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Properties

Label 18.8.11480578918...8688.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,2^{6}\cdot 3^{33}\cdot 19^{9}$
Root discriminant $41.16$
Ramified primes $2, 3, 19$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T585

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![127, 399, -456, -2187, 57, 3612, 210, -1986, 993, 358, -597, 15, 9, 42, -30, 6, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 + 6*x^15 - 30*x^14 + 42*x^13 + 9*x^12 + 15*x^11 - 597*x^10 + 358*x^9 + 993*x^8 - 1986*x^7 + 210*x^6 + 3612*x^5 + 57*x^4 - 2187*x^3 - 456*x^2 + 399*x + 127)
 
gp: K = bnfinit(x^18 - 3*x^17 + 6*x^16 + 6*x^15 - 30*x^14 + 42*x^13 + 9*x^12 + 15*x^11 - 597*x^10 + 358*x^9 + 993*x^8 - 1986*x^7 + 210*x^6 + 3612*x^5 + 57*x^4 - 2187*x^3 - 456*x^2 + 399*x + 127, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 6 x^{16} + 6 x^{15} - 30 x^{14} + 42 x^{13} + 9 x^{12} + 15 x^{11} - 597 x^{10} + 358 x^{9} + 993 x^{8} - 1986 x^{7} + 210 x^{6} + 3612 x^{5} + 57 x^{4} - 2187 x^{3} - 456 x^{2} + 399 x + 127 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-114805789186292807734226138688=-\,2^{6}\cdot 3^{33}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.16$
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:  $2, 3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{5}{12} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{12}$, $\frac{1}{36} a^{14} + \frac{1}{36} a^{13} - \frac{1}{18} a^{12} + \frac{1}{18} a^{11} - \frac{1}{36} a^{10} - \frac{1}{36} a^{9} + \frac{5}{12} a^{8} - \frac{1}{12} a^{7} + \frac{1}{6} a^{6} + \frac{17}{36} a^{5} + \frac{2}{9} a^{4} + \frac{11}{36} a^{3} - \frac{2}{9} a^{2} + \frac{13}{36} a + \frac{13}{36}$, $\frac{1}{72} a^{15} - \frac{1}{24} a^{13} + \frac{1}{18} a^{12} - \frac{1}{24} a^{11} - \frac{1}{6} a^{10} + \frac{1}{18} a^{9} + \frac{1}{4} a^{8} - \frac{3}{8} a^{7} + \frac{11}{72} a^{6} - \frac{1}{8} a^{5} - \frac{11}{24} a^{4} - \frac{19}{72} a^{3} - \frac{5}{24} a^{2} - \frac{1}{3} a + \frac{35}{72}$, $\frac{1}{144} a^{16} - \frac{1}{144} a^{15} - \frac{1}{144} a^{14} - \frac{1}{48} a^{13} + \frac{1}{144} a^{12} + \frac{7}{144} a^{11} + \frac{1}{72} a^{10} - \frac{1}{12} a^{9} + \frac{19}{48} a^{8} - \frac{1}{36} a^{7} - \frac{11}{36} a^{6} - \frac{13}{72} a^{5} + \frac{1}{24} a^{4} - \frac{29}{72} a^{3} - \frac{37}{144} a^{2} + \frac{25}{144} a - \frac{19}{48}$, $\frac{1}{3961593753276274848} a^{17} - \frac{3224472077446469}{1980796876638137424} a^{16} - \frac{694931205378620}{123799804789883589} a^{15} + \frac{8964831923982497}{1980796876638137424} a^{14} + \frac{548651015007467}{247599609579767178} a^{13} + \frac{121379374798297447}{1980796876638137424} a^{12} - \frac{84444588904085909}{3961593753276274848} a^{11} - \frac{122055292992671753}{1980796876638137424} a^{10} + \frac{7934191335877361}{3961593753276274848} a^{9} - \frac{589164717658546321}{3961593753276274848} a^{8} - \frac{280805584339743857}{990398438319068712} a^{7} + \frac{150593077384359821}{1980796876638137424} a^{6} + \frac{354450035264338379}{990398438319068712} a^{5} - \frac{100203467374208429}{247599609579767178} a^{4} - \frac{863759131632318863}{3961593753276274848} a^{3} + \frac{369349708195284787}{1980796876638137424} a^{2} - \frac{471835552881038923}{1980796876638137424} a - \frac{1784218768253568323}{3961593753276274848}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 52507779.0436 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T585:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 13824
The 96 conjugacy class representatives for t18n585 are not computed
Character table for t18n585 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.5609891727441.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
3Data not computed
$19$19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.6.5.3$x^{6} - 4864$$6$$1$$5$$C_6$$[\ ]_{6}$