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Properties

Label 18.10.3036991852...3376.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{31}\cdot 3^{18}\cdot 7^{9}\cdot 67^{6}$
Root discriminant $106.37$
Ramified primes $2, 3, 7, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T903

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![512, 2304, -9792, -6656, 6576, -3024, 8708, 7848, -72, -1176, 285, -1404, -950, 372, 279, -20, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 30*x^16 - 20*x^15 + 279*x^14 + 372*x^13 - 950*x^12 - 1404*x^11 + 285*x^10 - 1176*x^9 - 72*x^8 + 7848*x^7 + 8708*x^6 - 3024*x^5 + 6576*x^4 - 6656*x^3 - 9792*x^2 + 2304*x + 512)
 
gp: K = bnfinit(x^18 - 30*x^16 - 20*x^15 + 279*x^14 + 372*x^13 - 950*x^12 - 1404*x^11 + 285*x^10 - 1176*x^9 - 72*x^8 + 7848*x^7 + 8708*x^6 - 3024*x^5 + 6576*x^4 - 6656*x^3 - 9792*x^2 + 2304*x + 512, 1)
 

Normalized defining polynomial

\( x^{18} - 30 x^{16} - 20 x^{15} + 279 x^{14} + 372 x^{13} - 950 x^{12} - 1404 x^{11} + 285 x^{10} - 1176 x^{9} - 72 x^{8} + 7848 x^{7} + 8708 x^{6} - 3024 x^{5} + 6576 x^{4} - 6656 x^{3} - 9792 x^{2} + 2304 x + 512 \)

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:  \(3036991852889853849525071814775013376=2^{31}\cdot 3^{18}\cdot 7^{9}\cdot 67^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.37$
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:  $2, 3, 7, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} + \frac{1}{8} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} + \frac{1}{8} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{10} + \frac{1}{16} a^{6} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{32} a^{15} - \frac{1}{16} a^{13} - \frac{1}{32} a^{11} - \frac{1}{8} a^{10} - \frac{1}{16} a^{9} - \frac{1}{4} a^{8} - \frac{3}{32} a^{7} + \frac{3}{8} a^{5} + \frac{3}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{16} - \frac{1}{32} a^{14} - \frac{1}{16} a^{13} - \frac{1}{64} a^{12} + \frac{1}{16} a^{11} - \frac{1}{32} a^{10} + \frac{1}{16} a^{9} - \frac{11}{64} a^{8} + \frac{1}{8} a^{7} - \frac{3}{16} a^{6} - \frac{3}{8} a^{5} - \frac{3}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{13715146078490334377019491456} a^{17} - \frac{5944876755618462917190921}{3428786519622583594254872864} a^{16} + \frac{67295392469913517246571939}{6857573039245167188509745728} a^{15} - \frac{53701196180944343685689931}{3428786519622583594254872864} a^{14} - \frac{457002200008594209010694881}{13715146078490334377019491456} a^{13} + \frac{13156073089653749439379579}{1714393259811291797127436432} a^{12} + \frac{768503178558905012226708955}{6857573039245167188509745728} a^{11} + \frac{38605827958273178998462839}{3428786519622583594254872864} a^{10} + \frac{1126023806513129426505876325}{13715146078490334377019491456} a^{9} + \frac{642072717673468445983850205}{3428786519622583594254872864} a^{8} + \frac{595640915541305985296723907}{3428786519622583594254872864} a^{7} + \frac{135956549393381205521366795}{1714393259811291797127436432} a^{6} + \frac{1419995777075954523462962085}{3428786519622583594254872864} a^{5} - \frac{45191864621328440161922933}{857196629905645898563718216} a^{4} + \frac{167484915155654556665703253}{428598314952822949281859108} a^{3} + \frac{211798243560961830310245617}{428598314952822949281859108} a^{2} + \frac{11188458811157267103044344}{107149578738205737320464777} a - \frac{26069621802363597888969510}{107149578738205737320464777}$

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:  \( 3756459757230 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T903:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 559872
The 174 conjugacy class representatives for t18n903 are not computed
Character table for t18n903 is not computed

Intermediate fields

\(\Q(\sqrt{7}) \), 3.3.469.1, 6.6.98542528.1

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 R R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ $18$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.9.1$x^{4} + 6 x^{2} + 2$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.8.16.21$x^{8} + 12 x^{7} + 20 x^{5} + 16 x^{4} + 40 x + 20$$4$$2$$16$$Q_8:C_2$$[2, 3, 3]^{2}$
$3$3.9.9.9$x^{9} + 18 x^{5} + 27 x^{2} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
3.9.9.4$x^{9} + 3 x^{6} + 9 x^{4} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.12.6.1$x^{12} + 8978 x^{8} + 7218312 x^{6} + 20151121 x^{4} + 31052877461 x^{2} + 13026007032336$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$