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Properties

Label 18.0.87651540491...2768.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{30}\cdot 3^{18}\cdot 7^{12}\cdot 13^{5}\cdot 41$
Root discriminant $87.35$
Ramified primes $2, 3, 7, 13, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T926

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![832, -7488, 28080, -56160, 63180, -37908, 9781, -1824, 4104, -4104, 1539, 0, 32, -96, 72, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 72*x^14 - 96*x^13 + 32*x^12 + 1539*x^10 - 4104*x^9 + 4104*x^8 - 1824*x^7 + 9781*x^6 - 37908*x^5 + 63180*x^4 - 56160*x^3 + 28080*x^2 - 7488*x + 832)
 
gp: K = bnfinit(x^18 + 72*x^14 - 96*x^13 + 32*x^12 + 1539*x^10 - 4104*x^9 + 4104*x^8 - 1824*x^7 + 9781*x^6 - 37908*x^5 + 63180*x^4 - 56160*x^3 + 28080*x^2 - 7488*x + 832, 1)
 

Normalized defining polynomial

\( x^{18} + 72 x^{14} - 96 x^{13} + 32 x^{12} + 1539 x^{10} - 4104 x^{9} + 4104 x^{8} - 1824 x^{7} + 9781 x^{6} - 37908 x^{5} + 63180 x^{4} - 56160 x^{3} + 28080 x^{2} - 7488 x + 832 \)

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:  \(-87651540491153387182800195727392768=-\,2^{30}\cdot 3^{18}\cdot 7^{12}\cdot 13^{5}\cdot 41\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.35$
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:  $2, 3, 7, 13, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} + \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{64} a^{14} + \frac{3}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{3}{64} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{5}{64} a^{2} - \frac{5}{16} a + \frac{1}{16}$, $\frac{1}{512} a^{15} + \frac{1}{256} a^{14} + \frac{3}{64} a^{13} + \frac{7}{32} a^{12} - \frac{25}{64} a^{11} - \frac{15}{32} a^{10} - \frac{7}{16} a^{9} + \frac{3}{8} a^{8} - \frac{125}{512} a^{7} + \frac{127}{256} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{5}{512} a^{3} - \frac{5}{256} a^{2} - \frac{57}{128} a + \frac{17}{64}$, $\frac{1}{4096} a^{16} - \frac{1}{1024} a^{15} + \frac{3}{1024} a^{14} - \frac{1}{128} a^{13} + \frac{19}{512} a^{12} - \frac{9}{64} a^{11} + \frac{27}{64} a^{10} - \frac{1}{8} a^{9} + \frac{771}{4096} a^{8} - \frac{261}{1024} a^{7} + \frac{275}{1024} a^{6} - \frac{1}{2} a^{5} + \frac{1285}{4096} a^{4} + \frac{251}{512} a^{3} + \frac{107}{512} a^{2} + \frac{63}{128} a + \frac{13}{256}$, $\frac{1}{32768} a^{17} - \frac{1}{16384} a^{16} + \frac{1}{8192} a^{15} - \frac{1}{4096} a^{14} + \frac{11}{4096} a^{13} - \frac{17}{2048} a^{12} + \frac{9}{512} a^{11} - \frac{9}{256} a^{10} + \frac{3843}{32768} a^{9} - \frac{5895}{16384} a^{8} - \frac{1271}{8192} a^{7} + \frac{1043}{4096} a^{6} - \frac{6907}{32768} a^{5} + \frac{4337}{16384} a^{4} + \frac{1633}{4096} a^{3} + \frac{1001}{2048} a^{2} - \frac{247}{2048} a + \frac{13}{1024}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -1 \) (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:  \( 5955889539.76 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T926:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1119744
The 267 conjugacy class representatives for t18n926 are not computed
Character table for t18n926 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.0.1997632.1

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 $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $18$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ $18$ R ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.12.24.45$x^{12} - 8 x^{11} - 6 x^{10} + 4 x^{9} - 6 x^{8} - 12 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{2} + 16 x - 8$$4$$3$$24$12T89$[2, 2, 2, 3, 3]^{6}$
3Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
41Data not computed