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Properties

Label 18.0.26188507747...1875.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 5^{9}\cdot 7^{15}$
Root discriminant $48.97$
Ramified primes $3, 5, 7$
Class number $648$ (GRH)
Class group $[2, 18, 18]$ (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![1100051, -729036, 726267, -443393, 474168, -297333, 236523, -120219, 65484, -23243, 8802, -2964, 1224, -405, 117, -19, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 9*x^16 - 19*x^15 + 117*x^14 - 405*x^13 + 1224*x^12 - 2964*x^11 + 8802*x^10 - 23243*x^9 + 65484*x^8 - 120219*x^7 + 236523*x^6 - 297333*x^5 + 474168*x^4 - 443393*x^3 + 726267*x^2 - 729036*x + 1100051)
 
gp: K = bnfinit(x^18 - 3*x^17 + 9*x^16 - 19*x^15 + 117*x^14 - 405*x^13 + 1224*x^12 - 2964*x^11 + 8802*x^10 - 23243*x^9 + 65484*x^8 - 120219*x^7 + 236523*x^6 - 297333*x^5 + 474168*x^4 - 443393*x^3 + 726267*x^2 - 729036*x + 1100051, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 9 x^{16} - 19 x^{15} + 117 x^{14} - 405 x^{13} + 1224 x^{12} - 2964 x^{11} + 8802 x^{10} - 23243 x^{9} + 65484 x^{8} - 120219 x^{7} + 236523 x^{6} - 297333 x^{5} + 474168 x^{4} - 443393 x^{3} + 726267 x^{2} - 729036 x + 1100051 \)

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:  \(-2618850774742652270958169921875=-\,3^{24}\cdot 5^{9}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.97$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(199,·)$, $\chi_{315}(139,·)$, $\chi_{315}(226,·)$, $\chi_{315}(16,·)$, $\chi_{315}(211,·)$, $\chi_{315}(151,·)$, $\chi_{315}(94,·)$, $\chi_{315}(34,·)$, $\chi_{315}(229,·)$, $\chi_{315}(106,·)$, $\chi_{315}(46,·)$, $\chi_{315}(304,·)$, $\chi_{315}(19,·)$, $\chi_{315}(244,·)$, $\chi_{315}(121,·)$, $\chi_{315}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{2}{13} a^{11} - \frac{3}{13} a^{10} + \frac{1}{13} a^{9} + \frac{6}{13} a^{7} + \frac{1}{13} a^{5} + \frac{6}{13} a^{4} + \frac{1}{13} a^{3} + \frac{2}{13} a^{2} - \frac{3}{13} a - \frac{4}{13}$, $\frac{1}{13} a^{13} + \frac{6}{13} a^{11} - \frac{6}{13} a^{10} - \frac{2}{13} a^{9} + \frac{6}{13} a^{8} + \frac{1}{13} a^{7} + \frac{1}{13} a^{6} + \frac{4}{13} a^{5} + \frac{2}{13} a^{4} + \frac{6}{13} a^{2} + \frac{2}{13} a - \frac{5}{13}$, $\frac{1}{13} a^{14} - \frac{5}{13} a^{11} + \frac{3}{13} a^{10} + \frac{1}{13} a^{8} + \frac{4}{13} a^{7} + \frac{4}{13} a^{6} - \frac{4}{13} a^{5} + \frac{3}{13} a^{4} + \frac{3}{13} a^{2} - \frac{2}{13}$, $\frac{1}{13} a^{15} - \frac{2}{13} a^{10} + \frac{6}{13} a^{9} + \frac{4}{13} a^{8} - \frac{5}{13} a^{7} - \frac{4}{13} a^{6} - \frac{5}{13} a^{5} + \frac{4}{13} a^{4} - \frac{5}{13} a^{3} - \frac{3}{13} a^{2} - \frac{4}{13} a + \frac{6}{13}$, $\frac{1}{923} a^{16} - \frac{25}{923} a^{15} + \frac{12}{923} a^{14} - \frac{27}{923} a^{13} - \frac{31}{923} a^{12} + \frac{5}{71} a^{11} - \frac{147}{923} a^{10} + \frac{215}{923} a^{9} - \frac{151}{923} a^{8} - \frac{70}{923} a^{7} - \frac{118}{923} a^{6} - \frac{97}{923} a^{5} - \frac{322}{923} a^{4} + \frac{18}{71} a^{3} + \frac{31}{71} a^{2} - \frac{258}{923} a - \frac{266}{923}$, $\frac{1}{157608935331780542931915764280155802136238053} a^{17} - \frac{83262021347818171794668454429771749729755}{157608935331780542931915764280155802136238053} a^{16} - \frac{1082633641371146176728372341360313295162377}{157608935331780542931915764280155802136238053} a^{15} - \frac{5438990815001052893779743803402089560554526}{157608935331780542931915764280155802136238053} a^{14} - \frac{2252730614028941877956620756284360169492000}{157608935331780542931915764280155802136238053} a^{13} - \frac{3054762016299619640770155731171895511469949}{157608935331780542931915764280155802136238053} a^{12} - \frac{328199853158869353192969074980484692193115}{2219844159602542858195996680002194396285043} a^{11} - \frac{35558002729441316685196965018231992169893470}{157608935331780542931915764280155802136238053} a^{10} - \frac{45563605802145279229689468094217621188225730}{157608935331780542931915764280155802136238053} a^{9} - \frac{40309822653676804310798940583671801676347656}{157608935331780542931915764280155802136238053} a^{8} + \frac{1059788061677618915295754976446079584718533}{157608935331780542931915764280155802136238053} a^{7} + \frac{20350570651919965230730043016881377007584071}{157608935331780542931915764280155802136238053} a^{6} - \frac{59812851948087964927505194486812832878412340}{157608935331780542931915764280155802136238053} a^{5} - \frac{27936125888808294255746552158545541567176788}{157608935331780542931915764280155802136238053} a^{4} + \frac{56912492873984832588198510279364910271962183}{157608935331780542931915764280155802136238053} a^{3} - \frac{35952387382728723977258856401765590007439811}{157608935331780542931915764280155802136238053} a^{2} + \frac{16470145516325234920067819294722941864916519}{157608935331780542931915764280155802136238053} a + \frac{27293405524257389174258709187725830504318313}{157608935331780542931915764280155802136238053}$

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

Class group and class number

$C_{2}\times C_{18}\times C_{18}$, which has order $648$ (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:  \( 54408.4888887 \) (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

\(\Q(\sqrt{-35}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, 6.0.281302875.3, 6.0.13783840875.2, 6.0.2100875.1, 6.0.13783840875.1, 9.9.62523502209.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed