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Properties

Label 18.10.8659616134...7136.2
Degree $18$
Signature $[10, 4]$
Discriminant $2^{18}\cdot 3^{6}\cdot 7^{12}\cdot 41^{9}$
Root discriminant $67.59$
Ramified primes $2, 3, 7, 41$
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![-999832, 1686168, -153716, -649120, -157848, 144262, 266857, -114360, -68772, 44230, 203, -6098, 1598, 14, -96, 64, -12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 12*x^16 + 64*x^15 - 96*x^14 + 14*x^13 + 1598*x^12 - 6098*x^11 + 203*x^10 + 44230*x^9 - 68772*x^8 - 114360*x^7 + 266857*x^6 + 144262*x^5 - 157848*x^4 - 649120*x^3 - 153716*x^2 + 1686168*x - 999832)
 
gp: K = bnfinit(x^18 - 4*x^17 - 12*x^16 + 64*x^15 - 96*x^14 + 14*x^13 + 1598*x^12 - 6098*x^11 + 203*x^10 + 44230*x^9 - 68772*x^8 - 114360*x^7 + 266857*x^6 + 144262*x^5 - 157848*x^4 - 649120*x^3 - 153716*x^2 + 1686168*x - 999832, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 12 x^{16} + 64 x^{15} - 96 x^{14} + 14 x^{13} + 1598 x^{12} - 6098 x^{11} + 203 x^{10} + 44230 x^{9} - 68772 x^{8} - 114360 x^{7} + 266857 x^{6} + 144262 x^{5} - 157848 x^{4} - 649120 x^{3} - 153716 x^{2} + 1686168 x - 999832 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(865961613414533621361938102747136=2^{18}\cdot 3^{6}\cdot 7^{12}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.59$
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:  $2, 3, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{44} a^{16} + \frac{1}{22} a^{15} + \frac{1}{22} a^{14} - \frac{2}{11} a^{13} - \frac{2}{11} a^{12} - \frac{3}{22} a^{11} + \frac{3}{22} a^{10} - \frac{1}{22} a^{9} + \frac{5}{44} a^{8} - \frac{2}{11} a^{7} + \frac{3}{22} a^{6} + \frac{4}{11} a^{5} - \frac{5}{44} a^{4} - \frac{3}{11} a^{3} + \frac{2}{11} a^{2} - \frac{2}{11} a - \frac{3}{11}$, $\frac{1}{437992431786049732282347602098899303844} a^{17} - \frac{1049699291867248067466365825736445406}{109498107946512433070586900524724825961} a^{16} + \frac{11892964194093022635895253253079650749}{109498107946512433070586900524724825961} a^{15} + \frac{11109889456507698488107049602063631391}{218996215893024866141173801049449651922} a^{14} + \frac{21597630855014234691033188126568503009}{109498107946512433070586900524724825961} a^{13} - \frac{40043564570020858919679637085679751173}{218996215893024866141173801049449651922} a^{12} - \frac{135536283363426307683209727182108211}{580891819344893544141044565117903586} a^{11} - \frac{18462091492086447614729846779764268807}{109498107946512433070586900524724825961} a^{10} - \frac{3832063435280332813433981676391884903}{39817493798731793843849782008990845804} a^{9} - \frac{8667984981148176050240832395980320737}{109498107946512433070586900524724825961} a^{8} - \frac{4272259226760290890605758184340072506}{109498107946512433070586900524724825961} a^{7} + \frac{37712112789544859129579258131465938818}{109498107946512433070586900524724825961} a^{6} + \frac{170033839006683817027211498319106672133}{437992431786049732282347602098899303844} a^{5} + \frac{82367771294311580037198871060031121547}{218996215893024866141173801049449651922} a^{4} - \frac{3984664350850439801273708096472922020}{9954373449682948460962445502247711451} a^{3} + \frac{44691380779785848819319884493132723823}{109498107946512433070586900524724825961} a^{2} - \frac{50891583715013027570708245916080123164}{109498107946512433070586900524724825961} a + \frac{29279735373912088919137518662401379074}{109498107946512433070586900524724825961}$

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:  $13$
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:  \( 19796223245.9 \) (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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
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$$[\ ]_{3}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.6.5.2$x^{6} + 246$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
41.6.4.1$x^{6} + 1435 x^{3} + 2904768$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$