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Properties

Label 18.6.12760912301...2421.2
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 83^{4}\cdot 181^{5}$
Root discriminant $41.40$
Ramified primes $7, 83, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T705

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1429, 4530, 3356, 7097, 8373, 3188, 3794, 1649, 2167, 590, -287, 402, -186, -14, 4, -27, 8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 8*x^16 - 27*x^15 + 4*x^14 - 14*x^13 - 186*x^12 + 402*x^11 - 287*x^10 + 590*x^9 + 2167*x^8 + 1649*x^7 + 3794*x^6 + 3188*x^5 + 8373*x^4 + 7097*x^3 + 3356*x^2 + 4530*x + 1429)
 
gp: K = bnfinit(x^18 - 3*x^17 + 8*x^16 - 27*x^15 + 4*x^14 - 14*x^13 - 186*x^12 + 402*x^11 - 287*x^10 + 590*x^9 + 2167*x^8 + 1649*x^7 + 3794*x^6 + 3188*x^5 + 8373*x^4 + 7097*x^3 + 3356*x^2 + 4530*x + 1429, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 8 x^{16} - 27 x^{15} + 4 x^{14} - 14 x^{13} - 186 x^{12} + 402 x^{11} - 287 x^{10} + 590 x^{9} + 2167 x^{8} + 1649 x^{7} + 3794 x^{6} + 3188 x^{5} + 8373 x^{4} + 7097 x^{3} + 3356 x^{2} + 4530 x + 1429 \)

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:  \(127609123015304468465889942421=7^{12}\cdot 83^{4}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.40$
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:  $7, 83, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{41} a^{16} + \frac{8}{41} a^{15} - \frac{13}{41} a^{14} - \frac{17}{41} a^{13} + \frac{4}{41} a^{12} - \frac{3}{41} a^{11} + \frac{1}{41} a^{10} + \frac{2}{41} a^{9} - \frac{5}{41} a^{8} - \frac{11}{41} a^{7} + \frac{8}{41} a^{6} - \frac{16}{41} a^{5} - \frac{1}{41} a^{4} + \frac{1}{41} a^{3} + \frac{6}{41} a^{2} + \frac{2}{41} a + \frac{18}{41}$, $\frac{1}{34455714313460930235657114613942181} a^{17} - \frac{668052260952601170593057620343}{2650439562573917710435162662610937} a^{16} - \frac{7981144735231432939187826172459962}{34455714313460930235657114613942181} a^{15} + \frac{4731349594337605313732923178953894}{34455714313460930235657114613942181} a^{14} + \frac{11400970940071624340402158040843606}{34455714313460930235657114613942181} a^{13} - \frac{11992277240087980827131335061227026}{34455714313460930235657114613942181} a^{12} - \frac{415848929107140129104213777116072}{2650439562573917710435162662610937} a^{11} - \frac{9856171657280536549888282461993811}{34455714313460930235657114613942181} a^{10} - \frac{6341855596355233194301865137879471}{34455714313460930235657114613942181} a^{9} + \frac{2257249686112253276319730266256968}{34455714313460930235657114613942181} a^{8} - \frac{4578771315513547403551600283018853}{34455714313460930235657114613942181} a^{7} - \frac{9294782120605042353401367998733187}{34455714313460930235657114613942181} a^{6} - \frac{3626537151276349140866068706738714}{34455714313460930235657114613942181} a^{5} - \frac{5792782664748351322123143055332965}{34455714313460930235657114613942181} a^{4} - \frac{237711918925701984516703395580744}{1188128079774514835712314297032489} a^{3} - \frac{360537052604062276276904752861768}{840383275938071469162368649120541} a^{2} - \frac{9401622940251537053654585804903204}{34455714313460930235657114613942181} a + \frac{10527907277032957942218939758989441}{34455714313460930235657114613942181}$

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:  \( 59904292.1623 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T705:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 64 conjugacy class representatives for t18n705 are not computed
Character table for t18n705 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.26552265046321.1

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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ $18$ $18$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$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}$
7.3.2.2$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$$[\ ]_{3}^{2}$
83Data not computed
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$