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Properties

Label 18.14.2414477422...1072.1
Degree $18$
Signature $[14, 2]$
Discriminant $2^{6}\cdot 3^{18}\cdot 7^{15}\cdot 29^{5}$
Root discriminant $48.75$
Ramified primes $2, 3, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 15, 171, -152, -1377, 1197, 2128, -1530, -549, -520, -552, 1485, 287, -693, -18, 116, -9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 9*x^16 + 116*x^15 - 18*x^14 - 693*x^13 + 287*x^12 + 1485*x^11 - 552*x^10 - 520*x^9 - 549*x^8 - 1530*x^7 + 2128*x^6 + 1197*x^5 - 1377*x^4 - 152*x^3 + 171*x^2 + 15*x - 1)
 
gp: K = bnfinit(x^18 - 6*x^17 - 9*x^16 + 116*x^15 - 18*x^14 - 693*x^13 + 287*x^12 + 1485*x^11 - 552*x^10 - 520*x^9 - 549*x^8 - 1530*x^7 + 2128*x^6 + 1197*x^5 - 1377*x^4 - 152*x^3 + 171*x^2 + 15*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 9 x^{16} + 116 x^{15} - 18 x^{14} - 693 x^{13} + 287 x^{12} + 1485 x^{11} - 552 x^{10} - 520 x^{9} - 549 x^{8} - 1530 x^{7} + 2128 x^{6} + 1197 x^{5} - 1377 x^{4} - 152 x^{3} + 171 x^{2} + 15 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2414477422103715329143346811072=2^{6}\cdot 3^{18}\cdot 7^{15}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.75$
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:  $2, 3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{10} + \frac{3}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} + \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{8} + \frac{3}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} + \frac{3}{7} a^{8} - \frac{2}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{14} + \frac{3}{7} a^{7} + \frac{3}{7}$, $\frac{1}{49} a^{15} + \frac{3}{49} a^{14} - \frac{2}{49} a^{13} + \frac{2}{49} a^{12} + \frac{2}{49} a^{11} + \frac{1}{49} a^{10} + \frac{2}{49} a^{9} + \frac{11}{49} a^{8} + \frac{10}{49} a^{7} - \frac{15}{49} a^{6} + \frac{1}{49} a^{5} - \frac{20}{49} a^{4} - \frac{3}{49} a^{3} + \frac{22}{49} a^{2} - \frac{3}{7} a + \frac{13}{49}$, $\frac{1}{49} a^{16} + \frac{3}{49} a^{14} + \frac{1}{49} a^{13} + \frac{3}{49} a^{12} + \frac{2}{49} a^{11} - \frac{1}{49} a^{10} - \frac{2}{49} a^{9} + \frac{19}{49} a^{8} + \frac{4}{49} a^{7} + \frac{18}{49} a^{6} + \frac{5}{49} a^{5} - \frac{13}{49} a^{4} - \frac{18}{49} a^{3} - \frac{17}{49} a^{2} - \frac{1}{49} a - \frac{18}{49}$, $\frac{1}{14789916303846047} a^{17} + \frac{34952767439011}{14789916303846047} a^{16} + \frac{32654131614375}{14789916303846047} a^{15} + \frac{535699083840675}{14789916303846047} a^{14} + \frac{984700168127124}{14789916303846047} a^{13} + \frac{1026594902677671}{14789916303846047} a^{12} + \frac{71836385624651}{2112845186263721} a^{11} + \frac{813114500865658}{14789916303846047} a^{10} + \frac{425286561554195}{14789916303846047} a^{9} + \frac{640472171020627}{14789916303846047} a^{8} - \frac{187090854611301}{1137685869526619} a^{7} - \frac{5275589492406047}{14789916303846047} a^{6} - \frac{2803283399485213}{14789916303846047} a^{5} - \frac{964727192855967}{2112845186263721} a^{4} + \frac{3397987831584455}{14789916303846047} a^{3} + \frac{2567490184158405}{14789916303846047} a^{2} + \frac{5300783509266456}{14789916303846047} a + \frac{621243111625616}{14789916303846047}$

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:  $15$
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:  \( 1880123451.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T766:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 144 conjugacy class representatives for t18n766 are not computed
Character table for t18n766 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.13632439166829.1

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

Sibling fields

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 $18$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
3Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.12.11.2$x^{12} + 56$$12$$1$$11$$D_4 \times C_3$$[\ ]_{12}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$