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Properties

Label 18.18.1045641474...9008.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{24}\cdot 3^{37}\cdot 7^{12}$
Root discriminant $88.21$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (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![-1323, 0, 5381964, 0, -7636356, 0, 4484403, 0, -1428840, 0, 269892, 0, -30789, 0, 2052, 0, -72, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 72*x^16 + 2052*x^14 - 30789*x^12 + 269892*x^10 - 1428840*x^8 + 4484403*x^6 - 7636356*x^4 + 5381964*x^2 - 1323)
 
gp: K = bnfinit(x^18 - 72*x^16 + 2052*x^14 - 30789*x^12 + 269892*x^10 - 1428840*x^8 + 4484403*x^6 - 7636356*x^4 + 5381964*x^2 - 1323, 1)
 

Normalized defining polynomial

\( x^{18} - 72 x^{16} + 2052 x^{14} - 30789 x^{12} + 269892 x^{10} - 1428840 x^{8} + 4484403 x^{6} - 7636356 x^{4} + 5381964 x^{2} - 1323 \)

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:  \(104564147423601601885887702402859008=2^{24}\cdot 3^{37}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.21$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{18} a^{10} - \frac{1}{6} a^{7} - \frac{1}{6} a^{4} - \frac{1}{2} a$, $\frac{1}{18} a^{11} - \frac{1}{6} a^{8} - \frac{1}{6} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{630} a^{12} + \frac{1}{126} a^{10} - \frac{8}{105} a^{8} - \frac{1}{6} a^{7} + \frac{1}{35} a^{6} + \frac{13}{30} a^{4} + \frac{2}{5} a^{2} - \frac{1}{2} a - \frac{3}{10}$, $\frac{1}{1890} a^{13} - \frac{1}{63} a^{11} - \frac{8}{315} a^{9} + \frac{38}{315} a^{7} - \frac{2}{15} a^{5} + \frac{2}{15} a^{3} + \frac{7}{30} a$, $\frac{1}{1890} a^{14} - \frac{1}{630} a^{10} + \frac{8}{315} a^{8} - \frac{1}{6} a^{7} + \frac{16}{105} a^{6} - \frac{11}{30} a^{4} + \frac{7}{30} a^{2} - \frac{1}{2} a$, $\frac{1}{1890} a^{15} - \frac{1}{630} a^{11} + \frac{8}{315} a^{9} - \frac{1}{6} a^{8} + \frac{16}{105} a^{7} - \frac{11}{30} a^{5} + \frac{7}{30} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{115290} a^{16} + \frac{13}{115290} a^{14} - \frac{1}{7686} a^{12} + \frac{107}{4270} a^{10} - \frac{802}{19215} a^{8} - \frac{1}{6} a^{7} - \frac{9}{61} a^{6} - \frac{254}{915} a^{4} + \frac{613}{1830} a^{2} - \frac{1}{2} a + \frac{257}{610}$, $\frac{1}{807030} a^{17} - \frac{17}{80703} a^{15} - \frac{38}{403515} a^{13} - \frac{419}{44835} a^{11} + \frac{407}{38430} a^{9} + \frac{2584}{19215} a^{7} - \frac{1}{6} a^{6} + \frac{6019}{12810} a^{5} - \frac{87}{610} a^{3} + \frac{286}{915} a - \frac{1}{2}$

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:  \( 2175092952180 \) (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

\(\Q(\sqrt{3}) \), 3.3.756.1, 6.6.27433728.1, 9.9.2917096519063104.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
3Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.12.10.2$x^{12} + 35 x^{6} + 441$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$