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Properties

Label 18.12.8659616134...7136.2
Degree $18$
Signature $[12, 3]$
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![6929, 0, -82041, 0, 264491, 0, -262504, 0, 35318, 0, 19399, 0, -1097, 0, -337, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 4*x^16 - 337*x^14 - 1097*x^12 + 19399*x^10 + 35318*x^8 - 262504*x^6 + 264491*x^4 - 82041*x^2 + 6929)
 
gp: K = bnfinit(x^18 + 4*x^16 - 337*x^14 - 1097*x^12 + 19399*x^10 + 35318*x^8 - 262504*x^6 + 264491*x^4 - 82041*x^2 + 6929, 1)
 

Normalized defining polynomial

\( x^{18} + 4 x^{16} - 337 x^{14} - 1097 x^{12} + 19399 x^{10} + 35318 x^{8} - 262504 x^{6} + 264491 x^{4} - 82041 x^{2} + 6929 \)

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:  \(-865961613414533621361938102747136=-\,2^{18}\cdot 3^{6}\cdot 7^{12}\cdot 41^{9}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{5}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a + \frac{3}{16}$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{5}{16} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{3}{16} a$, $\frac{1}{464} a^{14} + \frac{9}{464} a^{12} + \frac{7}{116} a^{10} + \frac{6}{29} a^{8} - \frac{1}{4} a^{7} - \frac{103}{464} a^{6} - \frac{191}{464} a^{4} - \frac{1}{4} a^{3} - \frac{213}{464} a^{2} - \frac{1}{4} a - \frac{27}{464}$, $\frac{1}{928} a^{15} + \frac{9}{928} a^{13} - \frac{1}{32} a^{12} - \frac{11}{116} a^{11} - \frac{1}{16} a^{10} - \frac{5}{232} a^{9} + \frac{3}{16} a^{8} - \frac{219}{928} a^{7} - \frac{3}{16} a^{6} - \frac{423}{928} a^{5} - \frac{11}{32} a^{4} + \frac{135}{928} a^{3} + \frac{1}{8} a^{2} - \frac{27}{928} a - \frac{3}{32}$, $\frac{1}{3506125335260741344} a^{16} + \frac{2504924071976269}{3506125335260741344} a^{14} - \frac{1}{32} a^{13} + \frac{50556383195406043}{1753062667630370672} a^{12} - \frac{1}{16} a^{11} - \frac{10398585262104117}{438265666907592668} a^{10} - \frac{1}{16} a^{9} + \frac{659022617619809729}{3506125335260741344} a^{8} - \frac{3}{16} a^{7} - \frac{30156766413014071}{3506125335260741344} a^{6} + \frac{13}{32} a^{5} - \frac{388559218892427015}{3506125335260741344} a^{4} - \frac{1}{8} a^{3} - \frac{261813031578650199}{3506125335260741344} a^{2} - \frac{3}{32} a - \frac{665163080740428691}{1753062667630370672}$, $\frac{1}{45579629358389637472} a^{17} + \frac{17617533275686361}{45579629358389637472} a^{15} - \frac{1}{928} a^{14} - \frac{538835375749287545}{22789814679194818736} a^{13} + \frac{5}{232} a^{12} - \frac{252668268341804815}{2848726834899352342} a^{11} - \frac{43}{464} a^{10} - \frac{4902417569345504127}{45579629358389637472} a^{9} - \frac{77}{464} a^{8} - \frac{1586755514395153547}{45579629358389637472} a^{7} - \frac{71}{928} a^{6} + \frac{7681574095888762113}{45579629358389637472} a^{5} - \frac{209}{464} a^{4} - \frac{10493049462490382483}{45579629358389637472} a^{3} + \frac{445}{928} a^{2} + \frac{3732606197539208081}{22789814679194818736} a - \frac{59}{464}$

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:  $14$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 14716221116.5 \) (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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$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$$[\ ]_{2}^{6}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
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}$