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Properties

Label 18.6.82105965698...7297.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{24}\cdot 73^{6}\cdot 577^{3}$
Root discriminant $52.18$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T767

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2539377, 10095192, -9791766, 6986556, -4760613, 875178, -38649, -292068, 248619, -61148, 25806, 3558, -2765, 1071, -405, 2, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 + 2*x^15 - 405*x^14 + 1071*x^13 - 2765*x^12 + 3558*x^11 + 25806*x^10 - 61148*x^9 + 248619*x^8 - 292068*x^7 - 38649*x^6 + 875178*x^5 - 4760613*x^4 + 6986556*x^3 - 9791766*x^2 + 10095192*x + 2539377)
 
gp: K = bnfinit(x^18 - 3*x^17 + 6*x^16 + 2*x^15 - 405*x^14 + 1071*x^13 - 2765*x^12 + 3558*x^11 + 25806*x^10 - 61148*x^9 + 248619*x^8 - 292068*x^7 - 38649*x^6 + 875178*x^5 - 4760613*x^4 + 6986556*x^3 - 9791766*x^2 + 10095192*x + 2539377, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 6 x^{16} + 2 x^{15} - 405 x^{14} + 1071 x^{13} - 2765 x^{12} + 3558 x^{11} + 25806 x^{10} - 61148 x^{9} + 248619 x^{8} - 292068 x^{7} - 38649 x^{6} + 875178 x^{5} - 4760613 x^{4} + 6986556 x^{3} - 9791766 x^{2} + 10095192 x + 2539377 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8210596569826381664214775617297=3^{24}\cdot 73^{6}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.18$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{4}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{13} - \frac{1}{3} a^{11} + \frac{1}{9} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{4}{9} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{111226658194964749831438456108617173921023306725271255929} a^{17} + \frac{3511524005841933138766237889785992147339448663147432019}{111226658194964749831438456108617173921023306725271255929} a^{16} - \frac{3990912657905763139608028894475093928814775840563877338}{111226658194964749831438456108617173921023306725271255929} a^{15} + \frac{16078940916520522246834766471725349994094197041860840871}{111226658194964749831438456108617173921023306725271255929} a^{14} + \frac{10614269988175925537105198522274167626202927283337811704}{111226658194964749831438456108617173921023306725271255929} a^{13} + \frac{7262294441642304706569513289184738682358815568180194118}{111226658194964749831438456108617173921023306725271255929} a^{12} - \frac{15559885612881546240391807951374148807548792911856132773}{111226658194964749831438456108617173921023306725271255929} a^{11} - \frac{40901445228394235991848393246293660499907743640628584622}{111226658194964749831438456108617173921023306725271255929} a^{10} + \frac{27238483675345859595736950733016509756701027150528740222}{111226658194964749831438456108617173921023306725271255929} a^{9} - \frac{52174283651987450580870432621907444731528529691186506985}{111226658194964749831438456108617173921023306725271255929} a^{8} + \frac{20867671517903919773826256794888417847575304366940173046}{111226658194964749831438456108617173921023306725271255929} a^{7} + \frac{12268922866652930439038896192537174830653928841919228456}{111226658194964749831438456108617173921023306725271255929} a^{6} + \frac{8584728358230905830740377192427704390082102189236847203}{37075552731654916610479485369539057973674435575090418643} a^{5} + \frac{11663613850790212701494316817114727807443988902107238609}{37075552731654916610479485369539057973674435575090418643} a^{4} - \frac{15052215839391162823551342174692961397120175140358761472}{37075552731654916610479485369539057973674435575090418643} a^{3} + \frac{3803170537133805137700187866299081529294933233939233614}{12358517577218305536826495123179685991224811858363472881} a^{2} + \frac{1675634763243363282706054774684655262444226452072578474}{12358517577218305536826495123179685991224811858363472881} a - \frac{2029857126398359358836576331167243182854000763851119553}{12358517577218305536826495123179685991224811858363472881}$

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:  $11$
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:  \( 226739290.669 \) (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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R $18$ $18$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
73Data not computed
577Data not computed