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Properties

Label 18.6.26957006544...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 5^{9}\cdot 7^{8}\cdot 197^{6}$
Root discriminant $49.04$
Ramified primes $2, 5, 7, 197$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_3:S_3:S_4$ (as 18T153)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![289, 3094, 13674, 31628, 39588, 24101, 3411, -2448, -56, 406, -349, -88, -19, -32, 22, 15, -11, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 11*x^16 + 15*x^15 + 22*x^14 - 32*x^13 - 19*x^12 - 88*x^11 - 349*x^10 + 406*x^9 - 56*x^8 - 2448*x^7 + 3411*x^6 + 24101*x^5 + 39588*x^4 + 31628*x^3 + 13674*x^2 + 3094*x + 289)
 
gp: K = bnfinit(x^18 - x^17 - 11*x^16 + 15*x^15 + 22*x^14 - 32*x^13 - 19*x^12 - 88*x^11 - 349*x^10 + 406*x^9 - 56*x^8 - 2448*x^7 + 3411*x^6 + 24101*x^5 + 39588*x^4 + 31628*x^3 + 13674*x^2 + 3094*x + 289, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 11 x^{16} + 15 x^{15} + 22 x^{14} - 32 x^{13} - 19 x^{12} - 88 x^{11} - 349 x^{10} + 406 x^{9} - 56 x^{8} - 2448 x^{7} + 3411 x^{6} + 24101 x^{5} + 39588 x^{4} + 31628 x^{3} + 13674 x^{2} + 3094 x + 289 \)

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:  \(2695700654465561346632000000000=2^{12}\cdot 5^{9}\cdot 7^{8}\cdot 197^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.04$
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:  $2, 5, 7, 197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{10} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{425} a^{16} - \frac{6}{425} a^{15} - \frac{3}{85} a^{14} + \frac{39}{425} a^{13} - \frac{3}{425} a^{12} + \frac{6}{25} a^{11} + \frac{83}{425} a^{10} + \frac{194}{425} a^{9} + \frac{109}{425} a^{8} + \frac{13}{85} a^{7} + \frac{163}{425} a^{6} - \frac{118}{425} a^{5} - \frac{96}{425} a^{4} + \frac{203}{425} a^{3} + \frac{11}{25} a^{2} - \frac{179}{425} a - \frac{12}{25}$, $\frac{1}{32900441240419577094080375} a^{17} + \frac{6979273775847044773183}{32900441240419577094080375} a^{16} - \frac{3013879089992076913994979}{32900441240419577094080375} a^{15} + \frac{1454189814347826250899794}{32900441240419577094080375} a^{14} + \frac{2882336258836845167294418}{32900441240419577094080375} a^{13} - \frac{14984591747193229737901}{6580088248083915418816075} a^{12} + \frac{10350145230279866281589406}{32900441240419577094080375} a^{11} + \frac{459567093418501635901361}{32900441240419577094080375} a^{10} + \frac{1677527725158595755900551}{6580088248083915418816075} a^{9} - \frac{8219891381098699260001809}{32900441240419577094080375} a^{8} - \frac{6314786128684298733524797}{32900441240419577094080375} a^{7} + \frac{6150323736344554785516354}{32900441240419577094080375} a^{6} + \frac{4373125311274361972184202}{32900441240419577094080375} a^{5} + \frac{13053231750947544249256039}{32900441240419577094080375} a^{4} + \frac{12589230861088845936764954}{32900441240419577094080375} a^{3} - \frac{9793361002823678085878606}{32900441240419577094080375} a^{2} - \frac{2572810279677572555420227}{6580088248083915418816075} a + \frac{942118035620067311208517}{1935320072965857476122375}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 356760479.82 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3:S_4$ (as 18T153):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_3:S_3:S_4$
Character table for $C_3:S_3:S_4$

Intermediate fields

3.3.985.1, 6.2.4851125.1, 9.9.734261622920000.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$197$197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.4.2.1$x^{4} + 985 x^{2} + 349281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
197.4.2.1$x^{4} + 985 x^{2} + 349281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
197.4.2.1$x^{4} + 985 x^{2} + 349281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$