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Properties

Label 18.10.1011167298...9776.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{39}\cdot 3^{18}\cdot 7^{15}$
Root discriminant $68.17$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T857

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3584, 0, 12096, -3584, -16296, 8064, 11599, -4536, -2394, 512, -1605, -288, 468, 96, 45, -8, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^16 - 8*x^15 + 45*x^14 + 96*x^13 + 468*x^12 - 288*x^11 - 1605*x^10 + 512*x^9 - 2394*x^8 - 4536*x^7 + 11599*x^6 + 8064*x^5 - 16296*x^4 - 3584*x^3 + 12096*x^2 - 3584)
 
gp: K = bnfinit(x^18 - 18*x^16 - 8*x^15 + 45*x^14 + 96*x^13 + 468*x^12 - 288*x^11 - 1605*x^10 + 512*x^9 - 2394*x^8 - 4536*x^7 + 11599*x^6 + 8064*x^5 - 16296*x^4 - 3584*x^3 + 12096*x^2 - 3584, 1)
 

Normalized defining polynomial

\( x^{18} - 18 x^{16} - 8 x^{15} + 45 x^{14} + 96 x^{13} + 468 x^{12} - 288 x^{11} - 1605 x^{10} + 512 x^{9} - 2394 x^{8} - 4536 x^{7} + 11599 x^{6} + 8064 x^{5} - 16296 x^{4} - 3584 x^{3} + 12096 x^{2} - 3584 \)

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:  \(1011167298805722181508656609099776=2^{39}\cdot 3^{18}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.17$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{5}$, $\frac{1}{128} a^{14} - \frac{1}{64} a^{12} - \frac{1}{16} a^{11} + \frac{13}{128} a^{10} - \frac{1}{4} a^{9} - \frac{7}{32} a^{8} - \frac{1}{4} a^{7} - \frac{5}{128} a^{6} + \frac{11}{64} a^{4} - \frac{7}{16} a^{3} - \frac{17}{128} a^{2} - \frac{7}{16}$, $\frac{1}{512} a^{15} - \frac{1}{256} a^{13} - \frac{1}{64} a^{12} - \frac{51}{512} a^{11} - \frac{1}{16} a^{10} - \frac{23}{128} a^{9} - \frac{1}{16} a^{8} - \frac{69}{512} a^{7} + \frac{43}{256} a^{5} - \frac{23}{64} a^{4} + \frac{239}{512} a^{3} + \frac{9}{64} a$, $\frac{1}{32768} a^{16} + \frac{27}{16384} a^{14} - \frac{1}{4096} a^{13} + \frac{3933}{32768} a^{12} - \frac{15}{1024} a^{11} - \frac{769}{8192} a^{10} - \frac{65}{1024} a^{9} + \frac{6299}{32768} a^{8} - \frac{7}{128} a^{7} - \frac{3809}{16384} a^{6} - \frac{311}{4096} a^{5} - \frac{12609}{32768} a^{4} + \frac{143}{512} a^{3} - \frac{927}{2048} a^{2} - \frac{113}{512}$, $\frac{1}{293084870759215726592} a^{17} + \frac{16563290313557}{146542435379607863296} a^{16} + \frac{60212457282329627}{146542435379607863296} a^{15} - \frac{98000994419027723}{73271217689803931648} a^{14} + \frac{7439069038383786509}{293084870759215726592} a^{13} + \frac{7670406378441499633}{146542435379607863296} a^{12} + \frac{3759436777587901263}{73271217689803931648} a^{11} - \frac{3883768037346608601}{36635608844901965824} a^{10} - \frac{61798926876408229029}{293084870759215726592} a^{9} - \frac{12723816414905253641}{146542435379607863296} a^{8} + \frac{8306222983555355167}{146542435379607863296} a^{7} + \frac{15809521303350908637}{73271217689803931648} a^{6} - \frac{28913972705630856561}{293084870759215726592} a^{5} + \frac{67822253946953103435}{146542435379607863296} a^{4} + \frac{8190082491168071865}{18317804422450982912} a^{3} + \frac{4437511952020533333}{9158902211225491456} a^{2} - \frac{316492243120906609}{4579451105612745728} a + \frac{126876531593586491}{2289725552806372864}$

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:  \( 12976547921.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T857:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 159 conjugacy class representatives for t18n857 are not computed
Character table for t18n857 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), 6.6.1229312.1

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 ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ $18$ $18$

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
$2$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.12.30.273$x^{12} + 2 x^{10} - 3 x^{8} + 4 x^{6} - 5 x^{4} - 6 x^{2} - 1$$4$$3$$30$12T134$[2, 2, 2, 3, 7/2, 7/2, 7/2]^{3}$
3Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.9.8.2$x^{9} - 7$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{3}$