↑

Properties

Label 15.7.69533750435...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 29^{4}\cdot 921789289^{2}$
Root discriminant $615.86$
Ramified primes $2, 5, 7, 29, 921789289$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T92

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![118784000, 415744000, 277657600, -302528000, -267264000, 29354240, 60653056, 10372064, -624384, -62424, 22688, 6684, -144, -180, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 180*x^13 - 144*x^12 + 6684*x^11 + 22688*x^10 - 62424*x^9 - 624384*x^8 + 10372064*x^7 + 60653056*x^6 + 29354240*x^5 - 267264000*x^4 - 302528000*x^3 + 277657600*x^2 + 415744000*x + 118784000)
 
gp: K = bnfinit(x^15 - 180*x^13 - 144*x^12 + 6684*x^11 + 22688*x^10 - 62424*x^9 - 624384*x^8 + 10372064*x^7 + 60653056*x^6 + 29354240*x^5 - 267264000*x^4 - 302528000*x^3 + 277657600*x^2 + 415744000*x + 118784000, 1)
 

Normalized defining polynomial

\( x^{15} - 180 x^{13} - 144 x^{12} + 6684 x^{11} + 22688 x^{10} - 62424 x^{9} - 624384 x^{8} + 10372064 x^{7} + 60653056 x^{6} + 29354240 x^{5} - 267264000 x^{4} - 302528000 x^{3} + 277657600 x^{2} + 415744000 x + 118784000 \)

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:  \(695337504354450262881345263439253504000000=2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 29^{4}\cdot 921789289^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $615.86$
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:  $2, 5, 7, 29, 921789289$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{16} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{8} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{9} - \frac{1}{16} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{128} a^{10} - \frac{1}{32} a^{6} - \frac{1}{16} a^{4}$, $\frac{1}{1280} a^{11} + \frac{1}{80} a^{8} - \frac{9}{320} a^{7} - \frac{1}{40} a^{6} - \frac{13}{160} a^{5} - \frac{1}{20} a^{4} + \frac{1}{20} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{2560} a^{12} + \frac{1}{160} a^{9} - \frac{9}{640} a^{8} - \frac{1}{80} a^{7} - \frac{13}{320} a^{6} + \frac{1}{10} a^{5} + \frac{1}{40} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{51200} a^{13} - \frac{1}{2560} a^{11} - \frac{9}{3200} a^{10} + \frac{71}{12800} a^{9} - \frac{11}{1600} a^{8} - \frac{123}{6400} a^{7} - \frac{9}{200} a^{6} + \frac{7}{1600} a^{5} - \frac{7}{100} a^{4} - \frac{9}{40} a^{3} - \frac{1}{5} a^{2} - \frac{1}{4} a$, $\frac{1}{11347371640771402354356993611304929484800} a^{14} + \frac{3667028577781902866344970629239941}{1418421455096425294294624201413116185600} a^{13} + \frac{109783568022751887501098354771461723}{567368582038570117717849680565246474240} a^{12} + \frac{146432733449984400559653769269443261}{709210727548212647147312100706558092800} a^{11} + \frac{3575606172110054581996130571764841063}{2836842910192850588589248402826232371200} a^{10} - \frac{121945910537171801993036596277365173}{17730268188705316178682802517663952320} a^{9} + \frac{2346932921651393531556839525167985689}{283684291019285058858924840282623237120} a^{8} - \frac{2738288793683172498252251778775106139}{177302681887053161786828025176639523200} a^{7} + \frac{4972773362312692639815031034146438651}{354605363774106323573656050353279046400} a^{6} + \frac{4807830589811361871950412700644239583}{44325670471763290446707006294159880800} a^{5} - \frac{2477744836065394967955843872794420787}{44325670471763290446707006294159880800} a^{4} + \frac{36198578944498839519134155647483853}{443256704717632904467070062941598808} a^{3} - \frac{520328492602093697418141614230070239}{4432567047176329044670700629415988080} a^{2} + \frac{7904794823889552797303791068930587}{221628352358816452233535031470799404} a + \frac{14529490867303563589578901805453083}{55407088089704113058383757867699851}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 658851832895000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T92:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 648000
The 55 conjugacy class representatives for [A(5)^3]3=A(5)wr3 are not computed
Character table for [A(5)^3]3=A(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 sibling: data not computed
Degree 36 sibling: 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 ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ $15$ $15$ $15$ R ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ $15$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.2$x^{6} + 4 x^{2} - 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.9.2$x^{6} + 4 x^{2} - 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
29.5.4.1$x^{5} - 29$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
29.5.0.1$x^{5} - x + 11$$1$$5$$0$$C_5$$[\ ]^{5}$
921789289Data not computed