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Properties

Label 18.0.54025729799...9959.7
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{12}\cdot 13^{15}$
Root discriminant $161.20$
Ramified primes $3, 7, 13$
Class number $31142748$ (GRH)
Class group $[19, 1639092]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1196406784, 741275136, 186484608, -242786912, 192346320, -129191472, 47843680, -16235718, 4076145, -685960, 197229, -25956, 10306, -2904, 306, -102, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 - 102*x^15 + 306*x^14 - 2904*x^13 + 10306*x^12 - 25956*x^11 + 197229*x^10 - 685960*x^9 + 4076145*x^8 - 16235718*x^7 + 47843680*x^6 - 129191472*x^5 + 192346320*x^4 - 242786912*x^3 + 186484608*x^2 + 741275136*x + 1196406784)
 
gp: K = bnfinit(x^18 - 3*x^16 - 102*x^15 + 306*x^14 - 2904*x^13 + 10306*x^12 - 25956*x^11 + 197229*x^10 - 685960*x^9 + 4076145*x^8 - 16235718*x^7 + 47843680*x^6 - 129191472*x^5 + 192346320*x^4 - 242786912*x^3 + 186484608*x^2 + 741275136*x + 1196406784, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{16} - 102 x^{15} + 306 x^{14} - 2904 x^{13} + 10306 x^{12} - 25956 x^{11} + 197229 x^{10} - 685960 x^{9} + 4076145 x^{8} - 16235718 x^{7} + 47843680 x^{6} - 129191472 x^{5} + 192346320 x^{4} - 242786912 x^{3} + 186484608 x^{2} + 741275136 x + 1196406784 \)

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:  \(-5402572979943138104302196852621075629959=-\,3^{27}\cdot 7^{12}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $161.20$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(819=3^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{819}(1,·)$, $\chi_{819}(389,·)$, $\chi_{819}(646,·)$, $\chi_{819}(779,·)$, $\chi_{819}(781,·)$, $\chi_{819}(725,·)$, $\chi_{819}(22,·)$, $\chi_{819}(218,·)$, $\chi_{819}(289,·)$, $\chi_{819}(802,·)$, $\chi_{819}(484,·)$, $\chi_{819}(625,·)$, $\chi_{819}(296,·)$, $\chi_{819}(445,·)$, $\chi_{819}(368,·)$, $\chi_{819}(680,·)$, $\chi_{819}(758,·)$, $\chi_{819}(701,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{32} a^{7} - \frac{1}{16} a^{5} - \frac{3}{32} a^{3} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{128} a^{8} + \frac{1}{64} a^{6} + \frac{1}{128} a^{4} - \frac{1}{4} a^{3} - \frac{1}{32} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{128} a^{9} - \frac{1}{64} a^{7} - \frac{7}{128} a^{5} + \frac{3}{16} a^{3} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{512} a^{10} + \frac{1}{512} a^{9} + \frac{1}{256} a^{7} - \frac{3}{512} a^{6} - \frac{31}{512} a^{5} + \frac{13}{256} a^{4} - \frac{25}{128} a^{3} + \frac{13}{64} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{512} a^{11} - \frac{1}{512} a^{9} - \frac{1}{256} a^{8} - \frac{5}{512} a^{7} - \frac{1}{128} a^{6} + \frac{25}{512} a^{5} + \frac{15}{256} a^{4} + \frac{27}{128} a^{3} + \frac{13}{64} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{8192} a^{12} - \frac{3}{4096} a^{11} + \frac{3}{8192} a^{10} + \frac{7}{2048} a^{9} + \frac{31}{8192} a^{8} - \frac{27}{4096} a^{7} - \frac{75}{8192} a^{6} + \frac{11}{1024} a^{4} + \frac{41}{256} a^{3} - \frac{19}{512} a^{2} + \frac{5}{32} a - \frac{9}{32}$, $\frac{1}{8192} a^{13} - \frac{1}{8192} a^{11} - \frac{1}{4096} a^{10} - \frac{9}{8192} a^{9} + \frac{1}{2048} a^{8} + \frac{113}{8192} a^{7} - \frac{25}{4096} a^{6} - \frac{39}{1024} a^{5} - \frac{3}{512} a^{4} - \frac{35}{512} a^{3} - \frac{29}{256} a^{2} - \frac{11}{32} a - \frac{7}{16}$, $\frac{1}{16384} a^{14} - \frac{1}{16384} a^{13} - \frac{1}{16384} a^{12} + \frac{15}{16384} a^{11} - \frac{7}{16384} a^{10} + \frac{61}{16384} a^{9} - \frac{51}{16384} a^{8} - \frac{115}{16384} a^{7} - \frac{35}{8192} a^{6} - \frac{101}{2048} a^{5} - \frac{21}{512} a^{4} - \frac{219}{1024} a^{3} - \frac{119}{512} a^{2} - \frac{19}{64} a - \frac{5}{32}$, $\frac{1}{327680} a^{15} - \frac{1}{65536} a^{14} - \frac{9}{327680} a^{13} + \frac{19}{327680} a^{12} + \frac{53}{65536} a^{11} + \frac{113}{327680} a^{10} + \frac{129}{65536} a^{9} + \frac{681}{327680} a^{8} - \frac{2179}{163840} a^{7} + \frac{1193}{40960} a^{6} + \frac{547}{10240} a^{5} + \frac{133}{4096} a^{4} + \frac{357}{2048} a^{3} - \frac{159}{640} a^{2} - \frac{9}{128} a + \frac{11}{80}$, $\frac{1}{10485760} a^{16} + \frac{7}{5242880} a^{15} + \frac{29}{2621440} a^{14} - \frac{113}{2621440} a^{13} + \frac{203}{5242880} a^{12} + \frac{333}{655360} a^{11} + \frac{2273}{2621440} a^{10} + \frac{8209}{2621440} a^{9} - \frac{27279}{10485760} a^{8} + \frac{59161}{5242880} a^{7} - \frac{3417}{131072} a^{6} - \frac{32619}{655360} a^{5} + \frac{6705}{131072} a^{4} + \frac{72441}{327680} a^{3} + \frac{14851}{81920} a^{2} - \frac{4497}{20480} a + \frac{403}{5120}$, $\frac{1}{4833319633611117081253900725644165120} a^{17} - \frac{102194691891778256233884459551}{4833319633611117081253900725644165120} a^{16} - \frac{2740794135774150134890735708097}{2416659816805558540626950362822082560} a^{15} - \frac{6830722792509661331174186021029}{604164954201389635156737590705520640} a^{14} - \frac{78846613670635454122477083732987}{2416659816805558540626950362822082560} a^{13} + \frac{131671902259765713185672300713529}{2416659816805558540626950362822082560} a^{12} + \frac{934056719926927297131513664961213}{1208329908402779270313475181411041280} a^{11} + \frac{113955433516318281111651564744881}{302082477100694817578368795352760320} a^{10} - \frac{133476401529416085039377822349699}{4833319633611117081253900725644165120} a^{9} - \frac{8405265440646510960526156566151083}{4833319633611117081253900725644165120} a^{8} - \frac{1936355385342593361767082953170665}{483331963361111708125390072564416512} a^{7} - \frac{3716555046381409620904498673698697}{151041238550347408789184397676380160} a^{6} + \frac{371018610177761909092838668416109}{15104123855034740878918439767638016} a^{5} + \frac{17394104258557403636054091258596577}{302082477100694817578368795352760320} a^{4} - \frac{26117321121923918485157851420544761}{151041238550347408789184397676380160} a^{3} - \frac{316181221685771502722619085816203}{37760309637586852197296099419095040} a^{2} + \frac{4151142447562233053449850495456777}{9440077409396713049324024854773760} a + \frac{97772192660046553461657288653205}{472003870469835652466201242738688}$

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

Class group and class number

$C_{19}\times C_{1639092}$, which has order $31142748$ (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:  \( 12689307150.302711 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-39}) \), 3.3.8281.1, 3.3.13689.1, 3.3.3969.1, 3.3.670761.2, 6.0.24069811311.1, 6.0.7308160119.1, 6.0.103827765951.7, 6.0.17546892445719.2, 9.9.301789003173921081.3

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{18}$

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
7Data not computed
13Data not computed