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Properties

Label 18.12.1248805752...3503.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,7^{15}\cdot 138041^{3}$
Root discriminant $36.38$
Ramified primes $7, 138041$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-419, -1781, -16, 11105, 19952, 4506, -22265, -22929, 132, 12015, 5221, -1828, -1816, -112, 262, 57, -22, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 22*x^16 + 57*x^15 + 262*x^14 - 112*x^13 - 1816*x^12 - 1828*x^11 + 5221*x^10 + 12015*x^9 + 132*x^8 - 22929*x^7 - 22265*x^6 + 4506*x^5 + 19952*x^4 + 11105*x^3 - 16*x^2 - 1781*x - 419)
 
gp: K = bnfinit(x^18 - 4*x^17 - 22*x^16 + 57*x^15 + 262*x^14 - 112*x^13 - 1816*x^12 - 1828*x^11 + 5221*x^10 + 12015*x^9 + 132*x^8 - 22929*x^7 - 22265*x^6 + 4506*x^5 + 19952*x^4 + 11105*x^3 - 16*x^2 - 1781*x - 419, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 22 x^{16} + 57 x^{15} + 262 x^{14} - 112 x^{13} - 1816 x^{12} - 1828 x^{11} + 5221 x^{10} + 12015 x^{9} + 132 x^{8} - 22929 x^{7} - 22265 x^{6} + 4506 x^{5} + 19952 x^{4} + 11105 x^{3} - 16 x^{2} - 1781 x - 419 \)

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:  \(-12488057521927227046014543503=-\,7^{15}\cdot 138041^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.38$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{39642161692051506950066501} a^{17} + \frac{14948064444599337570624671}{39642161692051506950066501} a^{16} - \frac{2019594628317800557626963}{39642161692051506950066501} a^{15} - \frac{4646170445763890728961920}{39642161692051506950066501} a^{14} + \frac{853984379448440021877162}{39642161692051506950066501} a^{13} + \frac{14940169133003940215259277}{39642161692051506950066501} a^{12} + \frac{17420226562933488118461272}{39642161692051506950066501} a^{11} - \frac{2907159725063019920770190}{39642161692051506950066501} a^{10} + \frac{2858816094060704556912373}{39642161692051506950066501} a^{9} + \frac{903862843530077817854496}{39642161692051506950066501} a^{8} - \frac{17118709468962035631786558}{39642161692051506950066501} a^{7} - \frac{2586989995143741480955052}{39642161692051506950066501} a^{6} - \frac{10906657332438876551819437}{39642161692051506950066501} a^{5} - \frac{460136273702863244673367}{39642161692051506950066501} a^{4} + \frac{2167180429178955765361863}{39642161692051506950066501} a^{3} - \frac{6764441543811187261225254}{39642161692051506950066501} a^{2} + \frac{10039002910501817972865610}{39642161692051506950066501} a + \frac{42188678255977720737761}{94611364420170660978679}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $14$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 20999582.8767 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n765 are not computed
Character table for t18n765 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ $18$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $18$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
7Data not computed
138041Data not computed