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Properties

Label 18.18.8071667041...8416.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{24}\cdot 7^{12}\cdot 71^{2}$
Root discriminant $40.36$
Ramified primes $2, 3, 7, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T178

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![631, -2151, -5235, 14118, 15195, -34770, -18260, 42888, 8121, -28242, 762, 9750, -1575, -1650, 396, 124, -36, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 36*x^16 + 124*x^15 + 396*x^14 - 1650*x^13 - 1575*x^12 + 9750*x^11 + 762*x^10 - 28242*x^9 + 8121*x^8 + 42888*x^7 - 18260*x^6 - 34770*x^5 + 15195*x^4 + 14118*x^3 - 5235*x^2 - 2151*x + 631)
 
gp: K = bnfinit(x^18 - 3*x^17 - 36*x^16 + 124*x^15 + 396*x^14 - 1650*x^13 - 1575*x^12 + 9750*x^11 + 762*x^10 - 28242*x^9 + 8121*x^8 + 42888*x^7 - 18260*x^6 - 34770*x^5 + 15195*x^4 + 14118*x^3 - 5235*x^2 - 2151*x + 631, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 36 x^{16} + 124 x^{15} + 396 x^{14} - 1650 x^{13} - 1575 x^{12} + 9750 x^{11} + 762 x^{10} - 28242 x^{9} + 8121 x^{8} + 42888 x^{7} - 18260 x^{6} - 34770 x^{5} + 15195 x^{4} + 14118 x^{3} - 5235 x^{2} - 2151 x + 631 \)

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:  \(80716670418377815414668988416=2^{12}\cdot 3^{24}\cdot 7^{12}\cdot 71^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.36$
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:  $2, 3, 7, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{3}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{16} a^{13} + \frac{1}{16} a^{9} + \frac{1}{8} a^{8} - \frac{5}{16} a^{7} - \frac{5}{16} a^{6} - \frac{1}{4} a^{5} - \frac{3}{16} a^{4} + \frac{5}{16} a^{3} - \frac{3}{16} a^{2} - \frac{3}{8} a + \frac{1}{16}$, $\frac{1}{16} a^{14} + \frac{1}{16} a^{10} + \frac{1}{8} a^{9} + \frac{3}{16} a^{8} + \frac{3}{16} a^{7} + \frac{1}{4} a^{6} + \frac{5}{16} a^{5} - \frac{3}{16} a^{4} + \frac{5}{16} a^{3} - \frac{3}{8} a^{2} - \frac{7}{16} a - \frac{1}{2}$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{13} + \frac{1}{32} a^{11} + \frac{1}{16} a^{10} - \frac{3}{16} a^{9} - \frac{7}{32} a^{8} - \frac{15}{32} a^{7} - \frac{7}{16} a^{6} - \frac{7}{32} a^{5} - \frac{1}{2} a^{4} - \frac{11}{32} a^{3} + \frac{1}{8} a^{2} + \frac{3}{16} a - \frac{1}{32}$, $\frac{1}{1760} a^{16} + \frac{1}{110} a^{15} - \frac{1}{1760} a^{14} + \frac{9}{440} a^{13} + \frac{29}{1760} a^{12} - \frac{29}{880} a^{11} - \frac{69}{880} a^{10} - \frac{19}{352} a^{9} - \frac{111}{1760} a^{8} + \frac{37}{80} a^{7} - \frac{839}{1760} a^{6} - \frac{49}{110} a^{5} + \frac{117}{352} a^{4} - \frac{91}{440} a^{3} - \frac{403}{880} a^{2} - \frac{137}{352} a - \frac{3}{55}$, $\frac{1}{5302828715377600} a^{17} - \frac{255167045779}{1325707178844400} a^{16} - \frac{16381338423007}{1325707178844400} a^{15} - \frac{17009468135353}{1325707178844400} a^{14} + \frac{4860564589613}{1325707178844400} a^{13} - \frac{42178743322913}{2651414357688800} a^{12} + \frac{390100712608063}{5302828715377600} a^{11} + \frac{482726768937331}{5302828715377600} a^{10} + \frac{6417969285729}{74687728385600} a^{9} - \frac{1236591088876909}{5302828715377600} a^{8} + \frac{1028578304787719}{2651414357688800} a^{7} + \frac{311730454608847}{2651414357688800} a^{6} + \frac{363037061411009}{2651414357688800} a^{5} + \frac{124504959741631}{331426794711100} a^{4} - \frac{1963688715034353}{5302828715377600} a^{3} - \frac{2216199357834893}{5302828715377600} a^{2} - \frac{43057208380063}{2651414357688800} a + \frac{2164252034145387}{5302828715377600}$

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:  \( 423732852.769 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T178:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 24 conjugacy class representatives for t18n178
Character table for t18n178 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{7})^+\), 9.9.62523502209.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
71Data not computed