↑

Properties

Label 18.0.51923518320...9867.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 107^{9}$
Root discriminant $44.76$
Ramified primes $3, 107$
Class number $36$ (GRH)
Class group $[6, 6]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![110764, -281418, 392661, -437878, 421431, -314298, 202515, -111702, 51336, -19902, 7284, -2430, 616, -312, 171, -36, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 - 36*x^15 + 171*x^14 - 312*x^13 + 616*x^12 - 2430*x^11 + 7284*x^10 - 19902*x^9 + 51336*x^8 - 111702*x^7 + 202515*x^6 - 314298*x^5 + 421431*x^4 - 437878*x^3 + 392661*x^2 - 281418*x + 110764)
 
gp: K = bnfinit(x^18 - 3*x^16 - 36*x^15 + 171*x^14 - 312*x^13 + 616*x^12 - 2430*x^11 + 7284*x^10 - 19902*x^9 + 51336*x^8 - 111702*x^7 + 202515*x^6 - 314298*x^5 + 421431*x^4 - 437878*x^3 + 392661*x^2 - 281418*x + 110764, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{16} - 36 x^{15} + 171 x^{14} - 312 x^{13} + 616 x^{12} - 2430 x^{11} + 7284 x^{10} - 19902 x^{9} + 51336 x^{8} - 111702 x^{7} + 202515 x^{6} - 314298 x^{5} + 421431 x^{4} - 437878 x^{3} + 392661 x^{2} - 281418 x + 110764 \)

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:  \(-519235183203048555634413069867=-\,3^{24}\cdot 107^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.76$
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:  $3, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{16} a^{11} - \frac{1}{8} a^{9} + \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{3}{16} a^{5} - \frac{3}{8} a^{3} + \frac{3}{16} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{3}{16} a^{6} + \frac{1}{8} a^{4} + \frac{3}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{13} + \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{16} a^{7} + \frac{3}{16} a^{4} + \frac{1}{8} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{6016} a^{14} - \frac{21}{3008} a^{13} - \frac{1}{47} a^{12} + \frac{79}{3008} a^{11} + \frac{11}{94} a^{10} - \frac{43}{752} a^{9} + \frac{39}{376} a^{8} - \frac{19}{376} a^{7} + \frac{295}{1504} a^{6} + \frac{629}{3008} a^{5} - \frac{49}{376} a^{4} + \frac{143}{752} a^{3} - \frac{1481}{6016} a^{2} + \frac{1389}{3008} a + \frac{125}{1504}$, $\frac{1}{42112} a^{15} + \frac{1}{42112} a^{14} - \frac{403}{21056} a^{13} + \frac{21}{3008} a^{12} - \frac{387}{21056} a^{11} + \frac{101}{1316} a^{10} + \frac{43}{658} a^{9} - \frac{269}{2632} a^{8} + \frac{505}{10528} a^{7} + \frac{1371}{21056} a^{6} + \frac{711}{21056} a^{5} - \frac{39}{376} a^{4} - \frac{9817}{42112} a^{3} + \frac{1}{6016} a^{2} + \frac{511}{3008} a + \frac{2367}{10528}$, $\frac{1}{360478720} a^{16} + \frac{2599}{360478720} a^{15} - \frac{127}{10299392} a^{14} + \frac{960713}{90119680} a^{13} + \frac{3814067}{180239360} a^{12} + \frac{1427619}{180239360} a^{11} + \frac{2362391}{22529920} a^{10} - \frac{149531}{4505984} a^{9} - \frac{8710811}{90119680} a^{8} - \frac{42386609}{180239360} a^{7} + \frac{40545823}{180239360} a^{6} + \frac{1240873}{180239360} a^{5} + \frac{59442471}{360478720} a^{4} - \frac{139459407}{360478720} a^{3} + \frac{101413}{219136} a^{2} + \frac{31944859}{180239360} a + \frac{17787617}{90119680}$, $\frac{1}{24714424546463054735360} a^{17} - \frac{172273463753}{386162883538485230240} a^{16} - \frac{2198106428478081}{262919410068755901440} a^{15} + \frac{1327270769616910127}{24714424546463054735360} a^{14} + \frac{156476072731399523701}{12357212273231527367680} a^{13} + \frac{182628163964114113591}{6178606136615763683840} a^{12} + \frac{40571814788991685317}{1765316039033075338240} a^{11} - \frac{2152699430479522091}{96540720884621307560} a^{10} - \frac{3210591883153082373}{57743982585194053120} a^{9} - \frac{8584240574308012241}{1765316039033075338240} a^{8} + \frac{62917588343418404411}{6178606136615763683840} a^{7} + \frac{15190360005356515401}{77232576707697046048} a^{6} - \frac{179692610149252960427}{4942884909292610947072} a^{5} + \frac{1951851008338641227}{27583063109891802160} a^{4} + \frac{5574368217062783595181}{12357212273231527367680} a^{3} + \frac{9043502014473732158063}{24714424546463054735360} a^{2} - \frac{112079178470779203721}{353063207806615067648} a + \frac{2059905981317131030393}{6178606136615763683840}$

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

Class group and class number

$C_{6}\times C_{6}$, which has order $36$ (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:  \( 9603674.43615 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

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 $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-107}) \), 3.1.8667.2 x3, 3.1.107.1 x3, 3.1.8667.3 x3, 3.1.8667.1 x3, 6.0.8037507123.3, 6.0.1225043.1, 6.0.8037507123.4, 6.0.8037507123.2, 9.1.69661074235041.6 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$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}$