↑

Properties

Label 18.0.36595840381...1871.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 31^{9}$
Root discriminant $20.37$
Ramified primes $7, 31$
Class number $3$
Class group $[3]$
Galois group $D_9$ (as 18T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, -54, -742, 1065, -1604, 1087, 656, -2694, 3830, -3592, 2823, -2031, 1199, -559, 232, -84, 22, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 22*x^16 - 84*x^15 + 232*x^14 - 559*x^13 + 1199*x^12 - 2031*x^11 + 2823*x^10 - 3592*x^9 + 3830*x^8 - 2694*x^7 + 656*x^6 + 1087*x^5 - 1604*x^4 + 1065*x^3 - 742*x^2 - 54*x + 729)
 
gp: K = bnfinit(x^18 - 5*x^17 + 22*x^16 - 84*x^15 + 232*x^14 - 559*x^13 + 1199*x^12 - 2031*x^11 + 2823*x^10 - 3592*x^9 + 3830*x^8 - 2694*x^7 + 656*x^6 + 1087*x^5 - 1604*x^4 + 1065*x^3 - 742*x^2 - 54*x + 729, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 22 x^{16} - 84 x^{15} + 232 x^{14} - 559 x^{13} + 1199 x^{12} - 2031 x^{11} + 2823 x^{10} - 3592 x^{9} + 3830 x^{8} - 2694 x^{7} + 656 x^{6} + 1087 x^{5} - 1604 x^{4} + 1065 x^{3} - 742 x^{2} - 54 x + 729 \)

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:  \(-365958403811771477871871=-\,7^{12}\cdot 31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.37$
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:  $7, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{93} a^{12} - \frac{5}{93} a^{11} + \frac{4}{93} a^{10} - \frac{2}{93} a^{9} - \frac{12}{31} a^{8} - \frac{12}{31} a^{7} - \frac{13}{31} a^{6} + \frac{2}{31} a^{5} + \frac{20}{93} a^{4} + \frac{44}{93} a^{3} - \frac{16}{93} a^{2} - \frac{4}{93} a - \frac{10}{31}$, $\frac{1}{93} a^{13} + \frac{10}{93} a^{11} - \frac{13}{93} a^{10} - \frac{5}{31} a^{9} - \frac{10}{31} a^{8} - \frac{11}{31} a^{7} - \frac{1}{31} a^{6} - \frac{43}{93} a^{5} - \frac{14}{31} a^{4} - \frac{13}{93} a^{3} + \frac{40}{93} a^{2} + \frac{4}{31} a + \frac{12}{31}$, $\frac{1}{279} a^{14} - \frac{1}{279} a^{12} + \frac{11}{279} a^{11} + \frac{34}{279} a^{10} + \frac{23}{279} a^{9} + \frac{28}{93} a^{8} + \frac{38}{93} a^{7} - \frac{79}{279} a^{6} - \frac{12}{31} a^{5} + \frac{46}{279} a^{4} - \frac{41}{279} a^{3} + \frac{2}{279} a^{2} - \frac{137}{279} a - \frac{15}{31}$, $\frac{1}{837} a^{15} - \frac{1}{837} a^{14} - \frac{4}{837} a^{13} + \frac{1}{279} a^{12} - \frac{55}{837} a^{11} - \frac{8}{837} a^{10} - \frac{2}{27} a^{9} - \frac{38}{279} a^{8} + \frac{230}{837} a^{7} + \frac{52}{837} a^{6} - \frac{50}{837} a^{5} + \frac{139}{279} a^{4} + \frac{58}{837} a^{3} + \frac{164}{837} a^{2} + \frac{188}{837} a + \frac{11}{31}$, $\frac{1}{77841} a^{16} + \frac{35}{77841} a^{15} - \frac{94}{77841} a^{14} + \frac{97}{25947} a^{13} - \frac{190}{77841} a^{12} - \frac{5984}{77841} a^{11} - \frac{191}{2511} a^{10} - \frac{4181}{25947} a^{9} - \frac{34114}{77841} a^{8} - \frac{9758}{77841} a^{7} - \frac{19616}{77841} a^{6} - \frac{10883}{25947} a^{5} + \frac{7753}{77841} a^{4} - \frac{7522}{77841} a^{3} - \frac{8533}{77841} a^{2} - \frac{1064}{8649} a + \frac{369}{961}$, $\frac{1}{50439008975553} a^{17} + \frac{2353357}{5604334330617} a^{16} - \frac{23199026261}{50439008975553} a^{15} - \frac{86803845679}{50439008975553} a^{14} + \frac{258019514354}{50439008975553} a^{13} - \frac{17805692350}{5604334330617} a^{12} + \frac{5152665155510}{50439008975553} a^{11} - \frac{3567758982845}{50439008975553} a^{10} - \frac{3958200167032}{50439008975553} a^{9} + \frac{7362004977694}{16813002991851} a^{8} + \frac{3945907644410}{50439008975553} a^{7} + \frac{8509441762924}{50439008975553} a^{6} + \frac{12360424855138}{50439008975553} a^{5} + \frac{223841007869}{542354935221} a^{4} + \frac{13468090461706}{50439008975553} a^{3} - \frac{15690891037006}{50439008975553} a^{2} - \frac{414319745770}{5604334330617} a - \frac{91688154820}{207567938171}$

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

Class group and class number

$C_{3}$, which has order $3$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 12668.0200523 \)
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

\(\Q(\sqrt{-31}) \), 3.1.31.1 x3, 6.0.29791.1, 9.1.108651322129.1 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$7$7.9.6.2$x^{9} - 49 x^{3} + 686$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
7.9.6.2$x^{9} - 49 x^{3} + 686$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$