↑

Properties

Label 18.6.12999413056...0657.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{15}\cdot 1399^{3}$
Root discriminant $16.93$
Ramified primes $7, 1399$
Class number $1$
Class group Trivial
Galois group 18T285

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 16, -16, 67, -134, 89, -66, 26, 138, -236, 140, 1, -46, 35, -24, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 - 24*x^15 + 35*x^14 - 46*x^13 + x^12 + 140*x^11 - 236*x^10 + 138*x^9 + 26*x^8 - 66*x^7 + 89*x^6 - 134*x^5 + 67*x^4 - 16*x^3 + 16*x^2 + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 15*x^16 - 24*x^15 + 35*x^14 - 46*x^13 + x^12 + 140*x^11 - 236*x^10 + 138*x^9 + 26*x^8 - 66*x^7 + 89*x^6 - 134*x^5 + 67*x^4 - 16*x^3 + 16*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 15 x^{16} - 24 x^{15} + 35 x^{14} - 46 x^{13} + x^{12} + 140 x^{11} - 236 x^{10} + 138 x^{9} + 26 x^{8} - 66 x^{7} + 89 x^{6} - 134 x^{5} + 67 x^{4} - 16 x^{3} + 16 x^{2} + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12999413056615907410657=7^{15}\cdot 1399^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.93$
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:  $7, 1399$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{86} a^{15} - \frac{5}{86} a^{14} + \frac{5}{43} a^{13} + \frac{11}{86} a^{12} + \frac{7}{86} a^{11} - \frac{2}{43} a^{10} + \frac{7}{43} a^{9} - \frac{15}{86} a^{8} + \frac{18}{43} a^{7} - \frac{35}{86} a^{6} - \frac{20}{43} a^{5} - \frac{23}{86} a^{4} + \frac{4}{43} a^{3} + \frac{35}{86} a^{2} - \frac{1}{2} a + \frac{6}{43}$, $\frac{1}{86} a^{16} - \frac{15}{86} a^{14} + \frac{9}{43} a^{13} + \frac{19}{86} a^{12} - \frac{6}{43} a^{11} - \frac{3}{43} a^{10} + \frac{6}{43} a^{9} + \frac{2}{43} a^{8} - \frac{27}{86} a^{7} + \frac{35}{86} a^{5} - \frac{21}{86} a^{4} + \frac{16}{43} a^{3} - \frac{20}{43} a^{2} + \frac{6}{43} a - \frac{13}{43}$, $\frac{1}{3744526} a^{17} - \frac{14519}{3744526} a^{16} + \frac{19363}{3744526} a^{15} + \frac{433781}{3744526} a^{14} - \frac{175215}{3744526} a^{13} - \frac{334561}{3744526} a^{12} - \frac{191345}{1872263} a^{11} + \frac{282019}{3744526} a^{10} - \frac{349229}{3744526} a^{9} + \frac{4720}{43541} a^{8} + \frac{425593}{1872263} a^{7} + \frac{233577}{3744526} a^{6} + \frac{12592}{1872263} a^{5} + \frac{37355}{1872263} a^{4} - \frac{704837}{3744526} a^{3} + \frac{84347}{1872263} a^{2} + \frac{272129}{3744526} a - \frac{794624}{1872263}$

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:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 11536.976046 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T285:

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 t18n285
Character table for t18n285 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.6.23512993.1, 9.3.164590951.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ $18$

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
7Data not computed
1399Data not computed