↑

Properties

Label 18.6.15407387070...7393.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{24}\cdot 73^{4}\cdot 577^{3}$
Root discriminant $32.39$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T767

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1819, -1545, 2895, -4633, -132, 5811, -5820, 1227, 1275, -1794, 981, -303, -68, 129, -51, 12, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 12*x^15 - 51*x^14 + 129*x^13 - 68*x^12 - 303*x^11 + 981*x^10 - 1794*x^9 + 1275*x^8 + 1227*x^7 - 5820*x^6 + 5811*x^5 - 132*x^4 - 4633*x^3 + 2895*x^2 - 1545*x + 1819)
 
gp: K = bnfinit(x^18 - 3*x^17 + 12*x^15 - 51*x^14 + 129*x^13 - 68*x^12 - 303*x^11 + 981*x^10 - 1794*x^9 + 1275*x^8 + 1227*x^7 - 5820*x^6 + 5811*x^5 - 132*x^4 - 4633*x^3 + 2895*x^2 - 1545*x + 1819, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 12 x^{15} - 51 x^{14} + 129 x^{13} - 68 x^{12} - 303 x^{11} + 981 x^{10} - 1794 x^{9} + 1275 x^{8} + 1227 x^{7} - 5820 x^{6} + 5811 x^{5} - 132 x^{4} - 4633 x^{3} + 2895 x^{2} - 1545 x + 1819 \)

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:  \(1540738707041918120513187393=3^{24}\cdot 73^{4}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.39$
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:  $3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{339908988584488168452512575615303} a^{17} + \frac{13741411978414093001065682595158}{339908988584488168452512575615303} a^{16} - \frac{75769677398038513424293084815786}{339908988584488168452512575615303} a^{15} - \frac{14087582043783379750577796450742}{339908988584488168452512575615303} a^{14} + \frac{8965842526435545959995795395139}{339908988584488168452512575615303} a^{13} - \frac{52803552009546610198875036313262}{339908988584488168452512575615303} a^{12} - \frac{73085711975697797300006149305014}{339908988584488168452512575615303} a^{11} + \frac{159893562602390629440425820767132}{339908988584488168452512575615303} a^{10} + \frac{147025799551051677911757280244516}{339908988584488168452512575615303} a^{9} + \frac{49552498952163981940035459205031}{339908988584488168452512575615303} a^{8} - \frac{142659610195198146791924763661384}{339908988584488168452512575615303} a^{7} - \frac{95165221829960552296937456028777}{339908988584488168452512575615303} a^{6} + \frac{19751339759076941460275039588345}{339908988584488168452512575615303} a^{5} + \frac{159293074017104312080134304286210}{339908988584488168452512575615303} a^{4} + \frac{168640531577048331849666654034025}{339908988584488168452512575615303} a^{3} + \frac{69901948986666112012275825377177}{339908988584488168452512575615303} a^{2} + \frac{47182741768500036179626347012388}{339908988584488168452512575615303} a - \frac{84762240478711279147762149990466}{339908988584488168452512575615303}$

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:  \( 3320682.13172 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T767:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n767 are not computed
Character table for t18n767 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R $18$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
73Data not computed
577Data not computed