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Properties

Label 18.0.86390775685...4599.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 13^{15}\cdot 19^{12}$
Root discriminant $313.67$
Ramified primes $3, 13, 19$
Class number $552631872$ (GRH)
Class group $[2, 2, 2, 2, 34539492]$ (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![109314048000, -27902361600, -4097894400, 1597948416, -1952795136, 658281600, 105153016, -78893766, 21698505, -4183416, 583197, 206604, -55742, -5400, 3114, -54, -75, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 75*x^16 - 54*x^15 + 3114*x^14 - 5400*x^13 - 55742*x^12 + 206604*x^11 + 583197*x^10 - 4183416*x^9 + 21698505*x^8 - 78893766*x^7 + 105153016*x^6 + 658281600*x^5 - 1952795136*x^4 + 1597948416*x^3 - 4097894400*x^2 - 27902361600*x + 109314048000)
 
gp: K = bnfinit(x^18 - 75*x^16 - 54*x^15 + 3114*x^14 - 5400*x^13 - 55742*x^12 + 206604*x^11 + 583197*x^10 - 4183416*x^9 + 21698505*x^8 - 78893766*x^7 + 105153016*x^6 + 658281600*x^5 - 1952795136*x^4 + 1597948416*x^3 - 4097894400*x^2 - 27902361600*x + 109314048000, 1)
 

Normalized defining polynomial

\( x^{18} - 75 x^{16} - 54 x^{15} + 3114 x^{14} - 5400 x^{13} - 55742 x^{12} + 206604 x^{11} + 583197 x^{10} - 4183416 x^{9} + 21698505 x^{8} - 78893766 x^{7} + 105153016 x^{6} + 658281600 x^{5} - 1952795136 x^{4} + 1597948416 x^{3} - 4097894400 x^{2} - 27902361600 x + 109314048000 \)

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:  \(-863907756858622745144456333032042876658884599=-\,3^{27}\cdot 13^{15}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $313.67$
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:  $3, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2223=3^{2}\cdot 13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{2223}(1,·)$, $\chi_{2223}(391,·)$, $\chi_{2223}(524,·)$, $\chi_{2223}(140,·)$, $\chi_{2223}(1616,·)$, $\chi_{2223}(1426,·)$, $\chi_{2223}(919,·)$, $\chi_{2223}(1816,·)$, $\chi_{2223}(1793,·)$, $\chi_{2223}(1949,·)$, $\chi_{2223}(296,·)$, $\chi_{2223}(1388,·)$, $\chi_{2223}(368,·)$, $\chi_{2223}(818,·)$, $\chi_{2223}(1717,·)$, $\chi_{2223}(1654,·)$, $\chi_{2223}(1147,·)$, $\chi_{2223}(2044,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{32} a^{6} + \frac{1}{32} a^{5} + \frac{1}{32} a^{4} + \frac{7}{32} a^{3} + \frac{3}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{7} - \frac{1}{16} a^{5} + \frac{1}{32} a^{4} + \frac{11}{64} a^{3} + \frac{7}{32} a^{2} - \frac{3}{8} a$, $\frac{1}{128} a^{8} - \frac{1}{64} a^{6} + \frac{1}{32} a^{5} + \frac{5}{128} a^{4} - \frac{5}{32} a^{3} + \frac{7}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{256} a^{9} - \frac{1}{256} a^{8} + \frac{1}{128} a^{6} - \frac{11}{256} a^{5} - \frac{9}{256} a^{4} - \frac{27}{128} a^{3} + \frac{1}{32} a^{2} + \frac{1}{4} a$, $\frac{1}{1024} a^{10} + \frac{1}{1024} a^{9} - \frac{1}{512} a^{7} + \frac{13}{1024} a^{6} + \frac{17}{1024} a^{5} - \frac{7}{512} a^{4} + \frac{11}{64} a^{3} + \frac{3}{16} a^{2} - \frac{3}{8} a$, $\frac{1}{4096} a^{11} - \frac{1}{2048} a^{10} - \frac{3}{4096} a^{9} + \frac{7}{2048} a^{8} + \frac{3}{4096} a^{7} - \frac{11}{2048} a^{6} + \frac{223}{4096} a^{5} - \frac{107}{2048} a^{4} + \frac{9}{128} a^{3} - \frac{17}{128} a^{2} + \frac{1}{16} a$, $\frac{1}{4096} a^{12} + \frac{1}{4096} a^{10} + \frac{15}{4096} a^{8} - \frac{1}{128} a^{7} + \frac{59}{4096} a^{6} + \frac{5}{128} a^{5} - \frac{3}{1024} a^{4} + \frac{1}{32} a^{3} - \frac{5}{64} a^{2}$, $\frac{1}{327680} a^{13} + \frac{5}{65536} a^{12} + \frac{27}{327680} a^{11} - \frac{11}{327680} a^{10} + \frac{353}{327680} a^{9} + \frac{979}{327680} a^{8} + \frac{1481}{327680} a^{7} + \frac{4311}{327680} a^{6} + \frac{333}{163840} a^{5} + \frac{793}{40960} a^{4} - \frac{499}{2048} a^{3} + \frac{69}{512} a^{2} - \frac{43}{320} a$, $\frac{1}{327680} a^{14} - \frac{19}{163840} a^{12} + \frac{17}{163840} a^{11} + \frac{17}{81920} a^{10} + \frac{277}{163840} a^{9} + \frac{303}{163840} a^{8} + \frac{1043}{163840} a^{7} - \frac{3829}{327680} a^{6} - \frac{233}{163840} a^{5} - \frac{493}{8192} a^{4} - \frac{385}{2048} a^{3} - \frac{629}{2560} a^{2} + \frac{19}{64} a$, $\frac{1}{5242880} a^{15} + \frac{1}{2621440} a^{14} - \frac{93}{1310720} a^{12} - \frac{299}{2621440} a^{11} + \frac{17}{327680} a^{10} - \frac{609}{655360} a^{9} - \frac{39}{262144} a^{8} - \frac{10739}{5242880} a^{7} - \frac{8873}{2621440} a^{6} + \frac{7141}{327680} a^{5} + \frac{3563}{81920} a^{4} + \frac{3993}{20480} a^{3} - \frac{717}{10240} a^{2} + \frac{149}{1280} a - \frac{1}{2}$, $\frac{1}{10485760} a^{16} - \frac{1}{10485760} a^{15} - \frac{3}{5242880} a^{14} + \frac{3}{2621440} a^{13} + \frac{579}{5242880} a^{12} + \frac{457}{5242880} a^{11} + \frac{521}{1310720} a^{10} + \frac{5027}{2621440} a^{9} - \frac{10063}{10485760} a^{8} + \frac{3227}{2097152} a^{7} + \frac{36579}{5242880} a^{6} + \frac{35133}{655360} a^{5} - \frac{4773}{163840} a^{4} + \frac{10027}{40960} a^{3} + \frac{1823}{20480} a^{2} - \frac{163}{512} a - \frac{1}{4}$, $\frac{1}{18004831016951519765309512048857639917977600} a^{17} + \frac{93434172904634419833293108322966479}{3600966203390303953061902409771527983595520} a^{16} - \frac{85591535298193173376045298754275813}{1800483101695151976530951204885763991797760} a^{15} - \frac{6179732447203509996466639785910063071}{4501207754237879941327378012214409979494400} a^{14} + \frac{3963795668956123215437574617242985547}{9002415508475759882654756024428819958988800} a^{13} - \frac{5221267007921160446983347737724936203}{360096620339030395306190240977152798359552} a^{12} + \frac{2380950016820339251614142906521587759}{562650969279734992665922251526801247436800} a^{11} + \frac{942081456248506191428817277408980522971}{4501207754237879941327378012214409979494400} a^{10} - \frac{16812344403392761715481821715795011308863}{18004831016951519765309512048857639917977600} a^{9} - \frac{33219348665154095508630002155870460209581}{18004831016951519765309512048857639917977600} a^{8} + \frac{4739737111336969872015090683867037614777}{1800483101695151976530951204885763991797760} a^{7} - \frac{4893316650319252429322363462966337499237}{2250603877118939970663689006107204989747200} a^{6} + \frac{8511465622138188668175448853601723466057}{140662742319933748166480562881700311859200} a^{5} - \frac{26797364271508782022799309502672906397}{703313711599668740832402814408501559296} a^{4} + \frac{2972618518679411012553266772923378626977}{35165685579983437041620140720425077964800} a^{3} - \frac{60814732361281783654027900101500472153}{8791421394995859260405035180106269491200} a^{2} + \frac{42006501844340631915506028288830742159}{219785534874896481510125879502656737280} a + \frac{136768770474572031909820979980466329}{343414898242025752359571686722901152}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{34539492}$, which has order $552631872$ (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:  \( 931139254937.0632 \) (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

\(\Q(\sqrt{-39}) \), 3.3.29241.1, 3.3.4941729.3, 3.3.61009.1, 3.3.13689.1, 6.0.5635542809871.3, 6.0.952406734868199.1, 6.0.1306456426431.2, 6.0.7308160119.1, 9.9.120680409781884363489.9

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$3$3.6.9.11$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.11$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.11$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$[2]_{2}$
$13$13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
19Data not computed