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Properties

Label 18.10.7414090047...6601.1
Degree $18$
Signature $[10, 4]$
Discriminant $7^{13}\cdot 83^{5}\cdot 181^{5}$
Root discriminant $58.96$
Ramified primes $7, 83, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![511909, 68763, -1191588, -440590, 1030277, 589667, -406768, -348932, 54703, 107520, 10205, -17664, -4468, 1445, 612, -46, -39, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 39*x^16 - 46*x^15 + 612*x^14 + 1445*x^13 - 4468*x^12 - 17664*x^11 + 10205*x^10 + 107520*x^9 + 54703*x^8 - 348932*x^7 - 406768*x^6 + 589667*x^5 + 1030277*x^4 - 440590*x^3 - 1191588*x^2 + 68763*x + 511909)
 
gp: K = bnfinit(x^18 - 39*x^16 - 46*x^15 + 612*x^14 + 1445*x^13 - 4468*x^12 - 17664*x^11 + 10205*x^10 + 107520*x^9 + 54703*x^8 - 348932*x^7 - 406768*x^6 + 589667*x^5 + 1030277*x^4 - 440590*x^3 - 1191588*x^2 + 68763*x + 511909, 1)
 

Normalized defining polynomial

\( x^{18} - 39 x^{16} - 46 x^{15} + 612 x^{14} + 1445 x^{13} - 4468 x^{12} - 17664 x^{11} + 10205 x^{10} + 107520 x^{9} + 54703 x^{8} - 348932 x^{7} - 406768 x^{6} + 589667 x^{5} + 1030277 x^{4} - 440590 x^{3} - 1191588 x^{2} + 68763 x + 511909 \)

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:  \(74140900471891896178682056546601=7^{13}\cdot 83^{5}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.96$
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:  $7, 83, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} + \frac{1}{7} a^{12} + \frac{2}{7} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{9} + \frac{3}{7} a^{8} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{13} + \frac{2}{7} a^{12} - \frac{1}{7} a^{11} - \frac{2}{7} a^{10} + \frac{3}{7} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{4} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{7} a^{16} + \frac{2}{7} a^{13} - \frac{2}{7} a^{12} + \frac{3}{7} a^{11} - \frac{3}{7} a^{10} + \frac{1}{7} a^{8} + \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{4} + \frac{2}{7} a^{3} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{20905769209511743185491331063154747453} a^{17} - \frac{865194947291650436850950958488447519}{20905769209511743185491331063154747453} a^{16} - \frac{1337329871104956640181720851254909595}{20905769209511743185491331063154747453} a^{15} - \frac{95193448435084389651058686479600694}{20905769209511743185491331063154747453} a^{14} + \frac{96833989524119187590990250235611238}{2986538458501677597927333009022106779} a^{13} - \frac{9563995748084475959428745497342340081}{20905769209511743185491331063154747453} a^{12} - \frac{5405644314029796835282771523289565182}{20905769209511743185491331063154747453} a^{11} - \frac{8249834448711871954702351285406290370}{20905769209511743185491331063154747453} a^{10} - \frac{742840815151257976263017710243090263}{2986538458501677597927333009022106779} a^{9} - \frac{6695518678291210715253912870507698976}{20905769209511743185491331063154747453} a^{8} - \frac{8387635213843107536667841315878856084}{20905769209511743185491331063154747453} a^{7} + \frac{439020008066068712420045042341561076}{20905769209511743185491331063154747453} a^{6} - \frac{6416094034825341827706498219259859200}{20905769209511743185491331063154747453} a^{5} + \frac{5094958827701626524228189670037230935}{20905769209511743185491331063154747453} a^{4} - \frac{3854743918225025267613564374148838077}{20905769209511743185491331063154747453} a^{3} - \frac{6593680213630586761502713490545314856}{20905769209511743185491331063154747453} a^{2} - \frac{2437201552000403666375613405975993718}{20905769209511743185491331063154747453} a + \frac{8817801880662272937987106745576854887}{20905769209511743185491331063154747453}$

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:  \( 5114850075.51 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 192 conjugacy class representatives for t18n839 are not computed
Character table for t18n839 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.26552265046321.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

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$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
83Data not computed
181Data not computed