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Properties

Label 18.0.18777800570...3232.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 23^{9}\cdot 37^{14}$
Root discriminant $546.29$
Ramified primes $2, 3, 23, 37$
Class number $1197439362$ (GRH)
Class group $[3, 3, 3, 3, 14783202]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1343941011449, 2096104977033, 1596953609076, 703629239684, 169228552005, 6179192277, -9152309280, -2482374477, -17013828, 109842349, 17018340, -1579305, -634314, -13221, 12147, 742, -132, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 132*x^16 + 742*x^15 + 12147*x^14 - 13221*x^13 - 634314*x^12 - 1579305*x^11 + 17018340*x^10 + 109842349*x^9 - 17013828*x^8 - 2482374477*x^7 - 9152309280*x^6 + 6179192277*x^5 + 169228552005*x^4 + 703629239684*x^3 + 1596953609076*x^2 + 2096104977033*x + 1343941011449)
 
gp: K = bnfinit(x^18 - 9*x^17 - 132*x^16 + 742*x^15 + 12147*x^14 - 13221*x^13 - 634314*x^12 - 1579305*x^11 + 17018340*x^10 + 109842349*x^9 - 17013828*x^8 - 2482374477*x^7 - 9152309280*x^6 + 6179192277*x^5 + 169228552005*x^4 + 703629239684*x^3 + 1596953609076*x^2 + 2096104977033*x + 1343941011449, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 132 x^{16} + 742 x^{15} + 12147 x^{14} - 13221 x^{13} - 634314 x^{12} - 1579305 x^{11} + 17018340 x^{10} + 109842349 x^{9} - 17013828 x^{8} - 2482374477 x^{7} - 9152309280 x^{6} + 6179192277 x^{5} + 169228552005 x^{4} + 703629239684 x^{3} + 1596953609076 x^{2} + 2096104977033 x + 1343941011449 \)

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:  \(-18777800570904586987097565808700098548206552543232=-\,2^{12}\cdot 3^{24}\cdot 23^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $546.29$
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:  $2, 3, 23, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{37} a^{12} - \frac{6}{37} a^{11} + \frac{14}{37} a^{10} - \frac{15}{37} a^{9} - \frac{10}{37} a^{8} - \frac{10}{37} a^{7} + \frac{8}{37} a^{6} + \frac{14}{37} a^{5} + \frac{10}{37} a^{4} + \frac{16}{37} a^{3} + \frac{14}{37} a^{2} + \frac{1}{37} a - \frac{1}{37}$, $\frac{1}{37} a^{13} + \frac{15}{37} a^{11} - \frac{5}{37} a^{10} + \frac{11}{37} a^{9} + \frac{4}{37} a^{8} - \frac{15}{37} a^{7} - \frac{12}{37} a^{6} - \frac{17}{37} a^{5} + \frac{2}{37} a^{4} - \frac{1}{37} a^{3} + \frac{11}{37} a^{2} + \frac{5}{37} a - \frac{6}{37}$, $\frac{1}{37} a^{14} + \frac{11}{37} a^{11} - \frac{14}{37} a^{10} + \frac{7}{37} a^{9} - \frac{13}{37} a^{8} - \frac{10}{37} a^{7} + \frac{11}{37} a^{6} + \frac{14}{37} a^{5} - \frac{3}{37} a^{4} - \frac{7}{37} a^{3} + \frac{17}{37} a^{2} + \frac{16}{37} a + \frac{15}{37}$, $\frac{1}{37} a^{15} + \frac{15}{37} a^{11} + \frac{1}{37} a^{10} + \frac{4}{37} a^{9} - \frac{11}{37} a^{8} + \frac{10}{37} a^{7} - \frac{9}{37} a^{5} - \frac{6}{37} a^{4} - \frac{11}{37} a^{3} + \frac{10}{37} a^{2} + \frac{4}{37} a + \frac{11}{37}$, $\frac{1}{37} a^{16} + \frac{17}{37} a^{11} + \frac{16}{37} a^{10} - \frac{8}{37} a^{9} + \frac{12}{37} a^{8} + \frac{2}{37} a^{7} - \frac{18}{37} a^{6} + \frac{6}{37} a^{5} - \frac{13}{37} a^{4} - \frac{8}{37} a^{3} + \frac{16}{37} a^{2} - \frac{4}{37} a + \frac{15}{37}$, $\frac{1}{72234179172943325057563031319603948535392135643919185693580056703} a^{17} + \frac{629842605769142102337044341837215692182683479059815835884056158}{72234179172943325057563031319603948535392135643919185693580056703} a^{16} - \frac{874210986593457335992570971986043998190060216927469923065158840}{72234179172943325057563031319603948535392135643919185693580056703} a^{15} - \frac{575348982010698080313922808189950199801462427913595437155178211}{72234179172943325057563031319603948535392135643919185693580056703} a^{14} - \frac{159997467934957681665382902551604728755465800675640830408321641}{72234179172943325057563031319603948535392135643919185693580056703} a^{13} + \frac{312233198741960207466884939616997141837124881724028025968989563}{72234179172943325057563031319603948535392135643919185693580056703} a^{12} + \frac{29711675580852237631903873976450018287859268864379005722621500889}{72234179172943325057563031319603948535392135643919185693580056703} a^{11} + \frac{15804942329455848975773961215981721299669287528440660203771982665}{72234179172943325057563031319603948535392135643919185693580056703} a^{10} - \frac{16057149531138207710212231758021581076219429211212913006041970147}{72234179172943325057563031319603948535392135643919185693580056703} a^{9} - \frac{29833455049562198179397264888197346330807518871448680120643177684}{72234179172943325057563031319603948535392135643919185693580056703} a^{8} - \frac{25024678294745191589497492442267685796463239891061594174644318562}{72234179172943325057563031319603948535392135643919185693580056703} a^{7} + \frac{22925020886426572579995230454514134252347129292526026235856071886}{72234179172943325057563031319603948535392135643919185693580056703} a^{6} + \frac{1996690402655207810448307485946148775242656267366904262510674933}{72234179172943325057563031319603948535392135643919185693580056703} a^{5} + \frac{21502244295879438384279536965984389486429301429903768306461385015}{72234179172943325057563031319603948535392135643919185693580056703} a^{4} - \frac{15880448048572375770404826307403937549908141872852213437535310128}{72234179172943325057563031319603948535392135643919185693580056703} a^{3} + \frac{943888624226884067847824477887288948430597451786229129934003136}{1952275112782252028582784630259566176632219882268086099826488019} a^{2} - \frac{30817046421316415592951534460446583932510038529808139667946793366}{72234179172943325057563031319603948535392135643919185693580056703} a - \frac{48970436476472942915877365458697155343899986299156867164332079}{478372047502935927533530008739098996923126726118670103930993753}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{14783202}$, which has order $1197439362$ (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:  \( 118546543.87559307 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.3.110889.1, 3.3.148.1, 6.0.149609937695607.1, 6.0.266505968.3, 9.9.3228844269788073792.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$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}$
$23$23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.12.6.1$x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$