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Properties

Label 18.12.5109629454...4375.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,3^{36}\cdot 5^{9}\cdot 19^{3}\cdot 71^{4}$
Root discriminant $84.77$
Ramified primes $3, 5, 19, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T857

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![603649, 4107591, 5595606, -8094474, -13877136, 3497886, 9189660, 954900, -1868265, -662256, 86868, 35649, 16536, 2664, -855, -135, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^16 - 135*x^15 - 855*x^14 + 2664*x^13 + 16536*x^12 + 35649*x^11 + 86868*x^10 - 662256*x^9 - 1868265*x^8 + 954900*x^7 + 9189660*x^6 + 3497886*x^5 - 13877136*x^4 - 8094474*x^3 + 5595606*x^2 + 4107591*x + 603649)
 
gp: K = bnfinit(x^18 - 18*x^16 - 135*x^15 - 855*x^14 + 2664*x^13 + 16536*x^12 + 35649*x^11 + 86868*x^10 - 662256*x^9 - 1868265*x^8 + 954900*x^7 + 9189660*x^6 + 3497886*x^5 - 13877136*x^4 - 8094474*x^3 + 5595606*x^2 + 4107591*x + 603649, 1)
 

Normalized defining polynomial

\( x^{18} - 18 x^{16} - 135 x^{15} - 855 x^{14} + 2664 x^{13} + 16536 x^{12} + 35649 x^{11} + 86868 x^{10} - 662256 x^{9} - 1868265 x^{8} + 954900 x^{7} + 9189660 x^{6} + 3497886 x^{5} - 13877136 x^{4} - 8094474 x^{3} + 5595606 x^{2} + 4107591 x + 603649 \)

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:  \(-51096294546839412676148707927734375=-\,3^{36}\cdot 5^{9}\cdot 19^{3}\cdot 71^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.77$
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:  $3, 5, 19, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{12} a^{14} - \frac{1}{6} a^{13} - \frac{1}{12} a^{12} - \frac{1}{12} a^{11} + \frac{1}{12} a^{10} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} + \frac{1}{12} a^{7} + \frac{5}{12} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{5}{12} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{13} + \frac{1}{12} a^{12} - \frac{1}{12} a^{11} - \frac{1}{12} a^{10} + \frac{1}{12} a^{8} + \frac{1}{4} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{12} a^{4} - \frac{1}{12} a^{3} - \frac{5}{12} a^{2} - \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{48} a^{16} - \frac{1}{48} a^{15} - \frac{1}{24} a^{14} - \frac{1}{12} a^{13} - \frac{1}{48} a^{12} - \frac{1}{16} a^{11} - \frac{1}{12} a^{10} - \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{6} a^{7} - \frac{17}{48} a^{6} - \frac{1}{16} a^{5} + \frac{1}{3} a^{4} + \frac{17}{48} a^{3} - \frac{17}{48} a^{2} - \frac{5}{24} a + \frac{17}{48}$, $\frac{1}{16902745997180959432083948435264614621558983760039514600895248} a^{17} + \frac{158232775486701887150985638619180959850338563687639827055067}{16902745997180959432083948435264614621558983760039514600895248} a^{16} - \frac{69397672549429374907427311571767885436872151196526014513417}{2817124332863493238680658072544102436926497293339919100149208} a^{15} - \frac{108567076653112797890289390071822332490438869768199160948125}{4225686499295239858020987108816153655389745940009878650223812} a^{14} + \frac{1350909201594541979691603041369697375441827539083268613571415}{16902745997180959432083948435264614621558983760039514600895248} a^{13} - \frac{900281456283361520243195684244243543120371638123279727053381}{5634248665726986477361316145088204873852994586679838200298416} a^{12} - \frac{942022131019941106400886768797364163529267042852314572323}{9919451876279905770002317156845431115938370751196898239962} a^{11} + \frac{43439145624601551783487830593019513254366066608362271423147}{2112843249647619929010493554408076827694872970004939325111906} a^{10} + \frac{832285996178392085579177589618826782296360452282973876420215}{2112843249647619929010493554408076827694872970004939325111906} a^{9} + \frac{113485793593547272016289763099941296849178205200583221063973}{1056421624823809964505246777204038413847436485002469662555953} a^{8} + \frac{2371149814565798992186308941142886673820053191773850297188629}{5634248665726986477361316145088204873852994586679838200298416} a^{7} + \frac{1228447110551263316326547678513715249642450425553624462551977}{16902745997180959432083948435264614621558983760039514600895248} a^{6} + \frac{823357549895801766206209568329275492598800633958741721865191}{4225686499295239858020987108816153655389745940009878650223812} a^{5} + \frac{1681086029851165936328618905032970615596354226247255574016035}{5634248665726986477361316145088204873852994586679838200298416} a^{4} - \frac{8086788326186993442610117858447395365668304063240606233223557}{16902745997180959432083948435264614621558983760039514600895248} a^{3} + \frac{611923633023188603758961961313559229310804321209754980783135}{2817124332863493238680658072544102436926497293339919100149208} a^{2} + \frac{14136021182259118028413096764269882063343859679170686242627}{889618210377945233267576233434979716924157040002079715836592} a + \frac{45471772282888118000770262492234841188596695999540436822411}{222404552594486308316894058358744929231039260000519928959148}$

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:  \( 104818991244 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T857:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 159 conjugacy class representatives for t18n857 are not computed
Character table for t18n857 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 6.6.820125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 sibling: 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R $18$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ $18$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
3Data not computed
5Data not computed
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.3.2.1$x^{3} - 71$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
71.3.2.1$x^{3} - 71$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$