↑

Properties

Label 18.0.18535932472...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 5^{9}\cdot 7^{15}$
Root discriminant $117.61$
Ramified primes $2, 3, 5, 7$
Class number $2527616$ (GRH)
Class group $[2, 2, 2, 2, 2, 14, 5642]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18087890625, 0, 21705468750, 0, 9405703125, 0, 2039625000, 0, 248062500, 0, 17640000, 0, 735000, 0, 17325, 0, 210, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 210*x^16 + 17325*x^14 + 735000*x^12 + 17640000*x^10 + 248062500*x^8 + 2039625000*x^6 + 9405703125*x^4 + 21705468750*x^2 + 18087890625)
 
gp: K = bnfinit(x^18 + 210*x^16 + 17325*x^14 + 735000*x^12 + 17640000*x^10 + 248062500*x^8 + 2039625000*x^6 + 9405703125*x^4 + 21705468750*x^2 + 18087890625, 1)
 

Normalized defining polynomial

\( x^{18} + 210 x^{16} + 17325 x^{14} + 735000 x^{12} + 17640000 x^{10} + 248062500 x^{8} + 2039625000 x^{6} + 9405703125 x^{4} + 21705468750 x^{2} + 18087890625 \)

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:  \(-18535932472341721596787579392000000000=-\,2^{18}\cdot 3^{27}\cdot 5^{9}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $117.61$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1260=2^{2}\cdot 3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1260}(1,·)$, $\chi_{1260}(899,·)$, $\chi_{1260}(961,·)$, $\chi_{1260}(841,·)$, $\chi_{1260}(781,·)$, $\chi_{1260}(719,·)$, $\chi_{1260}(541,·)$, $\chi_{1260}(479,·)$, $\chi_{1260}(419,·)$, $\chi_{1260}(421,·)$, $\chi_{1260}(361,·)$, $\chi_{1260}(299,·)$, $\chi_{1260}(1201,·)$, $\chi_{1260}(839,·)$, $\chi_{1260}(1139,·)$, $\chi_{1260}(1259,·)$, $\chi_{1260}(121,·)$, $\chi_{1260}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{2625} a^{6}$, $\frac{1}{2625} a^{7}$, $\frac{1}{13125} a^{8}$, $\frac{1}{13125} a^{9}$, $\frac{1}{65625} a^{10}$, $\frac{1}{65625} a^{11}$, $\frac{1}{6890625} a^{12}$, $\frac{1}{6890625} a^{13}$, $\frac{1}{137812500} a^{14} + \frac{1}{4}$, $\frac{1}{137812500} a^{15} + \frac{1}{4} a$, $\frac{1}{2756250000} a^{16} + \frac{1}{551250000} a^{14} - \frac{1}{27562500} a^{12} - \frac{1}{262500} a^{10} + \frac{1}{52500} a^{8} - \frac{1}{50} a^{4} - \frac{7}{80} a^{2} - \frac{3}{16}$, $\frac{1}{2756250000} a^{17} + \frac{1}{551250000} a^{15} - \frac{1}{27562500} a^{13} - \frac{1}{262500} a^{11} + \frac{1}{52500} a^{9} - \frac{1}{50} a^{5} - \frac{7}{80} a^{3} - \frac{3}{16} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{14}\times C_{5642}$, which has order $2527616$ (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:  \( 54408.48888868202 \) (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

\(\Q(\sqrt{-105}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.2, 3.3.3969.1, 6.0.54010152000.17, 6.0.3630312000.2, 6.0.2646497448000.6, 6.0.2646497448000.7, 9.9.62523502209.1

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 R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed