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Properties

Label 18.6.11247392561...9689.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{24}\cdot 73^{5}\cdot 577^{3}$
Root discriminant $41.11$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T401

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10259, 46071, 15651, -68277, 684, 40296, -18363, -14829, 10632, 6973, -13365, 9033, -3554, 834, -99, -10, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 9*x^16 - 10*x^15 - 99*x^14 + 834*x^13 - 3554*x^12 + 9033*x^11 - 13365*x^10 + 6973*x^9 + 10632*x^8 - 14829*x^7 - 18363*x^6 + 40296*x^5 + 684*x^4 - 68277*x^3 + 15651*x^2 + 46071*x + 10259)
 
gp: K = bnfinit(x^18 - 3*x^17 + 9*x^16 - 10*x^15 - 99*x^14 + 834*x^13 - 3554*x^12 + 9033*x^11 - 13365*x^10 + 6973*x^9 + 10632*x^8 - 14829*x^7 - 18363*x^6 + 40296*x^5 + 684*x^4 - 68277*x^3 + 15651*x^2 + 46071*x + 10259, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 9 x^{16} - 10 x^{15} - 99 x^{14} + 834 x^{13} - 3554 x^{12} + 9033 x^{11} - 13365 x^{10} + 6973 x^{9} + 10632 x^{8} - 14829 x^{7} - 18363 x^{6} + 40296 x^{5} + 684 x^{4} - 68277 x^{3} + 15651 x^{2} + 46071 x + 10259 \)

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:  \(112473925614060022797462679689=3^{24}\cdot 73^{5}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{19} a^{16} + \frac{9}{19} a^{15} + \frac{4}{19} a^{14} + \frac{9}{19} a^{13} - \frac{6}{19} a^{12} - \frac{8}{19} a^{11} - \frac{8}{19} a^{10} - \frac{1}{19} a^{9} - \frac{9}{19} a^{8} + \frac{5}{19} a^{7} + \frac{5}{19} a^{6} - \frac{1}{19} a^{5} + \frac{3}{19} a^{4} - \frac{6}{19} a^{3} + \frac{7}{19} a^{2} - \frac{8}{19} a + \frac{1}{19}$, $\frac{1}{50540057190715295725597930510086143124262517} a^{17} + \frac{1259013506292272851001535129513785114789234}{50540057190715295725597930510086143124262517} a^{16} + \frac{14908362800523728485085833929476954510176122}{50540057190715295725597930510086143124262517} a^{15} - \frac{9237437066253766053717464676036839875107607}{50540057190715295725597930510086143124262517} a^{14} + \frac{13852144027491823257668941362277474857109245}{50540057190715295725597930510086143124262517} a^{13} + \frac{15765020849805993149347250785933689068718280}{50540057190715295725597930510086143124262517} a^{12} - \frac{7735303403933897805115770700754020477647643}{50540057190715295725597930510086143124262517} a^{11} - \frac{22618104773732442470013008263036947512847146}{50540057190715295725597930510086143124262517} a^{10} + \frac{8488099030585317925388864924546575637130576}{50540057190715295725597930510086143124262517} a^{9} + \frac{25230553513100403217764255081859126142872327}{50540057190715295725597930510086143124262517} a^{8} + \frac{6560117802163142188186670017260014151450194}{50540057190715295725597930510086143124262517} a^{7} + \frac{19368384616546313161238982868870919871990320}{50540057190715295725597930510086143124262517} a^{6} - \frac{73438741198563118356918427256345283264060}{50540057190715295725597930510086143124262517} a^{5} + \frac{8511914535412875964830775364174333255622212}{50540057190715295725597930510086143124262517} a^{4} + \frac{19484380019732861907462225650004340380493612}{50540057190715295725597930510086143124262517} a^{3} - \frac{25068293607969049484265605656268316180269935}{50540057190715295725597930510086143124262517} a^{2} - \frac{5454777680193448951441906138620474720611585}{50540057190715295725597930510086143124262517} a + \frac{4658951613768518384412503285063696461241501}{50540057190715295725597930510086143124262517}$

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:  \( 32893522.2121 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T401:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2592
The 44 conjugacy class representatives for t18n401
Character table for t18n401 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.2.276355881.2, 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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.6.3.1$x^{6} - 146 x^{4} + 5329 x^{2} - 76247332$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
577Data not computed