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Properties

Label 18.18.3830034706...5792.3
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 19^{16}\cdot 37^{3}$
Root discriminant $50.01$
Ramified primes $2, 19, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T264

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-50653, 0, 213564, 0, -369260, 0, 339994, 0, -181204, 0, 56951, 0, -10328, 0, 1023, 0, -51, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 51*x^16 + 1023*x^14 - 10328*x^12 + 56951*x^10 - 181204*x^8 + 339994*x^6 - 369260*x^4 + 213564*x^2 - 50653)
 
gp: K = bnfinit(x^18 - 51*x^16 + 1023*x^14 - 10328*x^12 + 56951*x^10 - 181204*x^8 + 339994*x^6 - 369260*x^4 + 213564*x^2 - 50653, 1)
 

Normalized defining polynomial

\( x^{18} - 51 x^{16} + 1023 x^{14} - 10328 x^{12} + 56951 x^{10} - 181204 x^{8} + 339994 x^{6} - 369260 x^{4} + 213564 x^{2} - 50653 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3830034706318154794675914145792=2^{18}\cdot 19^{16}\cdot 37^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.01$
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:  $2, 19, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{10} - \frac{1}{7} a^{8} - \frac{2}{7} a^{6} - \frac{1}{7} a^{2} - \frac{3}{7}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{11} - \frac{1}{7} a^{9} - \frac{2}{7} a^{7} - \frac{1}{7} a^{3} - \frac{3}{7} a$, $\frac{1}{259} a^{14} - \frac{2}{37} a^{12} + \frac{61}{259} a^{10} + \frac{69}{259} a^{8} - \frac{103}{259} a^{6} - \frac{15}{259} a^{4} + \frac{75}{259} a^{2} - \frac{2}{7}$, $\frac{1}{259} a^{15} - \frac{2}{37} a^{13} + \frac{61}{259} a^{11} + \frac{69}{259} a^{9} - \frac{103}{259} a^{7} - \frac{15}{259} a^{5} + \frac{75}{259} a^{3} - \frac{2}{7} a$, $\frac{1}{1999195877} a^{16} + \frac{1829747}{1999195877} a^{14} - \frac{2070432}{285599411} a^{12} + \frac{598167747}{1999195877} a^{10} - \frac{180344356}{1999195877} a^{8} + \frac{383742192}{1999195877} a^{6} - \frac{874513869}{1999195877} a^{4} + \frac{1855582}{54032321} a^{2} + \frac{295616}{1460333}$, $\frac{1}{1999195877} a^{17} + \frac{1829747}{1999195877} a^{15} - \frac{2070432}{285599411} a^{13} + \frac{598167747}{1999195877} a^{11} - \frac{180344356}{1999195877} a^{9} + \frac{383742192}{1999195877} a^{7} - \frac{874513869}{1999195877} a^{5} + \frac{1855582}{54032321} a^{3} + \frac{295616}{1460333} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 2205011395.69 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T264:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 32 conjugacy class representatives for t18n264
Character table for t18n264 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 R ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ $18$ $18$ R $18$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ 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
2Data not computed
19Data not computed
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$