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Properties

Label 18.0.23845365740...8816.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{27}\cdot 13^{12}$
Root discriminant $81.26$
Ramified primes $2, 3, 13$
Class number $13832$ (GRH)
Class group $[2, 2, 3458]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4634713, 4744224, 7609494, 3117170, 1834899, -228654, 55755, -177996, 91239, -23424, 21552, -8442, 3092, -1194, 441, -98, 27, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 27*x^16 - 98*x^15 + 441*x^14 - 1194*x^13 + 3092*x^12 - 8442*x^11 + 21552*x^10 - 23424*x^9 + 91239*x^8 - 177996*x^7 + 55755*x^6 - 228654*x^5 + 1834899*x^4 + 3117170*x^3 + 7609494*x^2 + 4744224*x + 4634713)
 
gp: K = bnfinit(x^18 - 6*x^17 + 27*x^16 - 98*x^15 + 441*x^14 - 1194*x^13 + 3092*x^12 - 8442*x^11 + 21552*x^10 - 23424*x^9 + 91239*x^8 - 177996*x^7 + 55755*x^6 - 228654*x^5 + 1834899*x^4 + 3117170*x^3 + 7609494*x^2 + 4744224*x + 4634713, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 27 x^{16} - 98 x^{15} + 441 x^{14} - 1194 x^{13} + 3092 x^{12} - 8442 x^{11} + 21552 x^{10} - 23424 x^{9} + 91239 x^{8} - 177996 x^{7} + 55755 x^{6} - 228654 x^{5} + 1834899 x^{4} + 3117170 x^{3} + 7609494 x^{2} + 4744224 x + 4634713 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-23845365740806374055553449712418816=-\,2^{27}\cdot 3^{27}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.26$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(936=2^{3}\cdot 3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{936}(1,·)$, $\chi_{936}(581,·)$, $\chi_{936}(841,·)$, $\chi_{936}(269,·)$, $\chi_{936}(653,·)$, $\chi_{936}(913,·)$, $\chi_{936}(341,·)$, $\chi_{936}(601,·)$, $\chi_{936}(217,·)$, $\chi_{936}(29,·)$, $\chi_{936}(289,·)$, $\chi_{936}(677,·)$, $\chi_{936}(529,·)$, $\chi_{936}(365,·)$, $\chi_{936}(625,·)$, $\chi_{936}(53,·)$, $\chi_{936}(313,·)$, $\chi_{936}(893,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{212} a^{15} - \frac{15}{212} a^{14} + \frac{19}{212} a^{13} + \frac{17}{212} a^{12} - \frac{37}{212} a^{11} - \frac{29}{212} a^{10} + \frac{43}{212} a^{9} - \frac{29}{212} a^{8} - \frac{7}{106} a^{7} + \frac{11}{106} a^{6} - \frac{17}{53} a^{5} - \frac{31}{106} a^{4} + \frac{17}{212} a^{3} - \frac{37}{212} a^{2} - \frac{9}{212} a - \frac{21}{212}$, $\frac{1}{412900892923449957412} a^{16} + \frac{466182597906876253}{412900892923449957412} a^{15} + \frac{17357615771671590475}{206450446461724978706} a^{14} + \frac{19804506835678133257}{206450446461724978706} a^{13} - \frac{10077984408154379421}{412900892923449957412} a^{12} - \frac{30235241455941283031}{412900892923449957412} a^{11} + \frac{6226028962750323544}{103225223230862489353} a^{10} + \frac{31392673470413393997}{206450446461724978706} a^{9} - \frac{42765145845247041823}{206450446461724978706} a^{8} + \frac{20053758385054726643}{206450446461724978706} a^{7} - \frac{45611253147394622889}{206450446461724978706} a^{6} - \frac{1240857029422615203}{103225223230862489353} a^{5} - \frac{176804131786331899755}{412900892923449957412} a^{4} - \frac{116611739184352766199}{412900892923449957412} a^{3} - \frac{20795273052122474456}{103225223230862489353} a^{2} - \frac{35891914147663902813}{103225223230862489353} a - \frac{14346605368785333091}{103225223230862489353}$, $\frac{1}{5304283386863539501664252083508} a^{17} - \frac{1944500851}{5304283386863539501664252083508} a^{16} + \frac{390875682043349786674197473}{2652141693431769750832126041754} a^{15} + \frac{599507749336078807839725284033}{5304283386863539501664252083508} a^{14} + \frac{78699495697553321497406410589}{2652141693431769750832126041754} a^{13} - \frac{597523667732692492883179332095}{5304283386863539501664252083508} a^{12} - \frac{181511217884950130954027212402}{1326070846715884875416063020877} a^{11} - \frac{930888669211011899070412249737}{5304283386863539501664252083508} a^{10} + \frac{809148675946671149889810340567}{5304283386863539501664252083508} a^{9} - \frac{84712785420675795353641491417}{1326070846715884875416063020877} a^{8} - \frac{210886317686295403462766414495}{2652141693431769750832126041754} a^{7} - \frac{225804873025450273383889797159}{1326070846715884875416063020877} a^{6} - \frac{1486053811190990362318314156349}{5304283386863539501664252083508} a^{5} - \frac{2589494663599905602489574689875}{5304283386863539501664252083508} a^{4} + \frac{346085856236936053347024691692}{1326070846715884875416063020877} a^{3} - \frac{1038335077745632127648653567693}{5304283386863539501664252083508} a^{2} + \frac{1546457579909668887176341782761}{5304283386863539501664252083508} a + \frac{342276206284086585588168294821}{1326070846715884875416063020877}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{3458}$, which has order $13832$ (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:  $8$
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:  \( 400417.136445 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\zeta_{9})^+\), 3.3.13689.1, 3.3.13689.2, 3.3.169.1, 6.0.10077696.1, 6.0.287829075456.8, 6.0.287829075456.10, 6.0.394827264.1, 9.9.2565164201769.1

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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
13Data not computed