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Properties

Label 18.0.10134905874...1083.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{10}\cdot 67^{10}$
Root discriminant $52.79$
Ramified primes $3, 7, 67$
Class number $9$ (GRH)
Class group $[9]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![134848, 309680, 26068, -771736, 97727, 394695, 30115, -114078, -32814, 7658, 18452, 1716, -2956, -426, 354, 54, -27, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 27*x^16 + 54*x^15 + 354*x^14 - 426*x^13 - 2956*x^12 + 1716*x^11 + 18452*x^10 + 7658*x^9 - 32814*x^8 - 114078*x^7 + 30115*x^6 + 394695*x^5 + 97727*x^4 - 771736*x^3 + 26068*x^2 + 309680*x + 134848)
 
gp: K = bnfinit(x^18 - 3*x^17 - 27*x^16 + 54*x^15 + 354*x^14 - 426*x^13 - 2956*x^12 + 1716*x^11 + 18452*x^10 + 7658*x^9 - 32814*x^8 - 114078*x^7 + 30115*x^6 + 394695*x^5 + 97727*x^4 - 771736*x^3 + 26068*x^2 + 309680*x + 134848, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 27 x^{16} + 54 x^{15} + 354 x^{14} - 426 x^{13} - 2956 x^{12} + 1716 x^{11} + 18452 x^{10} + 7658 x^{9} - 32814 x^{8} - 114078 x^{7} + 30115 x^{6} + 394695 x^{5} + 97727 x^{4} - 771736 x^{3} + 26068 x^{2} + 309680 x + 134848 \)

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:  \(-10134905874162866002761240391083=-\,3^{9}\cdot 7^{10}\cdot 67^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.79$
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:  $3, 7, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{3}{16} a^{6} + \frac{1}{8} a^{4} - \frac{3}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{16} a^{7} - \frac{3}{16} a^{4} - \frac{3}{8} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{16} a^{8} - \frac{3}{16} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{30016} a^{15} - \frac{773}{30016} a^{14} + \frac{463}{30016} a^{13} - \frac{205}{30016} a^{12} - \frac{675}{30016} a^{11} + \frac{2745}{30016} a^{10} + \frac{3043}{30016} a^{9} - \frac{2547}{30016} a^{8} - \frac{173}{4288} a^{7} - \frac{393}{4288} a^{6} - \frac{257}{30016} a^{5} + \frac{2675}{30016} a^{4} + \frac{995}{3752} a^{3} - \frac{115}{1072} a^{2} - \frac{33}{67} a - \frac{26}{67}$, $\frac{1}{5823104} a^{16} - \frac{25}{2911552} a^{15} - \frac{18133}{1455776} a^{14} + \frac{2903}{207968} a^{13} + \frac{3837}{415936} a^{12} + \frac{78005}{1455776} a^{11} + \frac{19903}{415936} a^{10} - \frac{150645}{2911552} a^{9} - \frac{39219}{727888} a^{8} - \frac{50195}{207968} a^{7} + \frac{10635}{43456} a^{6} - \frac{318273}{1455776} a^{5} - \frac{738819}{5823104} a^{4} + \frac{220441}{1455776} a^{3} - \frac{89971}{207968} a^{2} + \frac{14541}{51992} a + \frac{4853}{12998}$, $\frac{1}{274414673487933659086398185223866752} a^{17} - \frac{17796519499999246199714221483}{274414673487933659086398185223866752} a^{16} + \frac{519403345654473920509549058583}{34301834185991707385799773152983344} a^{15} + \frac{16552759786545737687670559190043}{1414508626226462160239165903215808} a^{14} + \frac{159315463578582351259492862693899}{17150917092995853692899886576491672} a^{13} + \frac{110325495162287917593618560369023}{9800524053140487824514220900852384} a^{12} - \frac{110575717922404261156868998558270}{2143864636624481711612485822061459} a^{11} + \frac{14003272039727674103139650308525819}{137207336743966829543199092611933376} a^{10} - \frac{59160188536710535169023117225781}{34301834185991707385799773152983344} a^{9} + \frac{17140497923278835909758192339700787}{137207336743966829543199092611933376} a^{8} + \frac{291615047788535860164242776798703}{8575458546497926846449943288245836} a^{7} + \frac{11866342571200019842634002310114655}{68603668371983414771599546305966688} a^{6} - \frac{4139532900104878798193522760371209}{274414673487933659086398185223866752} a^{5} - \frac{43352396239848406344132523855233051}{274414673487933659086398185223866752} a^{4} - \frac{2999084669373442222181410466165043}{8575458546497926846449943288245836} a^{3} + \frac{4599565413500768176686152924283685}{9800524053140487824514220900852384} a^{2} - \frac{215249132796489554445646890217489}{2450131013285121956128555225213096} a + \frac{189328169629205284941752306359467}{612532753321280489032138806303274}$

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

Class group and class number

$C_{9}$, 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:  \( \frac{63070084499689997787}{10423054587882227395017856} a^{17} - \frac{86658308529260226485}{5211527293941113697508928} a^{16} - \frac{853965806703195287685}{5211527293941113697508928} a^{15} + \frac{1442003459319454295053}{5211527293941113697508928} a^{14} + \frac{2750692339809226946649}{1302881823485278424377232} a^{13} - \frac{10039484221963899594327}{5211527293941113697508928} a^{12} - \frac{22278914378801212976435}{1302881823485278424377232} a^{11} + \frac{14547823215578737742037}{2605763646970556848754464} a^{10} + \frac{536950199058747900668359}{5211527293941113697508928} a^{9} + \frac{365907263450373890309819}{5211527293941113697508928} a^{8} - \frac{159084510456769199338511}{1302881823485278424377232} a^{7} - \frac{3370177485543569033096669}{5211527293941113697508928} a^{6} - \frac{89408940586539597619547}{10423054587882227395017856} a^{5} + \frac{10234032071207460138334355}{5211527293941113697508928} a^{4} + \frac{2607184771854803094424141}{2605763646970556848754464} a^{3} - \frac{4363243709153271743475549}{1302881823485278424377232} a^{2} + \frac{75370685741910210491469}{162860227935659803047154} a + \frac{78214958976504701961857}{81430113967829901523577} \) (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:  \( 448034530.39626753 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.469.1, 6.0.5938947.2, 6.0.5938947.1

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

Sibling fields

Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$67$67.3.0.1$x^{3} - x + 16$$1$$3$$0$$C_3$$[\ ]^{3}$
67.3.2.1$x^{3} - 67$$3$$1$$2$$C_3$$[\ ]_{3}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
67.6.5.5$x^{6} + 4288$$6$$1$$5$$C_6$$[\ ]_{6}$