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Properties

Label 18.18.1303780692...0625.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{30}\cdot 5^{12}\cdot 11^{10}$
Root discriminant $69.14$
Ramified primes $3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3.C_2$ (as 18T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -6, 144, 57, -2355, -786, 13914, 6729, -31773, -15305, 28194, 9264, -9306, -1968, 1245, 147, -63, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 63*x^16 + 147*x^15 + 1245*x^14 - 1968*x^13 - 9306*x^12 + 9264*x^11 + 28194*x^10 - 15305*x^9 - 31773*x^8 + 6729*x^7 + 13914*x^6 - 786*x^5 - 2355*x^4 + 57*x^3 + 144*x^2 - 6*x - 1)
 
gp: K = bnfinit(x^18 - 3*x^17 - 63*x^16 + 147*x^15 + 1245*x^14 - 1968*x^13 - 9306*x^12 + 9264*x^11 + 28194*x^10 - 15305*x^9 - 31773*x^8 + 6729*x^7 + 13914*x^6 - 786*x^5 - 2355*x^4 + 57*x^3 + 144*x^2 - 6*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 63 x^{16} + 147 x^{15} + 1245 x^{14} - 1968 x^{13} - 9306 x^{12} + 9264 x^{11} + 28194 x^{10} - 15305 x^{9} - 31773 x^{8} + 6729 x^{7} + 13914 x^{6} - 786 x^{5} - 2355 x^{4} + 57 x^{3} + 144 x^{2} - 6 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1303780692070187898696301025390625=3^{30}\cdot 5^{12}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.14$
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:  $3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{15} a^{12} - \frac{2}{15} a^{11} + \frac{2}{15} a^{10} - \frac{2}{15} a^{9} - \frac{1}{15} a^{8} - \frac{2}{5} a^{7} - \frac{2}{15} a^{6} + \frac{1}{5} a^{5} + \frac{1}{3} a^{4} - \frac{7}{15} a^{3} + \frac{1}{3} a^{2} - \frac{1}{15} a + \frac{1}{15}$, $\frac{1}{15} a^{13} - \frac{2}{15} a^{11} + \frac{2}{15} a^{10} - \frac{1}{3} a^{9} + \frac{7}{15} a^{8} + \frac{1}{15} a^{7} - \frac{1}{15} a^{6} - \frac{4}{15} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{15} a + \frac{2}{15}$, $\frac{1}{30} a^{14} - \frac{1}{30} a^{13} + \frac{1}{15} a^{10} + \frac{1}{10} a^{9} + \frac{1}{15} a^{8} + \frac{11}{30} a^{7} + \frac{4}{15} a^{6} - \frac{7}{30} a^{5} + \frac{13}{30} a^{4} - \frac{1}{30} a^{3} + \frac{1}{6} a^{2} - \frac{2}{15} a - \frac{1}{6}$, $\frac{1}{30} a^{15} - \frac{1}{30} a^{13} + \frac{1}{15} a^{11} - \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{7}{30} a^{8} - \frac{1}{30} a^{7} + \frac{1}{30} a^{6} - \frac{7}{15} a^{5} + \frac{2}{5} a^{4} + \frac{7}{15} a^{3} - \frac{3}{10} a^{2} + \frac{1}{30} a + \frac{1}{6}$, $\frac{1}{60} a^{16} - \frac{1}{60} a^{13} - \frac{1}{30} a^{12} + \frac{1}{20} a^{11} + \frac{3}{20} a^{10} + \frac{1}{15} a^{9} + \frac{1}{12} a^{8} + \frac{1}{10} a^{7} - \frac{7}{15} a^{6} + \frac{23}{60} a^{5} + \frac{7}{60} a^{4} - \frac{1}{5} a^{3} - \frac{7}{30} a^{2} - \frac{5}{12} a - \frac{3}{20}$, $\frac{1}{3128510590723505270220} a^{17} - \frac{149124626755521321}{148976694796357393820} a^{16} - \frac{1086440315856168989}{74488347398178696910} a^{15} - \frac{176148569544369983}{446930084389072181460} a^{14} - \frac{29309133087563912441}{1042836863574501756740} a^{13} + \frac{12237128454140380297}{3128510590723505270220} a^{12} - \frac{260599333872013245337}{1564255295361752635110} a^{11} - \frac{70102934378161838093}{446930084389072181460} a^{10} + \frac{721719304775710286687}{3128510590723505270220} a^{9} + \frac{288222447089404827007}{625702118144701054044} a^{8} + \frac{197490859271754113683}{782127647680876317555} a^{7} - \frac{685138884102944083451}{3128510590723505270220} a^{6} + \frac{384367656598138457053}{782127647680876317555} a^{5} - \frac{196076964259729169051}{625702118144701054044} a^{4} - \frac{81720838399340175983}{521418431787250878370} a^{3} + \frac{5517262803393795917}{1042836863574501756740} a^{2} + \frac{238256398873319359821}{521418431787250878370} a + \frac{198456299804807353391}{3128510590723505270220}$

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:  \( 39669265431.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3.C_2$ (as 18T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 14 conjugacy class representatives for $C_3^2:S_3.C_2$
Character table for $C_3^2:S_3.C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 6.6.55130625.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$11$11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.9.6.1$x^{9} - 121 x^{3} + 3993$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$