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Properties

Label 18.14.1148057891...8688.1
Degree $18$
Signature $[14, 2]$
Discriminant $2^{6}\cdot 3^{33}\cdot 19^{9}$
Root discriminant $41.16$
Ramified primes $2, 3, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T585

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51, -126, -2493, -4086, 6048, 12348, -3600, -9162, -2502, 465, 3060, 1764, -1029, -603, 180, 69, -21, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 21*x^16 + 69*x^15 + 180*x^14 - 603*x^13 - 1029*x^12 + 1764*x^11 + 3060*x^10 + 465*x^9 - 2502*x^8 - 9162*x^7 - 3600*x^6 + 12348*x^5 + 6048*x^4 - 4086*x^3 - 2493*x^2 - 126*x + 51)
 
gp: K = bnfinit(x^18 - 3*x^17 - 21*x^16 + 69*x^15 + 180*x^14 - 603*x^13 - 1029*x^12 + 1764*x^11 + 3060*x^10 + 465*x^9 - 2502*x^8 - 9162*x^7 - 3600*x^6 + 12348*x^5 + 6048*x^4 - 4086*x^3 - 2493*x^2 - 126*x + 51, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 21 x^{16} + 69 x^{15} + 180 x^{14} - 603 x^{13} - 1029 x^{12} + 1764 x^{11} + 3060 x^{10} + 465 x^{9} - 2502 x^{8} - 9162 x^{7} - 3600 x^{6} + 12348 x^{5} + 6048 x^{4} - 4086 x^{3} - 2493 x^{2} - 126 x + 51 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(114805789186292807734226138688=2^{6}\cdot 3^{33}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.16$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{38} a^{15} - \frac{1}{2} a^{14} + \frac{7}{19} a^{13} + \frac{17}{38} a^{12} - \frac{1}{38} a^{11} - \frac{8}{19} a^{10} + \frac{7}{38} a^{9} - \frac{11}{38} a^{8} + \frac{7}{38} a^{7} + \frac{13}{38} a^{6} - \frac{3}{19} a^{5} - \frac{8}{19} a^{4} - \frac{7}{38} a^{3} + \frac{11}{38} a - \frac{13}{38}$, $\frac{1}{38} a^{16} - \frac{5}{38} a^{14} + \frac{17}{38} a^{13} + \frac{9}{19} a^{12} + \frac{3}{38} a^{11} + \frac{7}{38} a^{10} + \frac{4}{19} a^{9} - \frac{6}{19} a^{8} - \frac{3}{19} a^{7} + \frac{13}{38} a^{6} - \frac{8}{19} a^{5} - \frac{7}{38} a^{4} - \frac{1}{2} a^{3} + \frac{11}{38} a^{2} + \frac{3}{19} a - \frac{1}{2}$, $\frac{1}{176155058694080170305593386} a^{17} + \frac{609616162137421122745351}{176155058694080170305593386} a^{16} - \frac{165863982690166827589737}{176155058694080170305593386} a^{15} + \frac{25126852125725444559970948}{88077529347040085152796693} a^{14} - \frac{4251979618980464172797205}{9271318878635798437136494} a^{13} + \frac{351677354159424665356347}{176155058694080170305593386} a^{12} - \frac{31451179996147563508258910}{88077529347040085152796693} a^{11} - \frac{43111084841447411825243629}{176155058694080170305593386} a^{10} - \frac{13033776748416956448356471}{88077529347040085152796693} a^{9} - \frac{1024293485235101899646291}{4635659439317899218568247} a^{8} + \frac{48223304210080617527617139}{176155058694080170305593386} a^{7} + \frac{66970326385801681486620151}{176155058694080170305593386} a^{6} + \frac{23789795497154468205557101}{176155058694080170305593386} a^{5} - \frac{28658820171038454844106322}{88077529347040085152796693} a^{4} - \frac{1336166622119323545731572}{88077529347040085152796693} a^{3} - \frac{45247503878483134788948595}{176155058694080170305593386} a^{2} + \frac{79344001654237661565774485}{176155058694080170305593386} a + \frac{18014971638177199340900221}{176155058694080170305593386}$

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:  $15$
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:  \( 417741885.482 \) (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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$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}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
3Data not computed
$19$19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
19.6.5.3$x^{6} - 4864$$6$$1$$5$$C_6$$[\ ]_{6}$