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Properties

Label 18.0.24466858935...9659.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 59^{9}$
Root discriminant $33.23$
Ramified primes $3, 59$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_9$ (as 18T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![944, 0, -3924, 0, 4587, 0, 536, 0, 1458, 0, 345, 0, 119, 0, 18, 0, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 12*x^16 + 18*x^14 + 119*x^12 + 345*x^10 + 1458*x^8 + 536*x^6 + 4587*x^4 - 3924*x^2 + 944)
 
gp: K = bnfinit(x^18 + 12*x^16 + 18*x^14 + 119*x^12 + 345*x^10 + 1458*x^8 + 536*x^6 + 4587*x^4 - 3924*x^2 + 944, 1)
 

Normalized defining polynomial

\( x^{18} + 12 x^{16} + 18 x^{14} + 119 x^{12} + 345 x^{10} + 1458 x^{8} + 536 x^{6} + 4587 x^{4} - 3924 x^{2} + 944 \)

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:  \(-2446685893599555554651329659=-\,3^{24}\cdot 59^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.23$
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:  $3, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{3}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{16} a^{5} - \frac{7}{16} a^{3} - \frac{1}{2}$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{10} - \frac{1}{8} a^{8} - \frac{1}{32} a^{6} + \frac{1}{32} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{12} - \frac{1}{64} a^{11} + \frac{1}{64} a^{10} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} + \frac{15}{64} a^{7} - \frac{15}{64} a^{6} - \frac{15}{64} a^{5} + \frac{15}{64} a^{4} + \frac{5}{16} a^{3} + \frac{3}{16} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{256} a^{14} + \frac{1}{128} a^{12} + \frac{9}{256} a^{10} + \frac{19}{256} a^{8} - \frac{1}{4} a^{7} + \frac{31}{128} a^{6} - \frac{9}{256} a^{4} - \frac{1}{2} a^{3} + \frac{15}{64} a^{2} + \frac{1}{4} a + \frac{7}{16}$, $\frac{1}{512} a^{15} - \frac{1}{512} a^{14} + \frac{1}{256} a^{13} - \frac{1}{256} a^{12} + \frac{9}{512} a^{11} + \frac{23}{512} a^{10} + \frac{19}{512} a^{9} - \frac{51}{512} a^{8} + \frac{31}{256} a^{7} - \frac{31}{256} a^{6} - \frac{9}{512} a^{5} - \frac{23}{512} a^{4} + \frac{15}{128} a^{3} - \frac{7}{128} a^{2} - \frac{1}{32} a + \frac{1}{32}$, $\frac{1}{83573653504} a^{16} + \frac{87901841}{83573653504} a^{14} - \frac{328913305}{83573653504} a^{12} - \frac{1439048195}{41786826752} a^{10} - \frac{7665711813}{83573653504} a^{8} + \frac{559307557}{2258747392} a^{6} - \frac{778728363}{83573653504} a^{4} + \frac{9113507397}{20893413376} a^{2} - \frac{1}{2} a + \frac{500293441}{5223353344}$, $\frac{1}{167147307008} a^{17} - \frac{1}{167147307008} a^{16} + \frac{87901841}{167147307008} a^{15} - \frac{87901841}{167147307008} a^{14} - \frac{328913305}{167147307008} a^{13} + \frac{328913305}{167147307008} a^{12} - \frac{1439048195}{83573653504} a^{11} + \frac{1439048195}{83573653504} a^{10} - \frac{7665711813}{167147307008} a^{9} + \frac{7665711813}{167147307008} a^{8} + \frac{559307557}{4517494784} a^{7} - \frac{559307557}{4517494784} a^{6} + \frac{41008098389}{167147307008} a^{5} - \frac{41008098389}{167147307008} a^{4} + \frac{9113507397}{41786826752} a^{3} + \frac{11779905979}{41786826752} a^{2} + \frac{500293441}{10446706688} a + \frac{4723059903}{10446706688}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 4622263.54519 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 18T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-59}) \), 3.1.59.1 x3, 6.0.205379.1, 9.1.6439662447201.3 x9

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

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ R

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$3.9.12.2$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 54$$3$$3$$12$$C_9$$[2]^{3}$
3.9.12.2$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 54$$3$$3$$12$$C_9$$[2]^{3}$
$59$59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$