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Properties

Label 18.4.41599945090...9611.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,3^{27}\cdot 73^{4}\cdot 577^{3}$
Root discriminant $38.90$
Ramified primes $3, 73, 577$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 18T767

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6731, 12846, -16191, 21989, -13620, 8928, -3535, -2298, -1431, 876, -3612, 1836, -1326, 240, -54, -2, 15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 15*x^16 - 2*x^15 - 54*x^14 + 240*x^13 - 1326*x^12 + 1836*x^11 - 3612*x^10 + 876*x^9 - 1431*x^8 - 2298*x^7 - 3535*x^6 + 8928*x^5 - 13620*x^4 + 21989*x^3 - 16191*x^2 + 12846*x - 6731)
 
gp: K = bnfinit(x^18 + 15*x^16 - 2*x^15 - 54*x^14 + 240*x^13 - 1326*x^12 + 1836*x^11 - 3612*x^10 + 876*x^9 - 1431*x^8 - 2298*x^7 - 3535*x^6 + 8928*x^5 - 13620*x^4 + 21989*x^3 - 16191*x^2 + 12846*x - 6731, 1)
 

Normalized defining polynomial

\( x^{18} + 15 x^{16} - 2 x^{15} - 54 x^{14} + 240 x^{13} - 1326 x^{12} + 1836 x^{11} - 3612 x^{10} + 876 x^{9} - 1431 x^{8} - 2298 x^{7} - 3535 x^{6} + 8928 x^{5} - 13620 x^{4} + 21989 x^{3} - 16191 x^{2} + 12846 x - 6731 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-41599945090131789253856059611=-\,3^{27}\cdot 73^{4}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.90$
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:  $3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{127473410483478202985376023516086076815009} a^{17} - \frac{33971199183137538274969424419286535550962}{127473410483478202985376023516086076815009} a^{16} - \frac{10575194463389435485064811552656832269104}{127473410483478202985376023516086076815009} a^{15} - \frac{56830196974741934443798926040709234938458}{127473410483478202985376023516086076815009} a^{14} + \frac{22192908796499400079426272241104545436479}{127473410483478202985376023516086076815009} a^{13} - \frac{31654932359684304192852013537676611061526}{127473410483478202985376023516086076815009} a^{12} + \frac{27642545857370836494983145091903516851405}{127473410483478202985376023516086076815009} a^{11} + \frac{6495425716438985476817432841526221490039}{127473410483478202985376023516086076815009} a^{10} + \frac{35378714297544785229772687377768969688023}{127473410483478202985376023516086076815009} a^{9} - \frac{7712459484694843642749291937819543270958}{127473410483478202985376023516086076815009} a^{8} + \frac{5592382056781750089665911698789876933467}{127473410483478202985376023516086076815009} a^{7} - \frac{45024144040812852629888568055325861589085}{127473410483478202985376023516086076815009} a^{6} - \frac{44246812236923445206525953567599062638089}{127473410483478202985376023516086076815009} a^{5} + \frac{35987519967990797674214414207007369485252}{127473410483478202985376023516086076815009} a^{4} - \frac{3135243519206366298998365808793316671250}{127473410483478202985376023516086076815009} a^{3} - \frac{31534543819454845271653369788811868451362}{127473410483478202985376023516086076815009} a^{2} + \frac{9856002733275416585104901451711176463828}{127473410483478202985376023516086076815009} a + \frac{52356031392981872604680727638411765587539}{127473410483478202985376023516086076815009}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 3339413.71798 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T767:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n767 are not computed
Character table for t18n767 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: 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 $18$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
73Data not computed
577Data not computed