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Properties

Label 18.6.14610422921...5693.3
Degree $18$
Signature $[6, 6]$
Discriminant $19^{16}\cdot 37^{3}$
Root discriminant $25.01$
Ramified primes $19, 37$
Class number $1$
Class group Trivial
Galois group 18T460

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -11, 81, -213, 391, -424, 35, 584, -1081, 1351, -1401, 1199, -776, 338, -64, -23, 22, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 22*x^16 - 23*x^15 - 64*x^14 + 338*x^13 - 776*x^12 + 1199*x^11 - 1401*x^10 + 1351*x^9 - 1081*x^8 + 584*x^7 + 35*x^6 - 424*x^5 + 391*x^4 - 213*x^3 + 81*x^2 - 11*x - 1)
 
gp: K = bnfinit(x^18 - 7*x^17 + 22*x^16 - 23*x^15 - 64*x^14 + 338*x^13 - 776*x^12 + 1199*x^11 - 1401*x^10 + 1351*x^9 - 1081*x^8 + 584*x^7 + 35*x^6 - 424*x^5 + 391*x^4 - 213*x^3 + 81*x^2 - 11*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 22 x^{16} - 23 x^{15} - 64 x^{14} + 338 x^{13} - 776 x^{12} + 1199 x^{11} - 1401 x^{10} + 1351 x^{9} - 1081 x^{8} + 584 x^{7} + 35 x^{6} - 424 x^{5} + 391 x^{4} - 213 x^{3} + 81 x^{2} - 11 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14610422921440715006545693=19^{16}\cdot 37^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.01$
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:  $19, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2497} a^{16} + \frac{109}{2497} a^{15} + \frac{458}{2497} a^{14} + \frac{897}{2497} a^{13} + \frac{1130}{2497} a^{12} + \frac{343}{2497} a^{11} - \frac{5}{227} a^{10} - \frac{62}{2497} a^{9} + \frac{1142}{2497} a^{8} - \frac{710}{2497} a^{7} + \frac{672}{2497} a^{6} - \frac{775}{2497} a^{5} - \frac{1104}{2497} a^{4} - \frac{1074}{2497} a^{3} - \frac{47}{227} a^{2} - \frac{612}{2497} a + \frac{622}{2497}$, $\frac{1}{803941510323457} a^{17} + \frac{8971567029}{803941510323457} a^{16} - \frac{40069441133261}{803941510323457} a^{15} - \frac{1802838023680}{21728148927661} a^{14} - \frac{248516232911858}{803941510323457} a^{13} + \frac{26096246658971}{73085591847587} a^{12} - \frac{103284232613370}{803941510323457} a^{11} - \frac{39159567876502}{803941510323457} a^{10} - \frac{144713497534576}{803941510323457} a^{9} + \frac{5884375495363}{73085591847587} a^{8} + \frac{3470093422358}{21728148927661} a^{7} + \frac{215535878499307}{803941510323457} a^{6} + \frac{11513396769608}{73085591847587} a^{5} - \frac{341631750496020}{803941510323457} a^{4} + \frac{358353169098759}{803941510323457} a^{3} + \frac{352395879821429}{803941510323457} a^{2} - \frac{16314188110237}{803941510323457} a - \frac{199685629057757}{803941510323457}$

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:  $11$
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:  \( 336986.886015 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T460:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 80 conjugacy class representatives for t18n460 are not computed
Character table for t18n460 is not computed

Intermediate fields

3.3.361.1, \(\Q(\zeta_{19})^+\)

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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ $18$ $18$ R $18$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
19Data not computed
37Data not computed