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Properties

Label 18.10.1055737533...0144.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{24}\cdot 3^{46}\cdot 71$
Root discriminant $52.91$
Ramified primes $2, 3, 71$
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![64, 1728, -3024, -13104, -3816, 15444, 8169, -9036, -5346, 3704, 2079, -1080, -612, 180, 135, -12, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^16 - 12*x^15 + 135*x^14 + 180*x^13 - 612*x^12 - 1080*x^11 + 2079*x^10 + 3704*x^9 - 5346*x^8 - 9036*x^7 + 8169*x^6 + 15444*x^5 - 3816*x^4 - 13104*x^3 - 3024*x^2 + 1728*x + 64)
 
gp: K = bnfinit(x^18 - 18*x^16 - 12*x^15 + 135*x^14 + 180*x^13 - 612*x^12 - 1080*x^11 + 2079*x^10 + 3704*x^9 - 5346*x^8 - 9036*x^7 + 8169*x^6 + 15444*x^5 - 3816*x^4 - 13104*x^3 - 3024*x^2 + 1728*x + 64, 1)
 

Normalized defining polynomial

\( x^{18} - 18 x^{16} - 12 x^{15} + 135 x^{14} + 180 x^{13} - 612 x^{12} - 1080 x^{11} + 2079 x^{10} + 3704 x^{9} - 5346 x^{8} - 9036 x^{7} + 8169 x^{6} + 15444 x^{5} - 3816 x^{4} - 13104 x^{3} - 3024 x^{2} + 1728 x + 64 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10557375333191113763691266310144=2^{24}\cdot 3^{46}\cdot 71\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.91$
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:  $2, 3, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{8} a^{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{1}{16} a^{7} + \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{8} + \frac{1}{16} a^{4}$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{7} - \frac{1}{16} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{13} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} + \frac{1}{32} a^{6} - \frac{5}{32} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{86432} a^{15} - \frac{15}{86432} a^{13} - \frac{427}{43216} a^{12} + \frac{45}{43216} a^{11} - \frac{139}{21608} a^{10} - \frac{2571}{43216} a^{9} + \frac{247}{2701} a^{8} + \frac{1037}{86432} a^{7} - \frac{1959}{21608} a^{6} - \frac{2139}{86432} a^{5} - \frac{9885}{43216} a^{4} + \frac{1153}{10804} a^{3} - \frac{485}{10804} a^{2} + \frac{1327}{5402} a - \frac{1065}{2701}$, $\frac{1}{86432} a^{16} - \frac{15}{86432} a^{14} - \frac{427}{43216} a^{13} + \frac{45}{43216} a^{12} - \frac{139}{21608} a^{11} - \frac{2571}{43216} a^{10} - \frac{725}{21608} a^{9} + \frac{1037}{86432} a^{8} + \frac{371}{10804} a^{7} - \frac{2139}{86432} a^{6} - \frac{4483}{43216} a^{5} + \frac{1153}{10804} a^{4} - \frac{3671}{21608} a^{3} + \frac{1327}{5402} a^{2} - \frac{1065}{2701} a$, $\frac{1}{172864} a^{17} - \frac{1}{172864} a^{16} - \frac{1}{172864} a^{15} - \frac{839}{172864} a^{14} + \frac{367}{86432} a^{13} - \frac{899}{86432} a^{12} - \frac{1663}{86432} a^{11} + \frac{2631}{86432} a^{10} + \frac{7577}{172864} a^{9} + \frac{15351}{172864} a^{8} - \frac{1393}{172864} a^{7} + \frac{2313}{172864} a^{6} - \frac{119}{21608} a^{5} - \frac{2587}{10804} a^{4} + \frac{187}{10804} a^{3} + \frac{1251}{10804} a^{2} - \frac{225}{2701} a + \frac{648}{2701}$

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:  $13$
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:  \( 1120179139.24 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{36})^+\)

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 R R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.12.18.64$x^{12} + 14 x^{11} + 12 x^{10} + 4 x^{9} + 10 x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 16 x - 8$$4$$3$$18$12T51$[2, 2, 2, 2]^{6}$
3Data not computed
$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$$0$Trivial$[\ ]$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
71.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$