↑

Properties

Label 18.0.23611087497...2187.6
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{39}\cdot 17^{12}$
Root discriminant $71.46$
Ramified primes $3, 17$
Class number $729$ (GRH)
Class group $[3, 3, 9, 9]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![47523, 138402, 132030, 7146, -43209, -23787, 3276, 10323, -1323, -2565, -783, 54, 519, -117, 36, -36, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 36*x^15 + 36*x^14 - 117*x^13 + 519*x^12 + 54*x^11 - 783*x^10 - 2565*x^9 - 1323*x^8 + 10323*x^7 + 3276*x^6 - 23787*x^5 - 43209*x^4 + 7146*x^3 + 132030*x^2 + 138402*x + 47523)
 
gp: K = bnfinit(x^18 + 9*x^16 - 36*x^15 + 36*x^14 - 117*x^13 + 519*x^12 + 54*x^11 - 783*x^10 - 2565*x^9 - 1323*x^8 + 10323*x^7 + 3276*x^6 - 23787*x^5 - 43209*x^4 + 7146*x^3 + 132030*x^2 + 138402*x + 47523, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} - 36 x^{15} + 36 x^{14} - 117 x^{13} + 519 x^{12} + 54 x^{11} - 783 x^{10} - 2565 x^{9} - 1323 x^{8} + 10323 x^{7} + 3276 x^{6} - 23787 x^{5} - 43209 x^{4} + 7146 x^{3} + 132030 x^{2} + 138402 x + 47523 \)

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:  \(-2361108749748912380642242285082187=-\,3^{39}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.46$
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:  $3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} - \frac{2}{7} a^{13} - \frac{3}{7} a^{12} + \frac{3}{7} a^{11} + \frac{2}{7} a^{10} - \frac{3}{7} a^{9} - \frac{3}{7} a^{8} + \frac{3}{7} a^{6} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{7} a^{15} - \frac{3}{7} a^{12} + \frac{1}{7} a^{11} + \frac{1}{7} a^{10} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} + \frac{3}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{511} a^{16} + \frac{30}{511} a^{15} - \frac{2}{73} a^{14} + \frac{32}{511} a^{13} + \frac{121}{511} a^{12} + \frac{38}{511} a^{11} + \frac{19}{73} a^{10} + \frac{214}{511} a^{9} - \frac{30}{511} a^{8} + \frac{201}{511} a^{7} - \frac{234}{511} a^{6} - \frac{85}{511} a^{5} - \frac{29}{511} a^{4} + \frac{10}{73} a^{3} + \frac{39}{511} a^{2} + \frac{208}{511} a$, $\frac{1}{1556109241243725144844138366982159441} a^{17} + \frac{1098083755426785376237223440921494}{1556109241243725144844138366982159441} a^{16} - \frac{19957381885871104188991567654633343}{1556109241243725144844138366982159441} a^{15} - \frac{87603951227836390736802566302489469}{1556109241243725144844138366982159441} a^{14} + \frac{185570408053255312457099643743789080}{1556109241243725144844138366982159441} a^{13} + \frac{274450336528611230073718309644730063}{1556109241243725144844138366982159441} a^{12} + \frac{389116382326718412525934561422755272}{1556109241243725144844138366982159441} a^{11} + \frac{151285576333701579176437060185358745}{1556109241243725144844138366982159441} a^{10} + \frac{95629696611649878942799609901280347}{222301320177675020692019766711737063} a^{9} + \frac{152455672105748760662323193413611419}{1556109241243725144844138366982159441} a^{8} + \frac{548919760453238562405944042775386841}{1556109241243725144844138366982159441} a^{7} - \frac{14927587698389127357118313341947600}{31757331453953574384574252387391009} a^{6} - \frac{36114473539915156222964991025922801}{222301320177675020692019766711737063} a^{5} + \frac{10904580880539202143027727801800250}{1556109241243725144844138366982159441} a^{4} - \frac{67161731245942759671946860910352674}{222301320177675020692019766711737063} a^{3} - \frac{765707958205013782080822904525872832}{1556109241243725144844138366982159441} a^{2} - \frac{560559508921725787653390947065717974}{1556109241243725144844138366982159441} a + \frac{18541834876725393974729559711902}{3045223564077740009479722831667631}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{9}\times C_{9}$, which has order $729$ (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:  \( \frac{98341829242963439643}{4662767561449574267795051} a^{17} - \frac{13635527181601005507}{4662767561449574267795051} a^{16} + \frac{906640970062123785069}{4662767561449574267795051} a^{15} - \frac{3898740207748743769167}{4662767561449574267795051} a^{14} + \frac{4117924112337609848535}{4662767561449574267795051} a^{13} - \frac{14405795060208774129465}{4662767561449574267795051} a^{12} + \frac{58781548369563493837023}{4662767561449574267795051} a^{11} - \frac{4366984226524593477174}{4662767561449574267795051} a^{10} - \frac{78528392529730599627139}{4662767561449574267795051} a^{9} - \frac{261823920741671999214285}{4662767561449574267795051} a^{8} - \frac{310134246229795453617006}{4662767561449574267795051} a^{7} + \frac{1560646057140659831997888}{4662767561449574267795051} a^{6} - \frac{38180945157585972609798}{4662767561449574267795051} a^{5} - \frac{2301816576018416175847098}{4662767561449574267795051} a^{4} - \frac{5186863588807920512255007}{4662767561449574267795051} a^{3} + \frac{1814162193337450914214236}{4662767561449574267795051} a^{2} + \frac{19512053718313327533272709}{4662767561449574267795051} a + \frac{14019652277350094847266545}{4662767561449574267795051} \) (order $6$)
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:  \( 34420157.5786797 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.70227.1 x3, 6.0.14795494587.4, 6.0.177147.1, 6.0.1643943843.1, 6.0.14795494587.3

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

Sibling fields

Degree 18 siblings: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$17$17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$