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Properties

Label 18.4.29863270848...7519.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,7^{15}\cdot 41^{3}\cdot 97^{3}$
Root discriminant $20.15$
Ramified primes $7, 41, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T285

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27, 207, -633, 1177, -1543, 1998, -2117, 1858, -1498, 1083, -574, 234, -33, -49, 52, -34, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 17*x^16 - 34*x^15 + 52*x^14 - 49*x^13 - 33*x^12 + 234*x^11 - 574*x^10 + 1083*x^9 - 1498*x^8 + 1858*x^7 - 2117*x^6 + 1998*x^5 - 1543*x^4 + 1177*x^3 - 633*x^2 + 207*x - 27)
 
gp: K = bnfinit(x^18 - 6*x^17 + 17*x^16 - 34*x^15 + 52*x^14 - 49*x^13 - 33*x^12 + 234*x^11 - 574*x^10 + 1083*x^9 - 1498*x^8 + 1858*x^7 - 2117*x^6 + 1998*x^5 - 1543*x^4 + 1177*x^3 - 633*x^2 + 207*x - 27, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 17 x^{16} - 34 x^{15} + 52 x^{14} - 49 x^{13} - 33 x^{12} + 234 x^{11} - 574 x^{10} + 1083 x^{9} - 1498 x^{8} + 1858 x^{7} - 2117 x^{6} + 1998 x^{5} - 1543 x^{4} + 1177 x^{3} - 633 x^{2} + 207 x - 27 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-298632708486259154687519=-\,7^{15}\cdot 41^{3}\cdot 97^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.15$
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:  $7, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3758245235635798619343} a^{17} - \frac{157080598269995950262}{1252748411878599539781} a^{16} + \frac{507179888357617044782}{3758245235635798619343} a^{15} - \frac{482891283973301974279}{3758245235635798619343} a^{14} - \frac{1600222099678086197558}{3758245235635798619343} a^{13} + \frac{1447447871643919059137}{3758245235635798619343} a^{12} + \frac{621693993900807953032}{1252748411878599539781} a^{11} + \frac{129882355789939190054}{417582803959533179927} a^{10} + \frac{728742558228839888699}{3758245235635798619343} a^{9} + \frac{5461102107627422308}{1252748411878599539781} a^{8} + \frac{1751787080318098381280}{3758245235635798619343} a^{7} + \frac{1544028605648352443677}{3758245235635798619343} a^{6} + \frac{152620533779816582872}{3758245235635798619343} a^{5} - \frac{24131458209213732797}{417582803959533179927} a^{4} - \frac{1737450782637865701547}{3758245235635798619343} a^{3} - \frac{463300601337577515800}{3758245235635798619343} a^{2} + \frac{614424296317456283450}{1252748411878599539781} a - \frac{106499760105719945605}{417582803959533179927}$

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:  $10$
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:  \( 31972.2038925 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T285:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 34 conjugacy class representatives for t18n285
Character table for t18n285 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.66841439.1, 9.5.467890073.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ $18$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
7Data not computed
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.6.0.1$x^{6} - x + 7$$1$$6$$0$$C_6$$[\ ]^{6}$
97Data not computed