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Properties

Label 18.2.92737858824...4592.2
Degree $18$
Signature $[2, 8]$
Discriminant $2^{33}\cdot 3^{32}\cdot 17^{12}$
Root discriminant $166.11$
Ramified primes $2, 3, 17$
Class number $27$ (GRH)
Class group $[3, 9]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23887872, 0, 0, 0, 0, 0, 788004, 0, 0, 0, 0, 0, 1074, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 1074*x^12 + 788004*x^6 - 23887872)
 
gp: K = bnfinit(x^18 + 1074*x^12 + 788004*x^6 - 23887872, 1)
 

Normalized defining polynomial

\( x^{18} + 1074 x^{12} + 788004 x^{6} - 23887872 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9273785882460929937291206301359215214592=2^{33}\cdot 3^{32}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $166.11$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6}$, $\frac{1}{6} a^{7}$, $\frac{1}{6} a^{8}$, $\frac{1}{18} a^{9} + \frac{1}{3} a^{3}$, $\frac{1}{18} a^{10} + \frac{1}{3} a^{4}$, $\frac{1}{18} a^{11} + \frac{1}{3} a^{5}$, $\frac{1}{6936876} a^{12} - \frac{1361}{385382} a^{6} + \frac{81445}{192691}$, $\frac{1}{27747504} a^{13} + \frac{381299}{4624584} a^{7} + \frac{81445}{770764} a$, $\frac{1}{332970048} a^{14} + \frac{381299}{55495008} a^{8} - \frac{2230847}{9249168} a^{2}$, $\frac{1}{3995640576} a^{15} + \frac{381299}{665940096} a^{9} - \frac{39227519}{110990016} a^{3}$, $\frac{1}{15982562304} a^{16} + \frac{37377971}{2663760384} a^{10} + \frac{108759169}{443960064} a^{4}$, $\frac{1}{191790747648} a^{17} - \frac{406582093}{31965124608} a^{11} - \frac{335200895}{5327520768} a^{5}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}\times C_{9}$, which has order $27$ (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:  $9$
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:  \( 642409524585.5234 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), 3.1.867.1, 6.2.384864768.2, 9.1.66498764695119936.21

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$2$2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.12.22.27$x^{12} + 2 x^{6} + 4$$6$$2$$22$$C_6\times S_3$$[3]_{3}^{6}$
$3$3.6.10.3$x^{6} + 36$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
3.12.22.93$x^{12} + 81 x^{11} - 81 x^{10} + 51 x^{9} + 63 x^{8} - 108 x^{7} - 93 x^{6} + 108 x^{5} + 72 x^{3} + 108 x^{2} - 81 x + 90$$6$$2$$22$$C_6\times S_3$$[2, 5/2]_{2}^{2}$
$17$17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$