6575851453469603054632830095469999728263}$

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:  \( 19657974406200000 \) (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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$$[\ ]_{3}^{2}$
2.6.10.3$x^{6} + 2 x^{5} + 4 x + 2$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$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$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$67$$\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$Trivial$[\ ]$
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}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167Data not computed
1182491Data not computed