↑

Properties

Label 18.12.6042409957...8352.2
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{6}\cdot 3^{33}\cdot 19^{8}$
Root discriminant $34.95$
Ramified primes $2, 3, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T585

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -60, -96, 63, 588, 849, 399, 336, -1464, -1991, 1428, 1437, -675, -435, 168, 60, -21, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 21*x^16 + 60*x^15 + 168*x^14 - 435*x^13 - 675*x^12 + 1437*x^11 + 1428*x^10 - 1991*x^9 - 1464*x^8 + 336*x^7 + 399*x^6 + 849*x^5 + 588*x^4 + 63*x^3 - 96*x^2 - 60*x - 8)
 
gp: K = bnfinit(x^18 - 3*x^17 - 21*x^16 + 60*x^15 + 168*x^14 - 435*x^13 - 675*x^12 + 1437*x^11 + 1428*x^10 - 1991*x^9 - 1464*x^8 + 336*x^7 + 399*x^6 + 849*x^5 + 588*x^4 + 63*x^3 - 96*x^2 - 60*x - 8, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 21 x^{16} + 60 x^{15} + 168 x^{14} - 435 x^{13} - 675 x^{12} + 1437 x^{11} + 1428 x^{10} - 1991 x^{9} - 1464 x^{8} + 336 x^{7} + 399 x^{6} + 849 x^{5} + 588 x^{4} + 63 x^{3} - 96 x^{2} - 60 x - 8 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-6042409957173305670222428352=-\,2^{6}\cdot 3^{33}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.95$
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:  $2, 3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{12} a^{11} + \frac{1}{12} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{5}{12} a^{4} - \frac{1}{6} a^{3} - \frac{5}{12} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{36} a^{14} + \frac{1}{36} a^{13} + \frac{1}{36} a^{12} + \frac{1}{18} a^{11} + \frac{5}{36} a^{10} - \frac{1}{9} a^{9} + \frac{1}{6} a^{8} + \frac{5}{12} a^{7} + \frac{5}{12} a^{6} - \frac{1}{36} a^{5} + \frac{2}{9} a^{4} - \frac{1}{36} a^{3} - \frac{2}{9} a^{2} + \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{72} a^{15} + \frac{1}{72} a^{12} + \frac{1}{24} a^{11} - \frac{1}{8} a^{10} - \frac{1}{36} a^{9} + \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{5}{18} a^{6} + \frac{1}{8} a^{5} + \frac{3}{8} a^{4} + \frac{29}{72} a^{3} - \frac{1}{6} a^{2} + \frac{4}{9}$, $\frac{1}{144} a^{16} - \frac{1}{144} a^{15} + \frac{1}{144} a^{13} - \frac{5}{72} a^{12} + \frac{19}{144} a^{10} - \frac{13}{144} a^{9} - \frac{1}{16} a^{8} + \frac{7}{18} a^{7} + \frac{61}{144} a^{6} - \frac{1}{8} a^{5} + \frac{19}{72} a^{4} + \frac{43}{144} a^{3} - \frac{1}{4} a^{2} + \frac{5}{36} a - \frac{1}{18}$, $\frac{1}{95868329184} a^{17} - \frac{11765}{17519797} a^{16} - \frac{7784245}{95868329184} a^{15} - \frac{329585857}{31956109728} a^{14} - \frac{3969259897}{95868329184} a^{13} + \frac{38436185}{47934164592} a^{12} - \frac{1769969869}{95868329184} a^{11} + \frac{5618708015}{47934164592} a^{10} - \frac{96679373}{840950256} a^{9} - \frac{28379378173}{95868329184} a^{8} + \frac{12178871323}{31956109728} a^{7} - \frac{43250870069}{95868329184} a^{6} - \frac{104946191}{998628429} a^{5} - \frac{41301634703}{95868329184} a^{4} - \frac{33839449}{95868329184} a^{3} + \frac{1019493319}{23967082296} a^{2} - \frac{2706099661}{23967082296} a + \frac{449584021}{3994513716}$

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:  $14$
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:  \( 73115077.6827 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T585:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 13824
The 96 conjugacy class representatives for t18n585 are not computed
Character table for t18n585 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.5609891727441.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
3Data not computed
$19$19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.6.4.1$x^{6} + 57 x^{3} + 1444$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$