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Properties

Label 18.6.86055014859...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{24}\cdot 3^{32}\cdot 5^{6}\cdot 11^{6}$
Root discriminant $67.56$
Ramified primes $2, 3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T268

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1517613, -760338, 2309247, 18882, 326187, 482364, -323604, -38430, 3498, -26096, -3315, 5268, -2439, 372, 267, -12, 24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 24*x^16 - 12*x^15 + 267*x^14 + 372*x^13 - 2439*x^12 + 5268*x^11 - 3315*x^10 - 26096*x^9 + 3498*x^8 - 38430*x^7 - 323604*x^6 + 482364*x^5 + 326187*x^4 + 18882*x^3 + 2309247*x^2 - 760338*x - 1517613)
 
gp: K = bnfinit(x^18 + 24*x^16 - 12*x^15 + 267*x^14 + 372*x^13 - 2439*x^12 + 5268*x^11 - 3315*x^10 - 26096*x^9 + 3498*x^8 - 38430*x^7 - 323604*x^6 + 482364*x^5 + 326187*x^4 + 18882*x^3 + 2309247*x^2 - 760338*x - 1517613, 1)
 

Normalized defining polynomial

\( x^{18} + 24 x^{16} - 12 x^{15} + 267 x^{14} + 372 x^{13} - 2439 x^{12} + 5268 x^{11} - 3315 x^{10} - 26096 x^{9} + 3498 x^{8} - 38430 x^{7} - 323604 x^{6} + 482364 x^{5} + 326187 x^{4} + 18882 x^{3} + 2309247 x^{2} - 760338 x - 1517613 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(860550148596054437082169344000000=2^{24}\cdot 3^{32}\cdot 5^{6}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.56$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} - \frac{3}{11} a^{10} - \frac{2}{11} a^{9} - \frac{2}{11} a^{8} + \frac{3}{11} a^{7} - \frac{5}{11} a^{6} - \frac{5}{11} a^{5} - \frac{5}{11} a^{4} - \frac{1}{11} a^{3} + \frac{2}{11} a^{2} + \frac{2}{11} a - \frac{3}{11}$, $\frac{1}{11} a^{13} - \frac{3}{11} a^{11} - \frac{2}{11} a^{10} - \frac{2}{11} a^{9} + \frac{3}{11} a^{8} - \frac{5}{11} a^{7} - \frac{5}{11} a^{6} - \frac{5}{11} a^{5} - \frac{1}{11} a^{4} + \frac{2}{11} a^{3} + \frac{2}{11} a^{2} - \frac{3}{11} a$, $\frac{1}{11} a^{14} - \frac{2}{11} a^{11} - \frac{3}{11} a^{9} + \frac{4}{11} a^{7} + \frac{2}{11} a^{6} - \frac{5}{11} a^{5} - \frac{2}{11} a^{4} - \frac{1}{11} a^{3} + \frac{3}{11} a^{2} - \frac{5}{11} a + \frac{2}{11}$, $\frac{1}{11} a^{15} + \frac{2}{11} a^{10} - \frac{4}{11} a^{9} - \frac{3}{11} a^{7} - \frac{4}{11} a^{6} - \frac{1}{11} a^{5} + \frac{1}{11} a^{3} - \frac{1}{11} a^{2} - \frac{5}{11} a + \frac{5}{11}$, $\frac{1}{43571} a^{16} + \frac{656}{43571} a^{15} - \frac{2}{2563} a^{14} - \frac{895}{43571} a^{13} + \frac{837}{43571} a^{12} - \frac{21599}{43571} a^{11} - \frac{21402}{43571} a^{10} + \frac{4524}{43571} a^{9} - \frac{15472}{43571} a^{8} + \frac{10356}{43571} a^{7} + \frac{12854}{43571} a^{6} - \frac{8930}{43571} a^{5} + \frac{10012}{43571} a^{4} + \frac{361}{43571} a^{3} - \frac{3585}{43571} a^{2} - \frac{529}{3961} a - \frac{4326}{43571}$, $\frac{1}{9300552266891164341496820422498064380643107683220373} a^{17} + \frac{102244421811549642961366819145336920625893726724}{9300552266891164341496820422498064380643107683220373} a^{16} - \frac{318005194119781252563604043398728943442465069688041}{9300552266891164341496820422498064380643107683220373} a^{15} + \frac{331710132699812816064066366736931945321334934897605}{9300552266891164341496820422498064380643107683220373} a^{14} - \frac{80876707545256181214681958569882770525792992111352}{9300552266891164341496820422498064380643107683220373} a^{13} + \frac{196681507617299605583755578709317518936510778575838}{9300552266891164341496820422498064380643107683220373} a^{12} + \frac{415831454850815164964519850969788361217900604206704}{9300552266891164341496820422498064380643107683220373} a^{11} + \frac{2714178115843203894040245054581805109358326426367517}{9300552266891164341496820422498064380643107683220373} a^{10} + \frac{1899532327469386249081947510160660710244757635925}{17851347921096284724562035359881121651906156781613} a^{9} + \frac{3514322178866396603207009137540064099941054756201200}{9300552266891164341496820422498064380643107683220373} a^{8} + \frac{2404318374152506939849713008524221302798243845781247}{9300552266891164341496820422498064380643107683220373} a^{7} + \frac{3345812643535403926757559926773740088844738069090189}{9300552266891164341496820422498064380643107683220373} a^{6} - \frac{1391076118889865919931117552736966178770916465223605}{9300552266891164341496820422498064380643107683220373} a^{5} + \frac{396453333441302523020384658519152150412375035052482}{845504751535560394681529129318005852785737062110943} a^{4} - \frac{1146002075941109877325493967436276841017916386664865}{9300552266891164341496820422498064380643107683220373} a^{3} + \frac{990322400268894436955280335739207492758891877514659}{9300552266891164341496820422498064380643107683220373} a^{2} - \frac{1854952941534576786614127202839548002361091135358481}{9300552266891164341496820422498064380643107683220373} a - \frac{2121049703872594592897307796242739554453904835207503}{9300552266891164341496820422498064380643107683220373}$

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:  $11$
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:  \( 6619213471.22 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T268:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 24 conjugacy class representatives for t18n268
Character table for t18n268 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.1620.1, 9.9.344373768000.1

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

Sibling fields

Degree 12 siblings: data not computed
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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
2Data not computed
$3$3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$