↑

Properties

Label 18.0.44884258115...6875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,5^{12}\cdot 107^{9}$
Root discriminant $30.25$
Ramified primes $5, 107$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![81232, -28952, 104656, -45474, 69537, -50207, 29724, -28895, 14193, -8812, 5509, -2083, 1093, -460, 113, -53, 12, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 12*x^16 - 53*x^15 + 113*x^14 - 460*x^13 + 1093*x^12 - 2083*x^11 + 5509*x^10 - 8812*x^9 + 14193*x^8 - 28895*x^7 + 29724*x^6 - 50207*x^5 + 69537*x^4 - 45474*x^3 + 104656*x^2 - 28952*x + 81232)
 
gp: K = bnfinit(x^18 - x^17 + 12*x^16 - 53*x^15 + 113*x^14 - 460*x^13 + 1093*x^12 - 2083*x^11 + 5509*x^10 - 8812*x^9 + 14193*x^8 - 28895*x^7 + 29724*x^6 - 50207*x^5 + 69537*x^4 - 45474*x^3 + 104656*x^2 - 28952*x + 81232, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 12 x^{16} - 53 x^{15} + 113 x^{14} - 460 x^{13} + 1093 x^{12} - 2083 x^{11} + 5509 x^{10} - 8812 x^{9} + 14193 x^{8} - 28895 x^{7} + 29724 x^{6} - 50207 x^{5} + 69537 x^{4} - 45474 x^{3} + 104656 x^{2} - 28952 x + 81232 \)

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:  \(-448842581157264283935546875=-\,5^{12}\cdot 107^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.25$
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:  $5, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{32} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{3}{32} a^{7} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} + \frac{5}{32} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{32} a^{12} - \frac{1}{16} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} + \frac{1}{32} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{12} - \frac{1}{64} a^{11} - \frac{1}{32} a^{10} - \frac{3}{64} a^{9} + \frac{1}{64} a^{8} + \frac{5}{64} a^{7} - \frac{1}{32} a^{6} + \frac{15}{64} a^{5} + \frac{7}{64} a^{4} - \frac{15}{64} a^{3} - \frac{7}{16} a^{2} - \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{11} - \frac{1}{64} a^{10} + \frac{1}{32} a^{9} - \frac{1}{16} a^{8} + \frac{1}{64} a^{7} - \frac{7}{64} a^{6} + \frac{1}{32} a^{5} - \frac{7}{32} a^{4} - \frac{1}{64} a^{3} + \frac{5}{16} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{320} a^{15} - \frac{1}{64} a^{12} + \frac{3}{320} a^{11} + \frac{9}{160} a^{10} - \frac{1}{16} a^{9} + \frac{1}{64} a^{8} + \frac{13}{320} a^{7} + \frac{13}{160} a^{6} - \frac{31}{160} a^{5} + \frac{7}{64} a^{4} + \frac{7}{40} a^{3} - \frac{7}{20} a^{2} + \frac{11}{40} a + \frac{3}{10}$, $\frac{1}{2560} a^{16} + \frac{1}{1280} a^{15} + \frac{3}{512} a^{14} + \frac{1}{256} a^{13} + \frac{9}{1280} a^{12} + \frac{1}{640} a^{11} - \frac{149}{2560} a^{10} - \frac{3}{256} a^{9} + \frac{29}{1280} a^{8} - \frac{27}{640} a^{7} - \frac{29}{512} a^{6} - \frac{287}{1280} a^{5} + \frac{261}{2560} a^{4} - \frac{35}{256} a^{3} - \frac{27}{320} a^{2} - \frac{11}{320} a - \frac{53}{160}$, $\frac{1}{3979551745304338470400} a^{17} - \frac{201627169810369649}{3979551745304338470400} a^{16} + \frac{3102059817893259969}{3979551745304338470400} a^{15} + \frac{17211528186644499}{9589281313986357760} a^{14} - \frac{2647238455738929373}{994887936326084617600} a^{13} - \frac{921869528043776497}{1989775872652169235200} a^{12} - \frac{7320219018078806941}{795910349060867694080} a^{11} + \frac{100901662607093801297}{3979551745304338470400} a^{10} + \frac{20732078413309098907}{994887936326084617600} a^{9} - \frac{7497078674192573379}{284253696093167033600} a^{8} + \frac{118467795981010189571}{3979551745304338470400} a^{7} + \frac{17764151762270369011}{568507392186334067200} a^{6} - \frac{473169974012666116457}{3979551745304338470400} a^{5} + \frac{126913973914704018197}{568507392186334067200} a^{4} - \frac{12112057507029797227}{79591034906086769408} a^{3} - \frac{723614968074624057}{35531712011645879200} a^{2} + \frac{160050721727742087591}{497443968163042308800} a - \frac{90107846111365312901}{248721984081521154400}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 6601269.29293 \) (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

\(\Q(\sqrt{-107}) \), 3.1.107.1 x3, 6.0.1225043.1, 9.1.2048118765625.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$107$107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$