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Properties

Label 18.0.15452655501...8608.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{18}\cdot 7^{15}\cdot 29^{5}$
Root discriminant $61.42$
Ramified primes $2, 3, 7, 29$
Class number $384$ (GRH)
Class group $[2, 4, 48]$ (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4509, 6858, 18630, 31743, 41094, 50841, 56811, 49026, 35334, 23132, 12933, 4968, 869, -153, -48, 20, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 3*x^16 + 20*x^15 - 48*x^14 - 153*x^13 + 869*x^12 + 4968*x^11 + 12933*x^10 + 23132*x^9 + 35334*x^8 + 49026*x^7 + 56811*x^6 + 50841*x^5 + 41094*x^4 + 31743*x^3 + 18630*x^2 + 6858*x + 4509)
 
gp: K = bnfinit(x^18 - 3*x^17 + 3*x^16 + 20*x^15 - 48*x^14 - 153*x^13 + 869*x^12 + 4968*x^11 + 12933*x^10 + 23132*x^9 + 35334*x^8 + 49026*x^7 + 56811*x^6 + 50841*x^5 + 41094*x^4 + 31743*x^3 + 18630*x^2 + 6858*x + 4509, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 3 x^{16} + 20 x^{15} - 48 x^{14} - 153 x^{13} + 869 x^{12} + 4968 x^{11} + 12933 x^{10} + 23132 x^{9} + 35334 x^{8} + 49026 x^{7} + 56811 x^{6} + 50841 x^{5} + 41094 x^{4} + 31743 x^{3} + 18630 x^{2} + 6858 x + 4509 \)

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:  \(-154526555014637781065174195908608=-\,2^{12}\cdot 3^{18}\cdot 7^{15}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.42$
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 a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{4}{9} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{18} a^{16} - \frac{1}{18} a^{15} - \frac{1}{18} a^{13} - \frac{1}{9} a^{12} + \frac{5}{18} a^{10} + \frac{2}{9} a^{9} + \frac{1}{3} a^{8} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{22724686322699784739360776096511644} a^{17} - \frac{311176159654020451050632089298009}{11362343161349892369680388048255822} a^{16} + \frac{256946578421500260289342351537075}{7574895440899928246453592032170548} a^{15} + \frac{2487324626379338256658265211710297}{22724686322699784739360776096511644} a^{14} - \frac{2742812225305872832496185919599655}{22724686322699784739360776096511644} a^{13} + \frac{75877575202431764454467938180082}{631241286741660687204466002680879} a^{12} + \frac{3463446914190096598347888821668025}{22724686322699784739360776096511644} a^{11} + \frac{8501112286706457277051745542012429}{22724686322699784739360776096511644} a^{10} + \frac{1022833424901260180455982658861}{1262482573483321374408932005361758} a^{9} - \frac{5415153902538730077787759151561393}{11362343161349892369680388048255822} a^{8} - \frac{2251613017427886769214451226999433}{5681171580674946184840194024127911} a^{7} - \frac{1476477101836514241056836537838467}{3787447720449964123226796016085274} a^{6} - \frac{86205001130600539317132448230327}{2524965146966642748817864010723516} a^{5} + \frac{397319810004741771256024750904941}{1262482573483321374408932005361758} a^{4} + \frac{34885533954073577461342146719876}{1893723860224982061613398008042637} a^{3} - \frac{1237160155340140557946961349012573}{2524965146966642748817864010723516} a^{2} + \frac{396734430757577172148625920473881}{2524965146966642748817864010723516} a + \frac{248269280274148984947820543336299}{2524965146966642748817864010723516}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{48}$, which has order $384$ (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:  \( 1152993.7459 \) (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 ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.12.6$x^{12} - 18 x^{10} + 11 x^{8} - 52 x^{6} - x^{4} + 6 x^{2} - 11$$2$$6$$12$12T105$[2, 2, 2, 2]^{12}$
$3$3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.5.2$x^{6} + 58$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$