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Properties

Label 18.0.43281927256...2768.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 7^{12}\cdot 13^{12}$
Root discriminant $57.22$
Ramified primes $2, 7, 13$
Class number $1072$ (GRH)
Class group $[4, 268]$ (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![966337, 1805644, 3195846, 1890250, 742493, -23318, -133405, -86316, 22175, -2428, 9676, -4058, 1346, -742, 297, -66, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 21*x^16 - 66*x^15 + 297*x^14 - 742*x^13 + 1346*x^12 - 4058*x^11 + 9676*x^10 - 2428*x^9 + 22175*x^8 - 86316*x^7 - 133405*x^6 - 23318*x^5 + 742493*x^4 + 1890250*x^3 + 3195846*x^2 + 1805644*x + 966337)
 
gp: K = bnfinit(x^18 - 6*x^17 + 21*x^16 - 66*x^15 + 297*x^14 - 742*x^13 + 1346*x^12 - 4058*x^11 + 9676*x^10 - 2428*x^9 + 22175*x^8 - 86316*x^7 - 133405*x^6 - 23318*x^5 + 742493*x^4 + 1890250*x^3 + 3195846*x^2 + 1805644*x + 966337, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 21 x^{16} - 66 x^{15} + 297 x^{14} - 742 x^{13} + 1346 x^{12} - 4058 x^{11} + 9676 x^{10} - 2428 x^{9} + 22175 x^{8} - 86316 x^{7} - 133405 x^{6} - 23318 x^{5} + 742493 x^{4} + 1890250 x^{3} + 3195846 x^{2} + 1805644 x + 966337 \)

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:  \(-43281927256346630219321487392768=-\,2^{27}\cdot 7^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.22$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(728=2^{3}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{728}(1,·)$, $\chi_{728}(555,·)$, $\chi_{728}(211,·)$, $\chi_{728}(417,·)$, $\chi_{728}(9,·)$, $\chi_{728}(529,·)$, $\chi_{728}(659,·)$, $\chi_{728}(235,·)$, $\chi_{728}(347,·)$, $\chi_{728}(81,·)$, $\chi_{728}(289,·)$, $\chi_{728}(547,·)$, $\chi_{728}(625,·)$, $\chi_{728}(107,·)$, $\chi_{728}(113,·)$, $\chi_{728}(627,·)$, $\chi_{728}(393,·)$, $\chi_{728}(443,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4}$, $\frac{1}{996} a^{13} - \frac{17}{166} a^{12} + \frac{35}{249} a^{11} - \frac{32}{249} a^{10} - \frac{187}{996} a^{9} + \frac{47}{498} a^{8} + \frac{1}{166} a^{7} + \frac{1}{498} a^{6} + \frac{145}{498} a^{5} - \frac{27}{166} a^{4} - \frac{71}{166} a^{3} - \frac{74}{249} a^{2} - \frac{135}{332} a + \frac{65}{249}$, $\frac{1}{996} a^{14} - \frac{55}{996} a^{12} + \frac{52}{249} a^{11} + \frac{203}{996} a^{10} - \frac{14}{249} a^{9} - \frac{39}{332} a^{8} + \frac{29}{249} a^{7} - \frac{1}{249} a^{6} + \frac{3}{83} a^{5} + \frac{40}{83} a^{4} + \frac{19}{249} a^{3} + \frac{93}{332} a^{2} + \frac{71}{249} a - \frac{41}{332}$, $\frac{1}{996} a^{15} + \frac{19}{249} a^{12} - \frac{65}{996} a^{11} - \frac{31}{249} a^{10} + \frac{14}{249} a^{9} - \frac{16}{83} a^{8} - \frac{43}{249} a^{7} + \frac{73}{498} a^{6} + \frac{247}{498} a^{5} + \frac{65}{498} a^{4} + \frac{85}{332} a^{3} - \frac{5}{83} a^{2} - \frac{81}{166} a - \frac{71}{498}$, $\frac{1}{2131488588522372} a^{16} + \frac{236808415919}{1065744294261186} a^{15} + \frac{562767293075}{2131488588522372} a^{14} + \frac{39916912243}{532872147130593} a^{13} - \frac{19341289783164}{177624049043531} a^{12} - \frac{86774976589783}{532872147130593} a^{11} + \frac{434459075785325}{2131488588522372} a^{10} + \frac{69773185113292}{532872147130593} a^{9} + \frac{363119373961235}{2131488588522372} a^{8} + \frac{70678810759460}{532872147130593} a^{7} - \frac{36030261805427}{1065744294261186} a^{6} - \frac{41812287946687}{177624049043531} a^{5} + \frac{246002110862383}{2131488588522372} a^{4} - \frac{61035816426478}{532872147130593} a^{3} + \frac{263926905940709}{710496196174124} a^{2} - \frac{38171781913271}{355248098087062} a - \frac{558594063415243}{2131488588522372}$, $\frac{1}{87484882222451185018651919844} a^{17} - \frac{4743614840218}{21871220555612796254662979961} a^{16} + \frac{2316905519479629133597045}{21871220555612796254662979961} a^{15} - \frac{20783907495590266539150107}{43742441111225592509325959922} a^{14} - \frac{4937018753911368559275283}{43742441111225592509325959922} a^{13} - \frac{1963006849956712659230144741}{87484882222451185018651919844} a^{12} - \frac{5174850319300114924265868296}{21871220555612796254662979961} a^{11} + \frac{2637002679938285452472777833}{14580813703741864169775319974} a^{10} - \frac{13187878841123672113475848883}{87484882222451185018651919844} a^{9} - \frac{3715874562274749904276472129}{87484882222451185018651919844} a^{8} + \frac{4610543412318114129082519966}{21871220555612796254662979961} a^{7} - \frac{2949779740163884323450463966}{21871220555612796254662979961} a^{6} + \frac{12104669230247077570870788557}{29161627407483728339550639948} a^{5} - \frac{8354750207834813354270896997}{21871220555612796254662979961} a^{4} + \frac{7841222050124682724122374987}{43742441111225592509325959922} a^{3} + \frac{2681318926715011776932051153}{7290406851870932084887659987} a^{2} - \frac{10163970864223434847458081431}{29161627407483728339550639948} a + \frac{37569882647306744161231664671}{87484882222451185018651919844}$

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

Class group and class number

$C_{4}\times C_{268}$, which has order $1072$ (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:  \( 205236.825908 \) (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

\(\Q(\sqrt{-2}) \), 3.3.8281.1, 3.3.169.1, \(\Q(\zeta_{7})^+\), 3.3.8281.2, 6.0.35110380032.7, 6.0.14623232.1, 6.0.1229312.1, 6.0.35110380032.4, 9.9.567869252041.1

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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
7Data not computed
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$