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Properties

Label 18.2.53285018970...9952.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{18}\cdot 7^{15}\cdot 29^{4}$
Root discriminant $50.94$
Ramified primes $2, 3, 7, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 0, -3843, 0, -24339, 0, -38562, 0, -17973, 0, -495, 0, 1546, 0, 375, 0, 33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 33*x^16 + 375*x^14 + 1546*x^12 - 495*x^10 - 17973*x^8 - 38562*x^6 - 24339*x^4 - 3843*x^2 - 7)
 
gp: K = bnfinit(x^18 + 33*x^16 + 375*x^14 + 1546*x^12 - 495*x^10 - 17973*x^8 - 38562*x^6 - 24339*x^4 - 3843*x^2 - 7, 1)
 

Normalized defining polynomial

\( x^{18} + 33 x^{16} + 375 x^{14} + 1546 x^{12} - 495 x^{10} - 17973 x^{8} - 38562 x^{6} - 24339 x^{4} - 3843 x^{2} - 7 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5328501897056475209143937789952=2^{12}\cdot 3^{18}\cdot 7^{15}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.94$
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:  $2, 3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{29} a^{12} - \frac{9}{29} a^{10} - \frac{11}{29} a^{8} - \frac{11}{29} a^{6} + \frac{5}{29} a^{4} - \frac{7}{29} a^{2} - \frac{10}{29}$, $\frac{1}{58} a^{13} - \frac{1}{58} a^{12} + \frac{10}{29} a^{11} - \frac{10}{29} a^{10} + \frac{9}{29} a^{9} - \frac{9}{29} a^{8} - \frac{11}{58} a^{7} + \frac{11}{58} a^{6} - \frac{12}{29} a^{5} + \frac{12}{29} a^{4} + \frac{11}{29} a^{3} - \frac{11}{29} a^{2} + \frac{19}{58} a - \frac{19}{58}$, $\frac{1}{116} a^{14} - \frac{1}{116} a^{12} + \frac{1}{29} a^{10} - \frac{41}{116} a^{8} - \frac{25}{116} a^{6} - \frac{27}{58} a^{4} - \frac{37}{116} a^{2} - \frac{51}{116}$, $\frac{1}{232} a^{15} - \frac{1}{232} a^{14} - \frac{1}{232} a^{13} + \frac{1}{232} a^{12} + \frac{1}{58} a^{11} - \frac{1}{58} a^{10} - \frac{41}{232} a^{9} + \frac{41}{232} a^{8} - \frac{25}{232} a^{7} + \frac{25}{232} a^{6} - \frac{27}{116} a^{5} + \frac{27}{116} a^{4} + \frac{79}{232} a^{3} - \frac{79}{232} a^{2} - \frac{51}{232} a + \frac{51}{232}$, $\frac{1}{407816899184} a^{16} + \frac{31427105}{203908449592} a^{14} + \frac{774274873}{407816899184} a^{12} + \frac{199006748051}{407816899184} a^{10} - \frac{11290713455}{101954224796} a^{8} + \frac{136827351007}{407816899184} a^{6} - \frac{88986344355}{407816899184} a^{4} - \frac{9207652387}{203908449592} a^{2} + \frac{161693509543}{407816899184}$, $\frac{1}{815633798368} a^{17} - \frac{1}{815633798368} a^{16} + \frac{31427105}{407816899184} a^{15} - \frac{31427105}{407816899184} a^{14} + \frac{774274873}{815633798368} a^{13} - \frac{774274873}{815633798368} a^{12} - \frac{208810151133}{815633798368} a^{11} + \frac{208810151133}{815633798368} a^{10} + \frac{90663511341}{203908449592} a^{9} - \frac{90663511341}{203908449592} a^{8} + \frac{136827351007}{815633798368} a^{7} - \frac{136827351007}{815633798368} a^{6} - \frac{88986344355}{815633798368} a^{5} + \frac{88986344355}{815633798368} a^{4} + \frac{194700797205}{407816899184} a^{3} - \frac{194700797205}{407816899184} a^{2} + \frac{161693509543}{815633798368} a - \frac{161693509543}{815633798368}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}$, which has order $4$ (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:  $9$
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:  \( 22323896.0175 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T766:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 144 conjugacy class representatives for t18n766 are not computed
Character table for t18n766 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.13632439166829.1

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 R R $18$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ $18$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.16$x^{12} - 16 x^{10} - 23 x^{8} + 24 x^{6} - 29 x^{4} - 8 x^{2} - 13$$2$$6$$12$12T134$[2, 2, 2, 2, 2, 2]^{6}$
$3$3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$