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Properties

Label 18.0.20224369827...7312.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{31}\cdot 3^{18}\cdot 79^{6}$
Root discriminant $42.47$
Ramified primes $2, 3, 79$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T903

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3392, 1152, 4464, 3072, 588, -1512, -3527, -2088, 918, 2032, 687, -720, -496, 120, 135, -8, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^16 - 8*x^15 + 135*x^14 + 120*x^13 - 496*x^12 - 720*x^11 + 687*x^10 + 2032*x^9 + 918*x^8 - 2088*x^7 - 3527*x^6 - 1512*x^5 + 588*x^4 + 3072*x^3 + 4464*x^2 + 1152*x + 3392)
 
gp: K = bnfinit(x^18 - 18*x^16 - 8*x^15 + 135*x^14 + 120*x^13 - 496*x^12 - 720*x^11 + 687*x^10 + 2032*x^9 + 918*x^8 - 2088*x^7 - 3527*x^6 - 1512*x^5 + 588*x^4 + 3072*x^3 + 4464*x^2 + 1152*x + 3392, 1)
 

Normalized defining polynomial

\( x^{18} - 18 x^{16} - 8 x^{15} + 135 x^{14} + 120 x^{13} - 496 x^{12} - 720 x^{11} + 687 x^{10} + 2032 x^{9} + 918 x^{8} - 2088 x^{7} - 3527 x^{6} - 1512 x^{5} + 588 x^{4} + 3072 x^{3} + 4464 x^{2} + 1152 x + 3392 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-202243698273060960219338637312=-\,2^{31}\cdot 3^{18}\cdot 79^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.47$
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:  $2, 3, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{7} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{8} a^{5} - \frac{3}{16} a^{4} - \frac{1}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{32} a^{9} - \frac{1}{32} a^{7} - \frac{1}{16} a^{6} + \frac{3}{32} a^{5} + \frac{1}{8} a^{4} - \frac{7}{32} a^{3} - \frac{5}{16} a^{2} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{7} - \frac{1}{32} a^{6} - \frac{1}{8} a^{5} - \frac{7}{32} a^{4} - \frac{1}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{1}{64} a^{9} - \frac{1}{64} a^{8} - \frac{3}{64} a^{7} + \frac{1}{64} a^{6} + \frac{5}{64} a^{5} - \frac{3}{64} a^{4} - \frac{5}{32} a^{3} + \frac{1}{16} a^{2} + \frac{1}{8} a$, $\frac{1}{128} a^{12} + \frac{1}{64} a^{8} - \frac{1}{16} a^{6} + \frac{21}{128} a^{4} - \frac{1}{2} a^{2} + \frac{3}{8}$, $\frac{1}{128} a^{13} - \frac{1}{64} a^{9} - \frac{1}{32} a^{7} - \frac{1}{16} a^{6} + \frac{9}{128} a^{5} + \frac{1}{8} a^{4} + \frac{7}{32} a^{3} - \frac{5}{16} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{256} a^{14} - \frac{1}{256} a^{13} + \frac{1}{128} a^{10} - \frac{1}{128} a^{9} - \frac{1}{32} a^{8} + \frac{1}{32} a^{7} - \frac{11}{256} a^{6} - \frac{21}{256} a^{5} - \frac{1}{4} a^{3} + \frac{1}{16} a^{2} - \frac{3}{16} a - \frac{1}{2}$, $\frac{1}{2048} a^{15} + \frac{1}{2048} a^{13} - \frac{1}{1024} a^{12} - \frac{3}{1024} a^{11} - \frac{1}{256} a^{10} + \frac{9}{1024} a^{9} - \frac{3}{512} a^{8} + \frac{21}{2048} a^{7} + \frac{7}{256} a^{6} + \frac{93}{2048} a^{5} - \frac{1}{1024} a^{4} + \frac{29}{128} a^{3} - \frac{13}{32} a^{2} + \frac{27}{128} a + \frac{25}{64}$, $\frac{1}{2048} a^{16} + \frac{1}{2048} a^{14} - \frac{1}{1024} a^{13} - \frac{3}{1024} a^{12} - \frac{1}{256} a^{11} + \frac{9}{1024} a^{10} - \frac{3}{512} a^{9} + \frac{21}{2048} a^{8} + \frac{7}{256} a^{7} + \frac{93}{2048} a^{6} - \frac{1}{1024} a^{5} + \frac{29}{128} a^{4} + \frac{3}{32} a^{3} + \frac{27}{128} a^{2} - \frac{7}{64} a$, $\frac{1}{8192} a^{17} + \frac{1}{8192} a^{16} - \frac{1}{8192} a^{15} - \frac{1}{8192} a^{14} + \frac{11}{4096} a^{13} - \frac{5}{4096} a^{12} + \frac{11}{4096} a^{11} - \frac{53}{4096} a^{10} + \frac{37}{8192} a^{9} + \frac{229}{8192} a^{8} - \frac{405}{8192} a^{7} - \frac{405}{8192} a^{6} + \frac{109}{2048} a^{5} - \frac{123}{2048} a^{4} + \frac{29}{512} a^{3} + \frac{213}{512} a^{2} - \frac{25}{128} a - \frac{25}{128}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1}{2048} a^{15} + \frac{15}{2048} a^{13} + \frac{1}{1024} a^{12} - \frac{45}{1024} a^{11} - \frac{3}{256} a^{10} + \frac{151}{1024} a^{9} + \frac{27}{512} a^{8} - \frac{693}{2048} a^{7} - \frac{51}{256} a^{6} + \frac{1107}{2048} a^{5} + \frac{657}{1024} a^{4} - \frac{29}{128} a^{3} - \frac{27}{32} a^{2} - \frac{75}{128} a - \frac{41}{64} \) (order $4$)
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:  \( 1456016777.95 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T903:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 559872
The 174 conjugacy class representatives for t18n903 are not computed
Character table for t18n903 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.3.316.1, 6.0.399424.1

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 ${\href{/LocalNumberField/5.9.0.1}{9} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ $18$ ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.8.6$x^{4} + 6 x^{2} + 4 x + 2$$4$$1$$8$$D_{4}$$[2, 3]^{2}$
2.4.9.4$x^{4} + 2 x^{2} + 10$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
3Data not computed
$79$79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$
79.8.4.1$x^{8} + 37446 x^{4} - 493039 x^{2} + 350550729$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$