↑

Properties

Label 18.0.56527886235...2823.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,11^{12}\cdot 23^{9}$
Root discriminant $23.72$
Ramified primes $11, 23$
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![529, -989, 583, -525, 3861, 3862, 2387, 2808, 4131, 2799, 906, 314, 372, 245, 61, -13, -6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 6*x^16 - 13*x^15 + 61*x^14 + 245*x^13 + 372*x^12 + 314*x^11 + 906*x^10 + 2799*x^9 + 4131*x^8 + 2808*x^7 + 2387*x^6 + 3862*x^5 + 3861*x^4 - 525*x^3 + 583*x^2 - 989*x + 529)
 
gp: K = bnfinit(x^18 - x^17 - 6*x^16 - 13*x^15 + 61*x^14 + 245*x^13 + 372*x^12 + 314*x^11 + 906*x^10 + 2799*x^9 + 4131*x^8 + 2808*x^7 + 2387*x^6 + 3862*x^5 + 3861*x^4 - 525*x^3 + 583*x^2 - 989*x + 529, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 6 x^{16} - 13 x^{15} + 61 x^{14} + 245 x^{13} + 372 x^{12} + 314 x^{11} + 906 x^{10} + 2799 x^{9} + 4131 x^{8} + 2808 x^{7} + 2387 x^{6} + 3862 x^{5} + 3861 x^{4} - 525 x^{3} + 583 x^{2} - 989 x + 529 \)

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:  \(-5652788623542031943002823=-\,11^{12}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.72$
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:  $11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{115} a^{11} - \frac{7}{115} a^{10} + \frac{2}{115} a^{9} - \frac{1}{115} a^{8} - \frac{48}{115} a^{7} + \frac{28}{115} a^{6} + \frac{42}{115} a^{5} - \frac{4}{115} a^{4} + \frac{39}{115} a^{3} + \frac{3}{115} a^{2} + \frac{9}{115} a + \frac{2}{5}$, $\frac{1}{115} a^{12} - \frac{47}{115} a^{10} + \frac{13}{115} a^{9} - \frac{11}{23} a^{8} + \frac{37}{115} a^{7} + \frac{8}{115} a^{6} - \frac{11}{23} a^{5} + \frac{11}{115} a^{4} + \frac{2}{5} a^{3} + \frac{6}{23} a^{2} - \frac{6}{115} a - \frac{1}{5}$, $\frac{1}{115} a^{13} + \frac{29}{115} a^{10} + \frac{39}{115} a^{9} - \frac{2}{23} a^{8} + \frac{52}{115} a^{7} - \frac{4}{115} a^{6} + \frac{6}{23} a^{5} - \frac{27}{115} a^{4} + \frac{1}{5} a^{3} + \frac{4}{23} a^{2} + \frac{11}{23} a - \frac{1}{5}$, $\frac{1}{115} a^{14} + \frac{12}{115} a^{10} + \frac{47}{115} a^{9} - \frac{34}{115} a^{8} + \frac{8}{115} a^{7} + \frac{1}{5} a^{6} + \frac{4}{23} a^{5} + \frac{24}{115} a^{4} + \frac{39}{115} a^{3} - \frac{32}{115} a^{2} - \frac{54}{115} a + \frac{2}{5}$, $\frac{1}{805} a^{15} + \frac{3}{805} a^{14} - \frac{2}{805} a^{13} + \frac{1}{805} a^{12} + \frac{2}{805} a^{11} + \frac{48}{805} a^{10} - \frac{323}{805} a^{9} - \frac{234}{805} a^{8} + \frac{3}{7} a^{7} + \frac{11}{161} a^{6} + \frac{354}{805} a^{5} - \frac{359}{805} a^{4} + \frac{8}{161} a^{3} + \frac{31}{161} a^{2} + \frac{1}{35} a + \frac{17}{35}$, $\frac{1}{3092005} a^{16} + \frac{971}{3092005} a^{15} + \frac{26}{618401} a^{14} - \frac{2473}{618401} a^{13} - \frac{4252}{3092005} a^{12} - \frac{12261}{3092005} a^{11} + \frac{879071}{3092005} a^{10} - \frac{939083}{3092005} a^{9} + \frac{900931}{3092005} a^{8} + \frac{70359}{618401} a^{7} + \frac{6714}{134435} a^{6} + \frac{318163}{3092005} a^{5} + \frac{1329028}{3092005} a^{4} + \frac{183615}{618401} a^{3} + \frac{547222}{3092005} a^{2} + \frac{8828}{26887} a - \frac{277}{5845}$, $\frac{1}{87598326792481009925} a^{17} + \frac{7856155124668}{87598326792481009925} a^{16} + \frac{196415600983046}{544088986288701925} a^{15} + \frac{187074158022615706}{87598326792481009925} a^{14} + \frac{40234911636349907}{17519665358496201985} a^{13} + \frac{9817569919346076}{2502809336928028855} a^{12} + \frac{281679383685012722}{87598326792481009925} a^{11} - \frac{4985421698408238328}{87598326792481009925} a^{10} + \frac{86562145634379364}{544088986288701925} a^{9} - \frac{1631397365991108624}{3503933071699240397} a^{8} + \frac{27096760996359349576}{87598326792481009925} a^{7} + \frac{21806542860445405317}{87598326792481009925} a^{6} + \frac{505788784647135722}{3503933071699240397} a^{5} - \frac{14176065851399428458}{87598326792481009925} a^{4} + \frac{40169682017268010989}{87598326792481009925} a^{3} + \frac{1395918264017561178}{12514046684640144275} a^{2} + \frac{793058505240310409}{3808622904020913475} a - \frac{40432200822224069}{165592300174822325}$

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:  \( 99911.2053896 \) (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

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.12167.1, 9.1.495755401801.1 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$11$11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$