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Properties

Label 18.18.3192433564...7008.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 3^{24}\cdot 73^{3}\cdot 577^{4}$
Root discriminant $72.67$
Ramified primes $2, 3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-24303817, 0, 74806896, 0, -71080155, 0, 32196176, 0, -7993866, 0, 1145571, 0, -95631, 0, 4500, 0, -108, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 108*x^16 + 4500*x^14 - 95631*x^12 + 1145571*x^10 - 7993866*x^8 + 32196176*x^6 - 71080155*x^4 + 74806896*x^2 - 24303817)
 
gp: K = bnfinit(x^18 - 108*x^16 + 4500*x^14 - 95631*x^12 + 1145571*x^10 - 7993866*x^8 + 32196176*x^6 - 71080155*x^4 + 74806896*x^2 - 24303817, 1)
 

Normalized defining polynomial

\( x^{18} - 108 x^{16} + 4500 x^{14} - 95631 x^{12} + 1145571 x^{10} - 7993866 x^{8} + 32196176 x^{6} - 71080155 x^{4} + 74806896 x^{2} - 24303817 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3192433564329392175935038228267008=2^{18}\cdot 3^{24}\cdot 73^{3}\cdot 577^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.67$
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:  $2, 3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{73} a^{14} + \frac{28}{73} a^{12} + \frac{34}{73} a^{10} - \frac{19}{73} a^{8} - \frac{21}{73} a^{6} + \frac{27}{73} a^{4}$, $\frac{1}{73} a^{15} + \frac{28}{73} a^{13} + \frac{34}{73} a^{11} - \frac{19}{73} a^{9} - \frac{21}{73} a^{7} + \frac{27}{73} a^{5}$, $\frac{1}{8518846655995484749373753} a^{16} + \frac{34836969789487335767732}{8518846655995484749373753} a^{14} + \frac{3049778314372166006120760}{8518846655995484749373753} a^{12} + \frac{2339651643475434223707914}{8518846655995484749373753} a^{10} + \frac{3207653129538371519834451}{8518846655995484749373753} a^{8} - \frac{1562738791907646081573239}{8518846655995484749373753} a^{6} - \frac{1710835936635477493916820}{8518846655995484749373753} a^{4} - \frac{4570772183877821412950}{116696529534184722594161} a^{2} - \frac{17603535595906133932}{202247018256819276593}$, $\frac{1}{8518846655995484749373753} a^{17} + \frac{34836969789487335767732}{8518846655995484749373753} a^{15} + \frac{3049778314372166006120760}{8518846655995484749373753} a^{13} + \frac{2339651643475434223707914}{8518846655995484749373753} a^{11} + \frac{3207653129538371519834451}{8518846655995484749373753} a^{9} - \frac{1562738791907646081573239}{8518846655995484749373753} a^{7} - \frac{1710835936635477493916820}{8518846655995484749373753} a^{5} - \frac{4570772183877821412950}{116696529534184722594161} a^{3} - \frac{17603535595906133932}{202247018256819276593} a$

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:  $17$
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:  \( 83187809821.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T765:

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 t18n765 are not computed
Character table for t18n765 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 R R $18$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ $18$ $18$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
2Data not computed
$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}$
73.4.3.2$x^{4} - 1825$$4$$1$$3$$C_4$$[\ ]_{4}$
73.6.0.1$x^{6} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
577Data not computed