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Properties

Label 18.0.13275077566...7463.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 52879^{2}$
Root discriminant $16.95$
Ramified primes $7, 52879$
Class number $1$
Class group Trivial
Galois group 18T286

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 18, -49, 129, -210, 279, -221, 77, -11, 25, -59, 36, -12, 11, -9, 8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 8*x^16 - 9*x^15 + 11*x^14 - 12*x^13 + 36*x^12 - 59*x^11 + 25*x^10 - 11*x^9 + 77*x^8 - 221*x^7 + 279*x^6 - 210*x^5 + 129*x^4 - 49*x^3 + 18*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 8*x^16 - 9*x^15 + 11*x^14 - 12*x^13 + 36*x^12 - 59*x^11 + 25*x^10 - 11*x^9 + 77*x^8 - 221*x^7 + 279*x^6 - 210*x^5 + 129*x^4 - 49*x^3 + 18*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 8 x^{16} - 9 x^{15} + 11 x^{14} - 12 x^{13} + 36 x^{12} - 59 x^{11} + 25 x^{10} - 11 x^{9} + 77 x^{8} - 221 x^{7} + 279 x^{6} - 210 x^{5} + 129 x^{4} - 49 x^{3} + 18 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-13275077566551425157463=-\,7^{15}\cdot 52879^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.95$
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:  $7, 52879$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{37} a^{16} + \frac{10}{37} a^{15} - \frac{6}{37} a^{14} - \frac{10}{37} a^{13} + \frac{5}{37} a^{12} + \frac{13}{37} a^{11} + \frac{3}{37} a^{10} - \frac{5}{37} a^{9} + \frac{9}{37} a^{8} + \frac{12}{37} a^{7} + \frac{10}{37} a^{6} - \frac{6}{37} a^{5} - \frac{18}{37} a^{4} + \frac{13}{37} a^{3} + \frac{4}{37} a^{2} + \frac{18}{37} a + \frac{9}{37}$, $\frac{1}{195263000203597} a^{17} + \frac{984974936495}{195263000203597} a^{16} - \frac{7126382774595}{195263000203597} a^{15} + \frac{36000620654088}{195263000203597} a^{14} - \frac{32444170977527}{195263000203597} a^{13} + \frac{54814223395252}{195263000203597} a^{12} + \frac{26908720039723}{195263000203597} a^{11} + \frac{68482979142247}{195263000203597} a^{10} - \frac{205623455322}{195263000203597} a^{9} - \frac{80271009936064}{195263000203597} a^{8} + \frac{4880650100074}{195263000203597} a^{7} - \frac{52437167700565}{195263000203597} a^{6} + \frac{24855598095780}{195263000203597} a^{5} + \frac{36651346263573}{195263000203597} a^{4} + \frac{83224382093118}{195263000203597} a^{3} + \frac{79056107995938}{195263000203597} a^{2} + \frac{4242470255068}{195263000203597} a - \frac{84932514281449}{195263000203597}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{141701104654080}{195263000203597} a^{17} - \frac{390489024684488}{195263000203597} a^{16} + \frac{1022095806218124}{195263000203597} a^{15} - \frac{979268767855026}{195263000203597} a^{14} + \frac{1198654081137127}{195263000203597} a^{13} - \frac{1285643336878647}{195263000203597} a^{12} + \frac{4642173413759320}{195263000203597} a^{11} - \frac{7071507169810894}{195263000203597} a^{10} + \frac{1262508093158063}{195263000203597} a^{9} - \frac{396455982281341}{195263000203597} a^{8} + \frac{10575909826303834}{195263000203597} a^{7} - \frac{28739504491862254}{195263000203597} a^{6} + \frac{31240259654577404}{195263000203597} a^{5} - \frac{18676951334772072}{195263000203597} a^{4} + \frac{9865567172331900}{195263000203597} a^{3} - \frac{2109346033204980}{195263000203597} a^{2} + \frac{787060245924693}{195263000203597} a + \frac{54703304974880}{195263000203597} \) (order $14$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 18327.0613213 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T286:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 34 conjugacy class representatives for t18n286
Character table for t18n286 is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 9.7.6221161471.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ $18$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
52879Data not computed