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Properties

Label 18.12.1052266677...5647.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,7^{15}\cdot 53^{6}$
Root discriminant $19.01$
Ramified primes $7, 53$
Class number $1$
Class group Trivial
Galois group $C_6\times S_4$ (as 18T61)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -37, 62, 21, -93, 61, -99, 153, -131, 56, 80, -132, 82, -25, -15, 21, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 21*x^16 - 15*x^15 - 25*x^14 + 82*x^13 - 132*x^12 + 80*x^11 + 56*x^10 - 131*x^9 + 153*x^8 - 99*x^7 + 61*x^6 - 93*x^5 + 21*x^4 + 62*x^3 - 37*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^18 - 8*x^17 + 21*x^16 - 15*x^15 - 25*x^14 + 82*x^13 - 132*x^12 + 80*x^11 + 56*x^10 - 131*x^9 + 153*x^8 - 99*x^7 + 61*x^6 - 93*x^5 + 21*x^4 + 62*x^3 - 37*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 21 x^{16} - 15 x^{15} - 25 x^{14} + 82 x^{13} - 132 x^{12} + 80 x^{11} + 56 x^{10} - 131 x^{9} + 153 x^{8} - 99 x^{7} + 61 x^{6} - 93 x^{5} + 21 x^{4} + 62 x^{3} - 37 x^{2} + 3 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-105226667788517176205647=-\,7^{15}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.01$
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:  $7, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{1067} a^{16} - \frac{273}{1067} a^{15} + \frac{126}{1067} a^{14} - \frac{169}{1067} a^{13} + \frac{283}{1067} a^{12} - \frac{277}{1067} a^{11} + \frac{47}{97} a^{10} - \frac{387}{1067} a^{9} + \frac{300}{1067} a^{8} - \frac{260}{1067} a^{7} - \frac{465}{1067} a^{6} + \frac{420}{1067} a^{5} + \frac{35}{1067} a^{4} - \frac{420}{1067} a^{3} - \frac{324}{1067} a^{2} + \frac{150}{1067} a - \frac{260}{1067}$, $\frac{1}{107779615141} a^{17} + \frac{47524573}{107779615141} a^{16} - \frac{44195865862}{107779615141} a^{15} - \frac{51927440935}{107779615141} a^{14} - \frac{24954646260}{107779615141} a^{13} + \frac{31132339559}{107779615141} a^{12} + \frac{43906983405}{107779615141} a^{11} + \frac{1576926063}{107779615141} a^{10} + \frac{27277217706}{107779615141} a^{9} + \frac{4727526249}{9798146831} a^{8} + \frac{27835571928}{107779615141} a^{7} + \frac{45645975271}{107779615141} a^{6} - \frac{8621058905}{107779615141} a^{5} + \frac{31184786951}{107779615141} a^{4} - \frac{34923039597}{107779615141} a^{3} - \frac{3793738873}{107779615141} a^{2} + \frac{45437430370}{107779615141} a - \frac{35370864648}{107779615141}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $14$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 91563.2420757 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_4$ (as 18T61):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 30 conjugacy class representatives for $C_6\times S_4$
Character table for $C_6\times S_4$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 6.4.47210863.1, 9.9.17515230173.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 sibling: 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.12.6.1$x^{12} + 2382032 x^{6} - 418195493 x^{2} + 1418519112256$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$