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Properties

Label 18.10.1547512649...3125.1
Degree $18$
Signature $[10, 4]$
Discriminant $3^{12}\cdot 5^{13}\cdot 13^{4}\cdot 17^{4}$
Root discriminant $22.07$
Ramified primes $3, 5, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T888

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 3, -10, 29, 70, 19, -39, -144, -246, -195, -36, 101, 66, -6, -4, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 4*x^16 - 4*x^15 - 6*x^14 + 66*x^13 + 101*x^12 - 36*x^11 - 195*x^10 - 246*x^9 - 144*x^8 - 39*x^7 + 19*x^6 + 70*x^5 + 29*x^4 - 10*x^3 + 3*x^2 + 3*x - 1)
 
gp: K = bnfinit(x^18 - 2*x^17 - 4*x^16 - 4*x^15 - 6*x^14 + 66*x^13 + 101*x^12 - 36*x^11 - 195*x^10 - 246*x^9 - 144*x^8 - 39*x^7 + 19*x^6 + 70*x^5 + 29*x^4 - 10*x^3 + 3*x^2 + 3*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 4 x^{16} - 4 x^{15} - 6 x^{14} + 66 x^{13} + 101 x^{12} - 36 x^{11} - 195 x^{10} - 246 x^{9} - 144 x^{8} - 39 x^{7} + 19 x^{6} + 70 x^{5} + 29 x^{4} - 10 x^{3} + 3 x^{2} + 3 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1547512649777735595703125=3^{12}\cdot 5^{13}\cdot 13^{4}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.07$
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:  $3, 5, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{317831272705} a^{17} - \frac{19445536628}{317831272705} a^{16} + \frac{22101989581}{317831272705} a^{15} + \frac{109594271}{63566254541} a^{14} + \frac{19781719121}{317831272705} a^{13} + \frac{27191027904}{317831272705} a^{12} - \frac{4713813602}{63566254541} a^{11} + \frac{63308149674}{317831272705} a^{10} - \frac{20547943007}{317831272705} a^{9} - \frac{60287527421}{317831272705} a^{8} - \frac{158758654894}{317831272705} a^{7} - \frac{108961038664}{317831272705} a^{6} + \frac{15111846447}{317831272705} a^{5} - \frac{151893103423}{317831272705} a^{4} - \frac{104081507752}{317831272705} a^{3} + \frac{141894435638}{317831272705} a^{2} + \frac{9337143967}{63566254541} a + \frac{136409985778}{317831272705}$

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:  $13$
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:  \( 362373.375537 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T888:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 362880
The 36 conjugacy class representatives for t18n888
Character table for t18n888 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 9.5.22253180625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $18$ R R $18$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ R R ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $18$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.6.6.1$x^{6} + 3 x^{5} - 2$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
3.6.6.1$x^{6} + 3 x^{5} - 2$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.10.2$x^{12} + 15 x^{6} + 100$$6$$2$$10$$C_6\times S_3$$[\ ]_{6}^{6}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.6.4.1$x^{6} + 39 x^{3} + 676$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.10.0.1$x^{10} + 2 x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$17$17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$