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Properties

Label 18.2.60006020121...5793.1
Degree $18$
Signature $[2, 8]$
Discriminant $7^{12}\cdot 41^{6}\cdot 97^{3}$
Root discriminant $27.05$
Ramified primes $7, 41, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-631, 856, -1581, 4131, -5021, 9227, -4949, 7050, -1567, 3908, -219, 1052, 208, 18, 128, -31, 22, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 22*x^16 - 31*x^15 + 128*x^14 + 18*x^13 + 208*x^12 + 1052*x^11 - 219*x^10 + 3908*x^9 - 1567*x^8 + 7050*x^7 - 4949*x^6 + 9227*x^5 - 5021*x^4 + 4131*x^3 - 1581*x^2 + 856*x - 631)
 
gp: K = bnfinit(x^18 - 3*x^17 + 22*x^16 - 31*x^15 + 128*x^14 + 18*x^13 + 208*x^12 + 1052*x^11 - 219*x^10 + 3908*x^9 - 1567*x^8 + 7050*x^7 - 4949*x^6 + 9227*x^5 - 5021*x^4 + 4131*x^3 - 1581*x^2 + 856*x - 631, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 22 x^{16} - 31 x^{15} + 128 x^{14} + 18 x^{13} + 208 x^{12} + 1052 x^{11} - 219 x^{10} + 3908 x^{9} - 1567 x^{8} + 7050 x^{7} - 4949 x^{6} + 9227 x^{5} - 5021 x^{4} + 4131 x^{3} - 1581 x^{2} + 856 x - 631 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(60006020121228767347575793=7^{12}\cdot 41^{6}\cdot 97^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.05$
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:  $7, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{83} a^{16} - \frac{17}{83} a^{15} - \frac{22}{83} a^{14} + \frac{8}{83} a^{13} - \frac{5}{83} a^{12} - \frac{10}{83} a^{11} + \frac{15}{83} a^{10} + \frac{10}{83} a^{9} - \frac{24}{83} a^{8} + \frac{13}{83} a^{7} + \frac{39}{83} a^{6} + \frac{16}{83} a^{5} + \frac{14}{83} a^{4} + \frac{37}{83} a^{3} - \frac{25}{83} a^{2} + \frac{23}{83} a + \frac{1}{83}$, $\frac{1}{2034844795102833367577848674724673} a^{17} - \frac{7494140874578527519855593779836}{2034844795102833367577848674724673} a^{16} + \frac{491741244068159261210134893850475}{2034844795102833367577848674724673} a^{15} - \frac{326067457708121924614678456383331}{2034844795102833367577848674724673} a^{14} - \frac{689980072847281963434261054840933}{2034844795102833367577848674724673} a^{13} - \frac{849294523989936503165831079514529}{2034844795102833367577848674724673} a^{12} - \frac{969058451084997015404446473471309}{2034844795102833367577848674724673} a^{11} + \frac{134719703989585896668070169543470}{2034844795102833367577848674724673} a^{10} - \frac{766375832648433253779906669708887}{2034844795102833367577848674724673} a^{9} + \frac{955995402274051148548124701035493}{2034844795102833367577848674724673} a^{8} - \frac{624209175195689313224559909014751}{2034844795102833367577848674724673} a^{7} - \frac{738594430219542508977374174390997}{2034844795102833367577848674724673} a^{6} - \frac{80416548011200579148976057427937}{2034844795102833367577848674724673} a^{5} + \frac{148247375506211041886460619541473}{2034844795102833367577848674724673} a^{4} + \frac{347525894749580217337745450522664}{2034844795102833367577848674724673} a^{3} + \frac{781999037397122795278246304066643}{2034844795102833367577848674724673} a^{2} + \frac{316335278166359140333418729231144}{2034844795102833367577848674724673} a - \frac{964421890582111667148881846390342}{2034844795102833367577848674724673}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 151821.786372 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n765 are not computed
Character table for t18n765 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.5.467890073.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
7Data not computed
41Data not computed
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.3.3$x^{4} + 485$$4$$1$$3$$C_4$$[\ ]_{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$