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Properties

Label 18.12.3913507539...7771.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,7^{12}\cdot 71^{3}\cdot 181\cdot 7585403561^{3}$
Root discriminant $440.59$
Ramified primes $7, 71, 181, 7585403561$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T926

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7892777521, -3895392708, -7378892694, -3114283707, 630219135, 2627021727, 865071579, -381679857, -170197830, 24094476, 13734798, -776142, -592021, 12555, 14373, -81, -186, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 186*x^16 - 81*x^15 + 14373*x^14 + 12555*x^13 - 592021*x^12 - 776142*x^11 + 13734798*x^10 + 24094476*x^9 - 170197830*x^8 - 381679857*x^7 + 865071579*x^6 + 2627021727*x^5 + 630219135*x^4 - 3114283707*x^3 - 7378892694*x^2 - 3895392708*x + 7892777521)
 
gp: K = bnfinit(x^18 - 186*x^16 - 81*x^15 + 14373*x^14 + 12555*x^13 - 592021*x^12 - 776142*x^11 + 13734798*x^10 + 24094476*x^9 - 170197830*x^8 - 381679857*x^7 + 865071579*x^6 + 2627021727*x^5 + 630219135*x^4 - 3114283707*x^3 - 7378892694*x^2 - 3895392708*x + 7892777521, 1)
 

Normalized defining polynomial

\( x^{18} - 186 x^{16} - 81 x^{15} + 14373 x^{14} + 12555 x^{13} - 592021 x^{12} - 776142 x^{11} + 13734798 x^{10} + 24094476 x^{9} - 170197830 x^{8} - 381679857 x^{7} + 865071579 x^{6} + 2627021727 x^{5} + 630219135 x^{4} - 3114283707 x^{3} - 7378892694 x^{2} - 3895392708 x + 7892777521 \)

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:  \(-391350753932783462782732212010334985869953357771=-\,7^{12}\cdot 71^{3}\cdot 181\cdot 7585403561^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $440.59$
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:  $7, 71, 181, 7585403561$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{1}{9} a^{6} + \frac{1}{9} a^{4} + \frac{1}{3} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} + \frac{1}{9} a^{9} - \frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{3} a^{3} + \frac{1}{9} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{8} - \frac{1}{9} a^{6} + \frac{2}{9} a^{4} + \frac{1}{9} a^{2} - \frac{1}{9}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{9} - \frac{1}{9} a^{7} + \frac{2}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{9} a$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{9} a^{6} + \frac{4}{9} a^{4} - \frac{1}{9} a^{2} - \frac{1}{3}$, $\frac{1}{2559919958539493978210654147830292190506952900564044277624504747} a^{17} + \frac{86216921248242210809585023855988561387667042621137668353681556}{2559919958539493978210654147830292190506952900564044277624504747} a^{16} + \frac{110344104099766053909348402555829759039774838614553867284145257}{2559919958539493978210654147830292190506952900564044277624504747} a^{15} - \frac{25800206412563456771365690200584135914610996221502198941509772}{853306652846497992736884715943430730168984300188014759208168249} a^{14} + \frac{139902847434822874779362580568889533856400445616587623427417645}{2559919958539493978210654147830292190506952900564044277624504747} a^{13} + \frac{45835003185015927343934758826018745970243475226773843440716557}{2559919958539493978210654147830292190506952900564044277624504747} a^{12} + \frac{15585153150303845666220793627522100919908806060806250739144453}{2559919958539493978210654147830292190506952900564044277624504747} a^{11} + \frac{26797514156716101101725958765388160883359633963636087807186361}{284435550948832664245628238647810243389661433396004919736056083} a^{10} - \frac{417112773747461106075176930930517367399775326932540492414148748}{2559919958539493978210654147830292190506952900564044277624504747} a^{9} - \frac{6039505512508422209148571789162093971593716364169967991943326}{2559919958539493978210654147830292190506952900564044277624504747} a^{8} + \frac{82272551913995188569323512843283087731456496056067884441531642}{853306652846497992736884715943430730168984300188014759208168249} a^{7} + \frac{4521684408773303698000802540074645811827474041861574886399944}{853306652846497992736884715943430730168984300188014759208168249} a^{6} + \frac{155662177264365190997772947475209063963654099677898070218152678}{2559919958539493978210654147830292190506952900564044277624504747} a^{5} + \frac{30409885607531313251986790561171875691224080248808731702988049}{853306652846497992736884715943430730168984300188014759208168249} a^{4} + \frac{384717101703174372940064475656541639722930150786077991314843903}{853306652846497992736884715943430730168984300188014759208168249} a^{3} - \frac{976670433999229743873430276518614333452400455599721927581509116}{2559919958539493978210654147830292190506952900564044277624504747} a^{2} - \frac{146961564880739865674125515258840257406119458974111574138521086}{853306652846497992736884715943430730168984300188014759208168249} a + \frac{747902276512753220822538085909131708126738994452038065464158738}{2559919958539493978210654147830292190506952900564044277624504747}$

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:  \( 296005180121000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T926:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1119744
The 267 conjugacy class representatives for t18n926 are not computed
Character table for t18n926 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.1293091330447231.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.9.0.1}{9} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ $18$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ $18$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ $18$

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
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
71Data not computed
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
7585403561Data not computed