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Properties

Label 18.12.2214758615...8399.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,7^{12}\cdot 113^{3}\cdot 223^{3}$
Root discriminant $19.81$
Ramified primes $7, 113, 223$
Class number $1$
Class group Trivial
Galois group 18T207

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125, -850, 2040, -1504, -1547, 2830, -415, -992, -41, 272, 290, -240, 44, 17, -60, 33, 4, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 4*x^16 + 33*x^15 - 60*x^14 + 17*x^13 + 44*x^12 - 240*x^11 + 290*x^10 + 272*x^9 - 41*x^8 - 992*x^7 - 415*x^6 + 2830*x^5 - 1547*x^4 - 1504*x^3 + 2040*x^2 - 850*x + 125)
 
gp: K = bnfinit(x^18 - 6*x^17 + 4*x^16 + 33*x^15 - 60*x^14 + 17*x^13 + 44*x^12 - 240*x^11 + 290*x^10 + 272*x^9 - 41*x^8 - 992*x^7 - 415*x^6 + 2830*x^5 - 1547*x^4 - 1504*x^3 + 2040*x^2 - 850*x + 125, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 4 x^{16} + 33 x^{15} - 60 x^{14} + 17 x^{13} + 44 x^{12} - 240 x^{11} + 290 x^{10} + 272 x^{9} - 41 x^{8} - 992 x^{7} - 415 x^{6} + 2830 x^{5} - 1547 x^{4} - 1504 x^{3} + 2040 x^{2} - 850 x + 125 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-221475861541215709988399=-\,7^{12}\cdot 113^{3}\cdot 223^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.81$
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:  $7, 113, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{15} - \frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{104878351780708075} a^{17} - \frac{8044743384441406}{104878351780708075} a^{16} + \frac{26305526926599654}{104878351780708075} a^{15} - \frac{44946915773333667}{104878351780708075} a^{14} + \frac{10321976759760048}{20975670356141615} a^{13} - \frac{51539238866702033}{104878351780708075} a^{12} + \frac{44103142331833869}{104878351780708075} a^{11} - \frac{3319183587825773}{20975670356141615} a^{10} + \frac{3359843609561413}{20975670356141615} a^{9} + \frac{18079719662329872}{104878351780708075} a^{8} + \frac{22234733227516259}{104878351780708075} a^{7} - \frac{24770068316119617}{104878351780708075} a^{6} - \frac{8673419715533178}{20975670356141615} a^{5} + \frac{10454995483303596}{20975670356141615} a^{4} - \frac{23624739810799022}{104878351780708075} a^{3} + \frac{49181708423145671}{104878351780708075} a^{2} + \frac{5269293345289853}{20975670356141615} a + \frac{1083563391337945}{4195134071228323}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $14$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 136529.514651 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T207:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 17 conjugacy class representatives for t18n207
Character table for t18n207

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.60502799.2, 9.7.2964637151.1 x3

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

Sibling fields

Degree 9 sibling: data not computed
Degree 12 sibling: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$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}$
113Data not computed
223Data not computed