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Properties

Label 18.6.43006206619...0352.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{24}\cdot 3^{9}\cdot 7^{12}\cdot 97^{2}$
Root discriminant $26.55$
Ramified primes $2, 3, 7, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T366

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8168, -25040, -29336, 4216, 37392, 24128, -16084, -17480, 4708, 5596, -348, -936, -328, 76, 120, -2, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^16 - 2*x^15 + 120*x^14 + 76*x^13 - 328*x^12 - 936*x^11 - 348*x^10 + 5596*x^9 + 4708*x^8 - 17480*x^7 - 16084*x^6 + 24128*x^5 + 37392*x^4 + 4216*x^3 - 29336*x^2 - 25040*x - 8168)
 
gp: K = bnfinit(x^18 - 18*x^16 - 2*x^15 + 120*x^14 + 76*x^13 - 328*x^12 - 936*x^11 - 348*x^10 + 5596*x^9 + 4708*x^8 - 17480*x^7 - 16084*x^6 + 24128*x^5 + 37392*x^4 + 4216*x^3 - 29336*x^2 - 25040*x - 8168, 1)
 

Normalized defining polynomial

\( x^{18} - 18 x^{16} - 2 x^{15} + 120 x^{14} + 76 x^{13} - 328 x^{12} - 936 x^{11} - 348 x^{10} + 5596 x^{9} + 4708 x^{8} - 17480 x^{7} - 16084 x^{6} + 24128 x^{5} + 37392 x^{4} + 4216 x^{3} - 29336 x^{2} - 25040 x - 8168 \)

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:  \(43006206619464857762660352=2^{24}\cdot 3^{9}\cdot 7^{12}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.55$
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:  $2, 3, 7, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{13} + \frac{1}{12} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{15} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{36} a^{16} + \frac{1}{36} a^{15} - \frac{1}{12} a^{13} - \frac{1}{18} a^{11} - \frac{1}{6} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{18} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} + \frac{1}{9} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{566003523020423541117260398121844} a^{17} - \frac{500064394009245608616216750563}{188667841006807847039086799373948} a^{16} + \frac{19736553948917727835721433955181}{566003523020423541117260398121844} a^{15} + \frac{3881644069344798477927954810583}{188667841006807847039086799373948} a^{14} - \frac{10241328403007306374094679105515}{94333920503403923519543399686974} a^{13} + \frac{34115510449400682021288863305855}{566003523020423541117260398121844} a^{12} + \frac{42664505205841237029322878701185}{283001761510211770558630199060922} a^{11} - \frac{22594776466536006605584766942233}{141500880755105885279315099530461} a^{10} + \frac{2852116986772863148183344419230}{47166960251701961759771699843487} a^{9} + \frac{2888821864412958013697826934093}{283001761510211770558630199060922} a^{8} + \frac{20317400149581962849517593583209}{141500880755105885279315099530461} a^{7} - \frac{5370177743198403726986926345616}{47166960251701961759771699843487} a^{6} + \frac{32929550543254918444823210692238}{141500880755105885279315099530461} a^{5} - \frac{67493341768965700686754070004613}{141500880755105885279315099530461} a^{4} - \frac{36309470113692919055869124971706}{141500880755105885279315099530461} a^{3} - \frac{13097301704858326694250921677059}{47166960251701961759771699843487} a^{2} - \frac{27130141046581107121364085821839}{141500880755105885279315099530461} a - \frac{43107639407051459512178979358072}{141500880755105885279315099530461}$

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:  \( 583097.776439 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T366:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 48 conjugacy class representatives for t18n366
Character table for t18n366 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.1.588.1, 9.3.203297472.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
2Data not computed
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed
97Data not computed