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Properties

Label 18.18.5787001694...1409.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{24}\cdot 7^{12}\cdot 23^{6}$
Root discriminant $45.03$
Ramified primes $3, 7, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_4 \times C_3$ (as 18T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, 0, -945, -1512, 7182, 19026, -435, -32634, -13965, 20818, 12159, -5880, -4107, 714, 630, -28, -42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 42*x^16 - 28*x^15 + 630*x^14 + 714*x^13 - 4107*x^12 - 5880*x^11 + 12159*x^10 + 20818*x^9 - 13965*x^8 - 32634*x^7 - 435*x^6 + 19026*x^5 + 7182*x^4 - 1512*x^3 - 945*x^2 + 27)
 
gp: K = bnfinit(x^18 - 42*x^16 - 28*x^15 + 630*x^14 + 714*x^13 - 4107*x^12 - 5880*x^11 + 12159*x^10 + 20818*x^9 - 13965*x^8 - 32634*x^7 - 435*x^6 + 19026*x^5 + 7182*x^4 - 1512*x^3 - 945*x^2 + 27, 1)
 

Normalized defining polynomial

\( x^{18} - 42 x^{16} - 28 x^{15} + 630 x^{14} + 714 x^{13} - 4107 x^{12} - 5880 x^{11} + 12159 x^{10} + 20818 x^{9} - 13965 x^{8} - 32634 x^{7} - 435 x^{6} + 19026 x^{5} + 7182 x^{4} - 1512 x^{3} - 945 x^{2} + 27 \)

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:  \(578700169474787302846561871409=3^{24}\cdot 7^{12}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.03$
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:  $3, 7, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3}$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} + \frac{1}{12} a^{8} - \frac{1}{4} a^{6} - \frac{1}{3} a^{5} + \frac{5}{12} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{6} a^{8} + \frac{1}{12} a^{7} - \frac{1}{6} a^{6} + \frac{1}{4} a^{5} - \frac{5}{12} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{12} a^{14} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{72} a^{15} - \frac{1}{24} a^{14} - \frac{1}{24} a^{13} + \frac{1}{36} a^{12} - \frac{1}{24} a^{11} - \frac{1}{12} a^{10} + \frac{1}{24} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} + \frac{1}{18} a^{6} - \frac{1}{8} a^{5} + \frac{1}{6} a^{4} + \frac{5}{24} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{72} a^{16} - \frac{1}{72} a^{13} - \frac{1}{24} a^{12} + \frac{1}{24} a^{11} + \frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{5}{24} a^{8} - \frac{11}{72} a^{7} - \frac{5}{24} a^{6} + \frac{1}{24} a^{5} - \frac{1}{24} a^{4} - \frac{1}{24} a^{3} + \frac{1}{4} a^{2} - \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{7958311920849096} a^{17} - \frac{12695741748809}{1989577980212274} a^{16} + \frac{39824725527}{442128440047172} a^{15} + \frac{28997128818707}{7958311920849096} a^{14} - \frac{91159206680875}{7958311920849096} a^{13} + \frac{92487240617719}{2652770640283032} a^{12} - \frac{188682335503459}{2652770640283032} a^{11} - \frac{17417088275551}{2652770640283032} a^{10} + \frac{61362717526497}{884256880094344} a^{9} + \frac{1354105638193195}{7958311920849096} a^{8} + \frac{20189536976269}{7958311920849096} a^{7} + \frac{415336002190775}{2652770640283032} a^{6} + \frac{97318672677503}{2652770640283032} a^{5} + \frac{98690383790435}{884256880094344} a^{4} + \frac{175668518119577}{1326385320141516} a^{3} - \frac{369012179554251}{884256880094344} a^{2} - \frac{332462724931799}{884256880094344} a - \frac{103303287544771}{442128440047172}$

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:  \( 1768913822.57 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times A_4$ (as 18T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $A_4 \times C_3$
Character table for $A_4 \times C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.3969.1, \(\Q(\zeta_{9})^+\), 3.3.3969.2, 6.6.8333316369.1, 9.9.62523502209.1

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

Sibling fields

Galois closure: data not computed
Degree 12 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$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}$
$23$23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.1$x^{6} - 46 x^{4} + 529 x^{2} - 194672$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$