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Properties

Label 18.0.13266257117...2288.5
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 17^{12}$
Root discriminant $78.65$
Ramified primes $2, 3, 17$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19476, -109404, 360072, -814854, 1123398, -836352, 283375, -19323, 17625, -78, -8355, 5337, -1676, 225, 69, -90, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 90*x^15 + 69*x^14 + 225*x^13 - 1676*x^12 + 5337*x^11 - 8355*x^10 - 78*x^9 + 17625*x^8 - 19323*x^7 + 283375*x^6 - 836352*x^5 + 1123398*x^4 - 814854*x^3 + 360072*x^2 - 109404*x + 19476)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 90*x^15 + 69*x^14 + 225*x^13 - 1676*x^12 + 5337*x^11 - 8355*x^10 - 78*x^9 + 17625*x^8 - 19323*x^7 + 283375*x^6 - 836352*x^5 + 1123398*x^4 - 814854*x^3 + 360072*x^2 - 109404*x + 19476, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 90 x^{15} + 69 x^{14} + 225 x^{13} - 1676 x^{12} + 5337 x^{11} - 8355 x^{10} - 78 x^{9} + 17625 x^{8} - 19323 x^{7} + 283375 x^{6} - 836352 x^{5} + 1123398 x^{4} - 814854 x^{3} + 360072 x^{2} - 109404 x + 19476 \)

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:  \(-13266257117930788904129800273932288=-\,2^{12}\cdot 3^{33}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.65$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{5}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{6} a^{6} + \frac{1}{6} a^{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{6} a^{7} + \frac{1}{6} a^{4}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{11} - \frac{1}{6} a^{8} + \frac{1}{6} a^{5}$, $\frac{1}{666774} a^{15} - \frac{7244}{111129} a^{14} + \frac{21371}{333387} a^{13} - \frac{7919}{111129} a^{12} - \frac{32080}{333387} a^{11} + \frac{4134}{37043} a^{10} + \frac{13882}{333387} a^{9} - \frac{3200}{6537} a^{8} - \frac{31001}{333387} a^{7} - \frac{2486}{37043} a^{6} + \frac{9212}{19611} a^{5} + \frac{15805}{111129} a^{4} - \frac{9209}{74086} a^{3} + \frac{21616}{111129} a^{2} + \frac{12436}{37043} a + \frac{3003}{37043}$, $\frac{1}{30261358627794} a^{16} - \frac{19293313}{30261358627794} a^{15} + \frac{361539263423}{15130679313897} a^{14} + \frac{248029389731}{30261358627794} a^{13} + \frac{1072044585560}{15130679313897} a^{12} - \frac{734440106525}{15130679313897} a^{11} + \frac{254864350855}{1780079919282} a^{10} + \frac{1429619358977}{15130679313897} a^{9} + \frac{4927406798383}{15130679313897} a^{8} - \frac{62693874767}{4323051232542} a^{7} + \frac{2661769089697}{15130679313897} a^{6} + \frac{2283939956096}{15130679313897} a^{5} - \frac{547904489191}{5043559771299} a^{4} - \frac{182981019479}{1441017077514} a^{3} + \frac{259977509174}{720508538757} a^{2} + \frac{387761206480}{1681186590433} a + \frac{323275307135}{1681186590433}$, $\frac{1}{80542874982031950228013098} a^{17} + \frac{352134721570}{40271437491015975114006549} a^{16} - \frac{8074093794583882921}{11506124997433135746859014} a^{15} - \frac{2965185040049862636098921}{40271437491015975114006549} a^{14} - \frac{28551804524979066249553}{2368908087706822065529797} a^{13} + \frac{2196000875027568848498927}{80542874982031950228013098} a^{12} + \frac{530218361742140397116541}{4474604165668441679334061} a^{11} - \frac{2840888870587696348576981}{40271437491015975114006549} a^{10} + \frac{9083155814414427239050117}{80542874982031950228013098} a^{9} - \frac{13821461838669158688656566}{40271437491015975114006549} a^{8} - \frac{12722734613069453731219555}{40271437491015975114006549} a^{7} - \frac{16321558907081128897057}{55971421113295309400982} a^{6} - \frac{30185647810369735475115781}{80542874982031950228013098} a^{5} - \frac{6125622946877724736223543}{13423812497005325038002183} a^{4} + \frac{303430302488223731867888}{1917687499572189291143169} a^{3} - \frac{6534712436583635223971858}{13423812497005325038002183} a^{2} + \frac{665922767873751681117462}{4474604165668441679334061} a - \frac{645328623006648496291869}{4474604165668441679334061}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( -\frac{7565492309725}{52363709759479773} a^{17} + \frac{125698333061045}{104727419518959546} a^{16} - \frac{83633358010321}{17454569919826591} a^{15} + \frac{1004242874454683}{104727419518959546} a^{14} - \frac{309695853131737}{104727419518959546} a^{13} - \frac{3693467852961323}{104727419518959546} a^{12} + \frac{447373245513801}{2053478814097246} a^{11} - \frac{64660059209166881}{104727419518959546} a^{10} + \frac{4695867846647923}{6160436442291738} a^{9} + \frac{61371661794996901}{104727419518959546} a^{8} - \frac{229815221989710383}{104727419518959546} a^{7} + \frac{5217642523859}{4281053816742} a^{6} - \frac{4181956669140809485}{104727419518959546} a^{5} + \frac{1624831263197158182}{17454569919826591} a^{4} - \frac{9899857547914075265}{104727419518959546} a^{3} + \frac{796060347411334117}{17454569919826591} a^{2} - \frac{239525240485617696}{17454569919826591} a + \frac{63307529338576150}{17454569919826591} \) (order $6$)
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:  \( 5411532426.153941 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.867.1 x3, 6.0.236727913392.11, 6.0.2834352.4, 6.0.236727913392.10, 6.0.2255067.2

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

Sibling fields

Degree 18 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.11.14$x^{6} + 12 x^{3} + 18 x^{2} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.14$x^{6} + 12 x^{3} + 18 x^{2} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.14$x^{6} + 12 x^{3} + 18 x^{2} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
$17$17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$