↑

Properties

Label 18.12.8659616134...7136.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{18}\cdot 3^{6}\cdot 7^{12}\cdot 41^{9}$
Root discriminant $67.59$
Ramified primes $2, 3, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2563673, 4327624, 3621940, -6573762, -1822347, 3611310, 244572, -850430, 73884, 48204, -23453, 15544, 936, -3178, 290, 230, -33, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 33*x^16 + 230*x^15 + 290*x^14 - 3178*x^13 + 936*x^12 + 15544*x^11 - 23453*x^10 + 48204*x^9 + 73884*x^8 - 850430*x^7 + 244572*x^6 + 3611310*x^5 - 1822347*x^4 - 6573762*x^3 + 3621940*x^2 + 4327624*x - 2563673)
 
gp: K = bnfinit(x^18 - 6*x^17 - 33*x^16 + 230*x^15 + 290*x^14 - 3178*x^13 + 936*x^12 + 15544*x^11 - 23453*x^10 + 48204*x^9 + 73884*x^8 - 850430*x^7 + 244572*x^6 + 3611310*x^5 - 1822347*x^4 - 6573762*x^3 + 3621940*x^2 + 4327624*x - 2563673, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 33 x^{16} + 230 x^{15} + 290 x^{14} - 3178 x^{13} + 936 x^{12} + 15544 x^{11} - 23453 x^{10} + 48204 x^{9} + 73884 x^{8} - 850430 x^{7} + 244572 x^{6} + 3611310 x^{5} - 1822347 x^{4} - 6573762 x^{3} + 3621940 x^{2} + 4327624 x - 2563673 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-865961613414533621361938102747136=-\,2^{18}\cdot 3^{6}\cdot 7^{12}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} - \frac{3}{7} a^{13} - \frac{1}{7} a^{12} - \frac{3}{7} a^{11} + \frac{2}{7} a^{10} + \frac{1}{7} a^{9} - \frac{3}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} + \frac{3}{7} a^{5} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{15} - \frac{3}{7} a^{13} + \frac{1}{7} a^{12} - \frac{1}{7} a^{8} - \frac{1}{7} a^{7} - \frac{2}{7} a^{6} + \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3}$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{13} - \frac{3}{7} a^{12} - \frac{2}{7} a^{11} - \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{3}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3}$, $\frac{1}{1458995765257427350972926821443182179313901751} a^{17} + \frac{35853984145192964701345686549789393408899894}{1458995765257427350972926821443182179313901751} a^{16} - \frac{39503292717758170553010673757860436765051679}{1458995765257427350972926821443182179313901751} a^{15} - \frac{21478185202702294594876412735762103930714137}{1458995765257427350972926821443182179313901751} a^{14} + \frac{242091482405116862444908236608564421922107553}{1458995765257427350972926821443182179313901751} a^{13} + \frac{139822734153494817507518684523389219796652866}{1458995765257427350972926821443182179313901751} a^{12} - \frac{94983288482397022299582293136907556609953993}{208427966465346764424703831634740311330557393} a^{11} - \frac{582712101199485820773779512001503548157962839}{1458995765257427350972926821443182179313901751} a^{10} - \frac{375836866568940742474299298883012819126348868}{1458995765257427350972926821443182179313901751} a^{9} - \frac{105266458475816579572242007389560651030660321}{1458995765257427350972926821443182179313901751} a^{8} + \frac{271883771663130936938937033322397842305624979}{1458995765257427350972926821443182179313901751} a^{7} + \frac{94831847843831162707446032368411234666344549}{1458995765257427350972926821443182179313901751} a^{6} + \frac{715067481422572088540059157657584145844552140}{1458995765257427350972926821443182179313901751} a^{5} + \frac{591674205235869246133282163760821006787705122}{1458995765257427350972926821443182179313901751} a^{4} + \frac{5825868429579155469834203459209314584740224}{1458995765257427350972926821443182179313901751} a^{3} - \frac{47276146893847067818182069931084528477423651}{208427966465346764424703831634740311330557393} a^{2} - \frac{71146025783756689668669344300471216630931718}{208427966465346764424703831634740311330557393} a + \frac{48568832116466300455394223858748821613379704}{208427966465346764424703831634740311330557393}$

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:  $14$
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:  \( 25173934569.0 \) (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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
41Data not computed