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Properties

Label 18.4.21579122566...9147.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,3^{27}\cdot 1297^{4}$
Root discriminant $25.55$
Ramified primes $3, 1297$
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![-3833, 17055, -19401, 17622, -13098, 4692, -4292, 942, -1029, 695, -33, 315, 16, 45, -15, -2, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 - 2*x^15 - 15*x^14 + 45*x^13 + 16*x^12 + 315*x^11 - 33*x^10 + 695*x^9 - 1029*x^8 + 942*x^7 - 4292*x^6 + 4692*x^5 - 13098*x^4 + 17622*x^3 - 19401*x^2 + 17055*x - 3833)
 
gp: K = bnfinit(x^18 - 3*x^16 - 2*x^15 - 15*x^14 + 45*x^13 + 16*x^12 + 315*x^11 - 33*x^10 + 695*x^9 - 1029*x^8 + 942*x^7 - 4292*x^6 + 4692*x^5 - 13098*x^4 + 17622*x^3 - 19401*x^2 + 17055*x - 3833, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{16} - 2 x^{15} - 15 x^{14} + 45 x^{13} + 16 x^{12} + 315 x^{11} - 33 x^{10} + 695 x^{9} - 1029 x^{8} + 942 x^{7} - 4292 x^{6} + 4692 x^{5} - 13098 x^{4} + 17622 x^{3} - 19401 x^{2} + 17055 x - 3833 \)

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:  \(-21579122566787439849729147=-\,3^{27}\cdot 1297^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.55$
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:  $3, 1297$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{6181043812174723767649034436350318449} a^{17} - \frac{1027920207493674523324736731437692934}{6181043812174723767649034436350318449} a^{16} - \frac{1503058302737430058428674602140882903}{6181043812174723767649034436350318449} a^{15} + \frac{934281445360377108804655243547802161}{6181043812174723767649034436350318449} a^{14} + \frac{2917558059461748835780293760128372259}{6181043812174723767649034436350318449} a^{13} + \frac{910566445013215842466067360503100141}{6181043812174723767649034436350318449} a^{12} + \frac{2133405188026864538541864875207284067}{6181043812174723767649034436350318449} a^{11} - \frac{2313701053680089391183510278360323429}{6181043812174723767649034436350318449} a^{10} + \frac{253424811724502760114681416620076354}{6181043812174723767649034436350318449} a^{9} + \frac{882431036966111233788427857257533414}{6181043812174723767649034436350318449} a^{8} + \frac{2613798038551554605486777697947627671}{6181043812174723767649034436350318449} a^{7} - \frac{869337747382973491508464120641808699}{6181043812174723767649034436350318449} a^{6} + \frac{2115756090841512418516631781721374221}{6181043812174723767649034436350318449} a^{5} - \frac{71815065518093032205798868187689479}{6181043812174723767649034436350318449} a^{4} - \frac{886684231976362887386372315475864616}{6181043812174723767649034436350318449} a^{3} - \frac{542808493387231949522919882407707893}{6181043812174723767649034436350318449} a^{2} + \frac{1065100899628533530609080223414774661}{6181043812174723767649034436350318449} a - \frac{245946830811683940264060045118535684}{6181043812174723767649034436350318449}$

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:  \( 346991.351785 \) (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

\(\Q(\zeta_{9})^+\), 9.5.689278977.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
1297Data not computed