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Properties

Label 18.14.8659616134...7136.1
Degree $18$
Signature $[14, 2]$
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![-587, 14314, 109343, 183050, 64828, -124454, -209892, -142480, -7313, 46224, 24651, 286, -5877, -1332, 697, 140, -43, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 43*x^16 + 140*x^15 + 697*x^14 - 1332*x^13 - 5877*x^12 + 286*x^11 + 24651*x^10 + 46224*x^9 - 7313*x^8 - 142480*x^7 - 209892*x^6 - 124454*x^5 + 64828*x^4 + 183050*x^3 + 109343*x^2 + 14314*x - 587)
 
gp: K = bnfinit(x^18 - 4*x^17 - 43*x^16 + 140*x^15 + 697*x^14 - 1332*x^13 - 5877*x^12 + 286*x^11 + 24651*x^10 + 46224*x^9 - 7313*x^8 - 142480*x^7 - 209892*x^6 - 124454*x^5 + 64828*x^4 + 183050*x^3 + 109343*x^2 + 14314*x - 587, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 43 x^{16} + 140 x^{15} + 697 x^{14} - 1332 x^{13} - 5877 x^{12} + 286 x^{11} + 24651 x^{10} + 46224 x^{9} - 7313 x^{8} - 142480 x^{7} - 209892 x^{6} - 124454 x^{5} + 64828 x^{4} + 183050 x^{3} + 109343 x^{2} + 14314 x - 587 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 2]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{56} a^{15} - \frac{5}{56} a^{14} - \frac{1}{56} a^{13} + \frac{13}{56} a^{12} - \frac{1}{28} a^{11} - \frac{5}{28} a^{10} + \frac{1}{7} a^{9} - \frac{5}{28} a^{8} - \frac{1}{56} a^{7} + \frac{27}{56} a^{6} + \frac{5}{56} a^{5} - \frac{3}{56} a^{4} - \frac{13}{56} a^{3} - \frac{3}{56} a^{2} - \frac{25}{56} a + \frac{27}{56}$, $\frac{1}{112} a^{16} + \frac{1}{56} a^{14} - \frac{5}{28} a^{13} - \frac{3}{16} a^{12} - \frac{5}{28} a^{11} - \frac{1}{8} a^{10} + \frac{1}{56} a^{9} + \frac{5}{112} a^{8} + \frac{11}{56} a^{7} - \frac{17}{56} a^{5} - \frac{1}{4} a^{4} + \frac{1}{7} a^{3} - \frac{5}{14} a^{2} - \frac{1}{8} a - \frac{33}{112}$, $\frac{1}{48461150616529471399453634118688} a^{17} - \frac{211214346199972696866555278125}{48461150616529471399453634118688} a^{16} + \frac{156568779291233721160807090647}{24230575308264735699726817059344} a^{15} + \frac{820335996051277565807578423795}{24230575308264735699726817059344} a^{14} + \frac{10134916816759683760498768254851}{48461150616529471399453634118688} a^{13} + \frac{11314660630336906636781422257289}{48461150616529471399453634118688} a^{12} - \frac{1370082004851380662781253134137}{24230575308264735699726817059344} a^{11} + \frac{48440057803328484551642810225}{432688844790441708923693161774} a^{10} + \frac{4642211712044252779106659289715}{48461150616529471399453634118688} a^{9} + \frac{4500482895833244269498557773389}{48461150616529471399453634118688} a^{8} - \frac{4369204572425292918686804944113}{24230575308264735699726817059344} a^{7} - \frac{10715712306676488858121079062415}{24230575308264735699726817059344} a^{6} - \frac{3175558995824162636183583287663}{24230575308264735699726817059344} a^{5} + \frac{141041150970647609804643531123}{865377689580883417847386323548} a^{4} - \frac{3457686273729067160713887035133}{12115287654132367849863408529672} a^{3} - \frac{9858287272309731586433998424101}{24230575308264735699726817059344} a^{2} + \frac{20251657055752029388703225229601}{48461150616529471399453634118688} a + \frac{7417994375567949217924560343489}{48461150616529471399453634118688}$

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:  $15$
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:  \( 24933985535.3 \) (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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
$3$3.6.0.1$x^{6} - x + 2$$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$$[\ ]_{2}^{6}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
41Data not computed