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Properties

Label 18.10.6630018602...5776.2
Degree $18$
Signature $[10, 4]$
Discriminant $2^{12}\cdot 3^{6}\cdot 7^{14}\cdot 41^{9}$
Root discriminant $66.59$
Ramified primes $2, 3, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4509, -27351, 215019, -819117, 1413873, -1149513, 253555, 283982, -229417, 46545, 13513, -8967, 2764, -810, -18, 133, -29, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 29*x^16 + 133*x^15 - 18*x^14 - 810*x^13 + 2764*x^12 - 8967*x^11 + 13513*x^10 + 46545*x^9 - 229417*x^8 + 283982*x^7 + 253555*x^6 - 1149513*x^5 + 1413873*x^4 - 819117*x^3 + 215019*x^2 - 27351*x + 4509)
 
gp: K = bnfinit(x^18 - 3*x^17 - 29*x^16 + 133*x^15 - 18*x^14 - 810*x^13 + 2764*x^12 - 8967*x^11 + 13513*x^10 + 46545*x^9 - 229417*x^8 + 283982*x^7 + 253555*x^6 - 1149513*x^5 + 1413873*x^4 - 819117*x^3 + 215019*x^2 - 27351*x + 4509, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 29 x^{16} + 133 x^{15} - 18 x^{14} - 810 x^{13} + 2764 x^{12} - 8967 x^{11} + 13513 x^{10} + 46545 x^{9} - 229417 x^{8} + 283982 x^{7} + 253555 x^{6} - 1149513 x^{5} + 1413873 x^{4} - 819117 x^{3} + 215019 x^{2} - 27351 x + 4509 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(663001860270502303855233859915776=2^{12}\cdot 3^{6}\cdot 7^{14}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.59$
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:  $2, 3, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{3} a^{8} + \frac{1}{6} a^{6} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{15} + \frac{1}{12} a^{13} + \frac{1}{12} a^{12} - \frac{1}{4} a^{11} + \frac{1}{12} a^{9} - \frac{1}{4} a^{8} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{5}{12} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{5148} a^{16} + \frac{31}{858} a^{15} + \frac{133}{5148} a^{14} - \frac{623}{5148} a^{13} + \frac{5}{52} a^{12} - \frac{1}{13} a^{11} + \frac{685}{5148} a^{10} - \frac{259}{1716} a^{9} + \frac{19}{1287} a^{8} - \frac{211}{429} a^{7} + \frac{361}{2574} a^{6} - \frac{73}{234} a^{5} - \frac{1487}{5148} a^{4} + \frac{181}{429} a^{3} + \frac{17}{52} a^{2} + \frac{229}{572} a - \frac{37}{143}$, $\frac{1}{112160419672266034426893720326397239556} a^{17} + \frac{562468733657723874451897937082869}{56080209836133017213446860163198619778} a^{16} + \frac{1499894063584685577954648575801227}{50093979308738738020050790677265404} a^{15} - \frac{5395382003926063976899708757654371231}{112160419672266034426893720326397239556} a^{14} - \frac{9770452530571332160158140054479018385}{112160419672266034426893720326397239556} a^{13} - \frac{2243126299543586573276226623657714}{25748489364615710382666143325619201} a^{12} + \frac{13187295136720243964984525029264258555}{112160419672266034426893720326397239556} a^{11} + \frac{4444002037746782009193663291465673873}{112160419672266034426893720326397239556} a^{10} + \frac{11394313017722776756788239950999702337}{56080209836133017213446860163198619778} a^{9} + \frac{7679685039644963558475375100954875485}{56080209836133017213446860163198619778} a^{8} - \frac{4793681847767578040167670920032442817}{56080209836133017213446860163198619778} a^{7} - \frac{3592767932478009814058430560387388638}{28040104918066508606723430081599309889} a^{6} - \frac{395857280106625652555318775523132113}{958636065574923371170031797661514868} a^{5} - \frac{7423895589901570369098952011119652410}{28040104918066508606723430081599309889} a^{4} + \frac{5439979235231949801061534797290539727}{12462268852474003825210413369599693284} a^{3} - \frac{7082051958225604099802321006046833881}{37386806557422011475631240108799079852} a^{2} + \frac{103131424482290059473848132051014263}{3115567213118500956302603342399923321} a - \frac{1618188460175150121121170502345569433}{6231134426237001912605206684799846642}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 11811526373.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.574470067776192.1

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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
41Data not computed