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Properties

Label 18.0.90466292782...2183.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 138041^{2}$
Root discriminant $18.85$
Ramified primes $7, 138041$
Class number $2$
Class group $[2]$
Galois group 18T286

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -14, 75, -176, 389, -521, 672, -593, 472, -186, 69, -37, 41, -16, 13, 5, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 4*x^16 + 5*x^15 + 13*x^14 - 16*x^13 + 41*x^12 - 37*x^11 + 69*x^10 - 186*x^9 + 472*x^8 - 593*x^7 + 672*x^6 - 521*x^5 + 389*x^4 - 176*x^3 + 75*x^2 - 14*x + 1)
 
gp: K = bnfinit(x^18 - 2*x^17 + 4*x^16 + 5*x^15 + 13*x^14 - 16*x^13 + 41*x^12 - 37*x^11 + 69*x^10 - 186*x^9 + 472*x^8 - 593*x^7 + 672*x^6 - 521*x^5 + 389*x^4 - 176*x^3 + 75*x^2 - 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 4 x^{16} + 5 x^{15} + 13 x^{14} - 16 x^{13} + 41 x^{12} - 37 x^{11} + 69 x^{10} - 186 x^{9} + 472 x^{8} - 593 x^{7} + 672 x^{6} - 521 x^{5} + 389 x^{4} - 176 x^{3} + 75 x^{2} - 14 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-90466292782051905202183=-\,7^{15}\cdot 138041^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.85$
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:  $7, 138041$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6531246438175038513} a^{17} + \frac{931464178769619980}{6531246438175038513} a^{16} + \frac{87429576334092584}{2177082146058346171} a^{15} - \frac{326236891542603581}{2177082146058346171} a^{14} - \frac{174334883603684618}{2177082146058346171} a^{13} + \frac{256521342556548361}{2177082146058346171} a^{12} + \frac{647922219305653433}{6531246438175038513} a^{11} + \frac{733961953426405281}{2177082146058346171} a^{10} - \frac{1279602757500014492}{6531246438175038513} a^{9} - \frac{3227982328050994090}{6531246438175038513} a^{8} - \frac{11417584673749534}{2177082146058346171} a^{7} + \frac{124706503480878327}{2177082146058346171} a^{6} + \frac{425463774262945701}{2177082146058346171} a^{5} + \frac{148876985171158523}{6531246438175038513} a^{4} - \frac{2594598260087581102}{6531246438175038513} a^{3} + \frac{2026193310452020088}{6531246438175038513} a^{2} + \frac{3004196031686272049}{6531246438175038513} a + \frac{577968673645156053}{2177082146058346171}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{259342831713518151}{2177082146058346171} a^{17} + \frac{459630731190099236}{2177082146058346171} a^{16} - \frac{2944773866872619660}{6531246438175038513} a^{15} - \frac{1384032204875034442}{2177082146058346171} a^{14} - \frac{11736984314874132778}{6531246438175038513} a^{13} + \frac{9218801385610646651}{6531246438175038513} a^{12} - \frac{30624665720251911668}{6531246438175038513} a^{11} + \frac{26416311265897582859}{6531246438175038513} a^{10} - \frac{53943261266002365113}{6531246438175038513} a^{9} + \frac{139835447222656059148}{6531246438175038513} a^{8} - \frac{345162374807520876059}{6531246438175038513} a^{7} + \frac{412426255608845311115}{6531246438175038513} a^{6} - \frac{512228804703543861880}{6531246438175038513} a^{5} + \frac{405136398204612428458}{6531246438175038513} a^{4} - \frac{313573272722086409221}{6531246438175038513} a^{3} + \frac{143551287245058298858}{6531246438175038513} a^{2} - \frac{22941751901084899916}{2177082146058346171} a + \frac{13496060608073540563}{6531246438175038513} \) (order $14$)
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:  \( 22017.7641526 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T286:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 34 conjugacy class representatives for t18n286
Character table for t18n286 is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 9.9.16240385609.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
7Data not computed
138041Data not computed