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Properties

Label 18.0.76038033095...2571.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 11^{9}\cdot 13^{12}$
Root discriminant $67.10$
Ramified primes $7, 11, 13$
Class number $2548$ (GRH)
Class group $[14, 182]$ (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![7656713, -399099, 6350272, -160053, 2351913, 100280, 549956, 3818, 87170, -7614, 9498, -612, 338, -266, 202, 24, -11, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 11*x^16 + 24*x^15 + 202*x^14 - 266*x^13 + 338*x^12 - 612*x^11 + 9498*x^10 - 7614*x^9 + 87170*x^8 + 3818*x^7 + 549956*x^6 + 100280*x^5 + 2351913*x^4 - 160053*x^3 + 6350272*x^2 - 399099*x + 7656713)
 
gp: K = bnfinit(x^18 - 3*x^17 - 11*x^16 + 24*x^15 + 202*x^14 - 266*x^13 + 338*x^12 - 612*x^11 + 9498*x^10 - 7614*x^9 + 87170*x^8 + 3818*x^7 + 549956*x^6 + 100280*x^5 + 2351913*x^4 - 160053*x^3 + 6350272*x^2 - 399099*x + 7656713, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 11 x^{16} + 24 x^{15} + 202 x^{14} - 266 x^{13} + 338 x^{12} - 612 x^{11} + 9498 x^{10} - 7614 x^{9} + 87170 x^{8} + 3818 x^{7} + 549956 x^{6} + 100280 x^{5} + 2351913 x^{4} - 160053 x^{3} + 6350272 x^{2} - 399099 x + 7656713 \)

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:  \(-760380330951008959273099338892571=-\,7^{12}\cdot 11^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.10$
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:  $7, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1001=7\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1001}(1,·)$, $\chi_{1001}(386,·)$, $\chi_{1001}(835,·)$, $\chi_{1001}(263,·)$, $\chi_{1001}(716,·)$, $\chi_{1001}(848,·)$, $\chi_{1001}(529,·)$, $\chi_{1001}(274,·)$, $\chi_{1001}(659,·)$, $\chi_{1001}(989,·)$, $\chi_{1001}(991,·)$, $\chi_{1001}(144,·)$, $\chi_{1001}(802,·)$, $\chi_{1001}(100,·)$, $\chi_{1001}(417,·)$, $\chi_{1001}(562,·)$, $\chi_{1001}(373,·)$, $\chi_{1001}(120,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{6} a^{13} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{423046170} a^{15} + \frac{6296203}{84609234} a^{14} + \frac{6851251}{84609234} a^{13} - \frac{728689}{42304617} a^{12} - \frac{9542873}{423046170} a^{11} - \frac{417905}{84609234} a^{10} + \frac{3927409}{84609234} a^{9} - \frac{10377031}{42304617} a^{8} + \frac{16099313}{70507695} a^{7} + \frac{8291024}{42304617} a^{6} - \frac{20588257}{423046170} a^{5} + \frac{1891189}{28203078} a^{4} + \frac{61590647}{211523085} a^{3} - \frac{2108640}{4700513} a^{2} + \frac{18140062}{70507695} a - \frac{38073023}{423046170}$, $\frac{1}{153142713540} a^{16} + \frac{17}{25523785590} a^{15} + \frac{198634322}{7657135677} a^{14} - \frac{505844995}{7657135677} a^{13} - \frac{908492554}{12761892795} a^{12} - \frac{632597611}{25523785590} a^{11} - \frac{1140192347}{15314271354} a^{10} - \frac{534641795}{15314271354} a^{9} + \frac{6052806482}{38285678385} a^{8} + \frac{9039529273}{76571356770} a^{7} + \frac{18214019197}{38285678385} a^{6} - \frac{6566205076}{38285678385} a^{5} + \frac{11117386217}{76571356770} a^{4} - \frac{2578866998}{38285678385} a^{3} + \frac{10078794809}{51047571180} a^{2} - \frac{27591968107}{76571356770} a - \frac{48406320431}{153142713540}$, $\frac{1}{110691041042770670248786524108420} a^{17} + \frac{8382114744691445473}{3689701368092355674959550803614} a^{16} + \frac{13720660669757437941269}{27672760260692667562196631027105} a^{15} - \frac{42268544904240837664782022249}{1844850684046177837479775401807} a^{14} + \frac{1059725549372912077498676423783}{27672760260692667562196631027105} a^{13} + \frac{15060651359308037389820214041}{11069104104277067024878652410842} a^{12} - \frac{402334268031693103256595786787}{27672760260692667562196631027105} a^{11} - \frac{17255207861331563398708508653}{614950228015392612493258467269} a^{10} - \frac{2370534272573072988129596278711}{55345520521385335124393262054210} a^{9} - \frac{245054254582875302074185336256}{5534552052138533512439326205421} a^{8} + \frac{3818398629996407087173726013104}{27672760260692667562196631027105} a^{7} + \frac{1906281508596892461066822280427}{5534552052138533512439326205421} a^{6} + \frac{2011377182350970559825654920199}{6149502280153926124932584672690} a^{5} - \frac{3266861171411223228006652729547}{11069104104277067024878652410842} a^{4} - \frac{36840711326834070282387071672249}{110691041042770670248786524108420} a^{3} - \frac{2124291034198983585092624224489}{55345520521385335124393262054210} a^{2} + \frac{27657867728814024985769842023817}{110691041042770670248786524108420} a + \frac{23767845425706520914102870323351}{55345520521385335124393262054210}$

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

Class group and class number

$C_{14}\times C_{182}$, which has order $2548$ (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

\(\Q(\sqrt{-11}) \), 3.3.169.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 3.3.8281.1, 6.0.38014691.1, 6.0.91273273091.4, 6.0.3195731.1, 6.0.91273273091.5, 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
7Data not computed
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$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}$