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Properties

Label 18.8.55112648414...4288.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{8}\cdot 167$
Root discriminant $58.00$
Ramified primes $2, 3, 7, 41, 167$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-29261, 118258, -250469, 324269, -193454, -13206, 102391, -110341, 60028, -10670, 463, 3597, -2165, 714, -221, 6, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 7*x^16 + 6*x^15 - 221*x^14 + 714*x^13 - 2165*x^12 + 3597*x^11 + 463*x^10 - 10670*x^9 + 60028*x^8 - 110341*x^7 + 102391*x^6 - 13206*x^5 - 193454*x^4 + 324269*x^3 - 250469*x^2 + 118258*x - 29261)
 
gp: K = bnfinit(x^18 - 3*x^17 + 7*x^16 + 6*x^15 - 221*x^14 + 714*x^13 - 2165*x^12 + 3597*x^11 + 463*x^10 - 10670*x^9 + 60028*x^8 - 110341*x^7 + 102391*x^6 - 13206*x^5 - 193454*x^4 + 324269*x^3 - 250469*x^2 + 118258*x - 29261, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 7 x^{16} + 6 x^{15} - 221 x^{14} + 714 x^{13} - 2165 x^{12} + 3597 x^{11} + 463 x^{10} - 10670 x^{9} + 60028 x^{8} - 110341 x^{7} + 102391 x^{6} - 13206 x^{5} - 193454 x^{4} + 324269 x^{3} - 250469 x^{2} + 118258 x - 29261 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-55112648414720699225397737484288=-\,2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{8}\cdot 167\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.00$
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:  $2, 3, 7, 41, 167$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6474} a^{16} - \frac{11}{498} a^{15} + \frac{24}{83} a^{14} + \frac{431}{2158} a^{13} - \frac{1945}{6474} a^{12} - \frac{967}{2158} a^{11} + \frac{275}{3237} a^{10} - \frac{447}{1079} a^{9} + \frac{2425}{6474} a^{8} + \frac{30}{83} a^{7} - \frac{885}{2158} a^{6} + \frac{2087}{6474} a^{5} + \frac{92}{1079} a^{4} + \frac{2987}{6474} a^{3} - \frac{28}{249} a^{2} - \frac{283}{1079} a + \frac{1447}{6474}$, $\frac{1}{3096714112353085603426235057632014224656476} a^{17} + \frac{23544670797131232844395217706509989545}{516119018725514267237705842938669037442746} a^{16} - \frac{7942738909825432164678499870131552295697}{79402925957771425728877821990564467298884} a^{15} - \frac{1183333994585560960302615565545331451156557}{3096714112353085603426235057632014224656476} a^{14} - \frac{432320508939316331658629134276289159977969}{1548357056176542801713117528816007112328238} a^{13} - \frac{25312663650626035043065610402001371211397}{258059509362757133618852921469334518721373} a^{12} + \frac{970256267920881689074575177532512160665851}{3096714112353085603426235057632014224656476} a^{11} - \frac{15805375291932015901126994438218926743941}{258059509362757133618852921469334518721373} a^{10} - \frac{69190615331961903570427708564371705982527}{1032238037451028534475411685877338074885492} a^{9} + \frac{38283523286831497963638406000037456080093}{3096714112353085603426235057632014224656476} a^{8} + \frac{172020544473340852674038338481462352818975}{1032238037451028534475411685877338074885492} a^{7} + \frac{5871608717156246787040390878272555338475}{59552194468328569296658366492923350474163} a^{6} + \frac{272999252057299667841380986695846958574059}{3096714112353085603426235057632014224656476} a^{5} - \frac{648239020823807253170528939408439966364523}{3096714112353085603426235057632014224656476} a^{4} - \frac{472312295770153062902889032365946456802365}{3096714112353085603426235057632014224656476} a^{3} - \frac{53821970511508046216537071918793243219474}{258059509362757133618852921469334518721373} a^{2} + \frac{446670693249933369389260584109454143579313}{1032238037451028534475411685877338074885492} a - \frac{1116892706717567204691807322377975401073215}{3096714112353085603426235057632014224656476}$

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:  $12$
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:  \( 2547340760.88 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.574470067776192.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
41Data not computed
167Data not computed