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Properties

Label 18.4.29863270848...7519.2
Degree $18$
Signature $[4, 7]$
Discriminant $-\,7^{15}\cdot 41^{3}\cdot 97^{3}$
Root discriminant $20.15$
Ramified primes $7, 41, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T879

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 12, 39, 82, 90, 68, -57, -30, -120, 38, -88, 86, -7, 32, -10, -8, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^16 - 8*x^15 - 10*x^14 + 32*x^13 - 7*x^12 + 86*x^11 - 88*x^10 + 38*x^9 - 120*x^8 - 30*x^7 - 57*x^6 + 68*x^5 + 90*x^4 + 82*x^3 + 39*x^2 + 12*x + 1)
 
gp: K = bnfinit(x^18 - 2*x^16 - 8*x^15 - 10*x^14 + 32*x^13 - 7*x^12 + 86*x^11 - 88*x^10 + 38*x^9 - 120*x^8 - 30*x^7 - 57*x^6 + 68*x^5 + 90*x^4 + 82*x^3 + 39*x^2 + 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{16} - 8 x^{15} - 10 x^{14} + 32 x^{13} - 7 x^{12} + 86 x^{11} - 88 x^{10} + 38 x^{9} - 120 x^{8} - 30 x^{7} - 57 x^{6} + 68 x^{5} + 90 x^{4} + 82 x^{3} + 39 x^{2} + 12 x + 1 \)

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:  \(-298632708486259154687519=-\,7^{15}\cdot 41^{3}\cdot 97^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.15$
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:  $7, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{13} + \frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{18} a^{16} + \frac{1}{18} a^{15} + \frac{1}{9} a^{13} + \frac{1}{18} a^{12} - \frac{1}{18} a^{11} + \frac{1}{9} a^{10} - \frac{1}{6} a^{9} - \frac{4}{9} a^{8} - \frac{1}{2} a^{7} + \frac{7}{18} a^{6} + \frac{2}{9} a^{5} - \frac{1}{18} a^{4} - \frac{1}{18} a^{3} + \frac{7}{18} a^{2} - \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{18950849838} a^{17} + \frac{476714311}{18950849838} a^{16} + \frac{221294354}{3158474973} a^{15} - \frac{221916925}{18950849838} a^{14} + \frac{3773989621}{18950849838} a^{13} + \frac{1115348671}{9475424919} a^{12} + \frac{1222256765}{18950849838} a^{11} - \frac{142103935}{1052824991} a^{10} + \frac{4285447477}{18950849838} a^{9} + \frac{331752578}{1052824991} a^{8} + \frac{4038543020}{9475424919} a^{7} + \frac{2900962433}{9475424919} a^{6} - \frac{613051901}{9475424919} a^{5} - \frac{4443602749}{18950849838} a^{4} - \frac{3700350835}{9475424919} a^{3} + \frac{1702692763}{18950849838} a^{2} + \frac{6926149301}{18950849838} a - \frac{787811927}{3158474973}$

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:  \( 31423.1321737 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T879:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 331776
The 360 conjugacy class representatives for t18n879 are not computed
Character table for t18n879 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.5.467890073.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ $18$ $18$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
7Data not computed
41Data not computed
97Data not computed