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Properties

Label 18.16.3426722037...0000.1
Degree $18$
Signature $[16, 1]$
Discriminant $-\,2^{12}\cdot 5^{9}\cdot 37^{6}\cdot 166946971879$
Root discriminant $49.70$
Ramified primes $2, 5, 37, 166946971879$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T903

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 17, 110, 325, 341, -371, -1177, -682, 591, 973, 256, -440, -316, 85, 110, -6, -17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 17*x^16 - 6*x^15 + 110*x^14 + 85*x^13 - 316*x^12 - 440*x^11 + 256*x^10 + 973*x^9 + 591*x^8 - 682*x^7 - 1177*x^6 - 371*x^5 + 341*x^4 + 325*x^3 + 110*x^2 + 17*x + 1)
 
gp: K = bnfinit(x^18 - 17*x^16 - 6*x^15 + 110*x^14 + 85*x^13 - 316*x^12 - 440*x^11 + 256*x^10 + 973*x^9 + 591*x^8 - 682*x^7 - 1177*x^6 - 371*x^5 + 341*x^4 + 325*x^3 + 110*x^2 + 17*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 17 x^{16} - 6 x^{15} + 110 x^{14} + 85 x^{13} - 316 x^{12} - 440 x^{11} + 256 x^{10} + 973 x^{9} + 591 x^{8} - 682 x^{7} - 1177 x^{6} - 371 x^{5} + 341 x^{4} + 325 x^{3} + 110 x^{2} + 17 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3426722037220245220088000000000=-\,2^{12}\cdot 5^{9}\cdot 37^{6}\cdot 166946971879\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.70$
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:  $2, 5, 37, 166946971879$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{17} a^{15} - \frac{4}{17} a^{13} - \frac{5}{17} a^{12} + \frac{7}{17} a^{11} - \frac{1}{17} a^{10} + \frac{8}{17} a^{9} - \frac{4}{17} a^{8} + \frac{2}{17} a^{7} - \frac{6}{17} a^{6} + \frac{1}{17} a^{5} + \frac{7}{17} a^{4} + \frac{3}{17} a^{3} - \frac{7}{17} a^{2} - \frac{4}{17} a - \frac{1}{17}$, $\frac{1}{17} a^{16} - \frac{4}{17} a^{14} - \frac{5}{17} a^{13} + \frac{7}{17} a^{12} - \frac{1}{17} a^{11} + \frac{8}{17} a^{10} - \frac{4}{17} a^{9} + \frac{2}{17} a^{8} - \frac{6}{17} a^{7} + \frac{1}{17} a^{6} + \frac{7}{17} a^{5} + \frac{3}{17} a^{4} - \frac{7}{17} a^{3} - \frac{4}{17} a^{2} - \frac{1}{17} a$, $\frac{1}{17} a^{17} - \frac{5}{17} a^{14} + \frac{8}{17} a^{13} - \frac{4}{17} a^{12} + \frac{2}{17} a^{11} - \frac{8}{17} a^{10} - \frac{5}{17} a^{8} - \frac{8}{17} a^{7} + \frac{7}{17} a^{5} + \frac{4}{17} a^{4} + \frac{8}{17} a^{3} + \frac{5}{17} a^{2} + \frac{1}{17} a - \frac{4}{17}$

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:  $16$
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:  \( 1304272731.89 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T903:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 559872
The 174 conjugacy class representatives for t18n903 are not computed
Character table for t18n903 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.148.1, 6.6.2738000.1

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

Sibling fields

Degree 18 sibling: 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$37$37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
166946971879Data not computed