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Properties

Label 18.0.21433361072...3968.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 29^{8}$
Root discriminant $25.54$
Ramified primes $2, 3, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 54, -199, 555, -993, 1431, -1272, 1437, -993, 924, -372, 254, -33, 48, -2, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 2*x^15 + 48*x^14 - 33*x^13 + 254*x^12 - 372*x^11 + 924*x^10 - 993*x^9 + 1437*x^8 - 1272*x^7 + 1431*x^6 - 993*x^5 + 555*x^4 - 199*x^3 + 54*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^18 + 9*x^16 - 2*x^15 + 48*x^14 - 33*x^13 + 254*x^12 - 372*x^11 + 924*x^10 - 993*x^9 + 1437*x^8 - 1272*x^7 + 1431*x^6 - 993*x^5 + 555*x^4 - 199*x^3 + 54*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} - 2 x^{15} + 48 x^{14} - 33 x^{13} + 254 x^{12} - 372 x^{11} + 924 x^{10} - 993 x^{9} + 1437 x^{8} - 1272 x^{7} + 1431 x^{6} - 993 x^{5} + 555 x^{4} - 199 x^{3} + 54 x^{2} - 9 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-21433361072561590534483968=-\,2^{12}\cdot 3^{21}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.54$
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:  $2, 3, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{8091} a^{15} + \frac{16}{261} a^{14} + \frac{1001}{8091} a^{13} - \frac{748}{8091} a^{12} + \frac{575}{8091} a^{11} - \frac{17}{2697} a^{10} + \frac{263}{2697} a^{9} + \frac{896}{8091} a^{8} - \frac{865}{8091} a^{7} + \frac{2575}{8091} a^{6} - \frac{3703}{8091} a^{5} - \frac{1309}{8091} a^{4} + \frac{3938}{8091} a^{3} - \frac{301}{2697} a^{2} - \frac{1415}{8091} a - \frac{3394}{8091}$, $\frac{1}{8091} a^{16} + \frac{412}{8091} a^{14} - \frac{332}{2697} a^{13} - \frac{67}{899} a^{12} + \frac{631}{8091} a^{11} - \frac{295}{2697} a^{10} + \frac{617}{8091} a^{9} - \frac{92}{2697} a^{8} + \frac{95}{8091} a^{7} + \frac{2869}{8091} a^{6} - \frac{142}{899} a^{5} + \frac{1075}{2697} a^{4} + \frac{1174}{8091} a^{3} - \frac{3926}{8091} a^{2} + \frac{2620}{8091} a - \frac{71}{261}$, $\frac{1}{290895723} a^{17} + \frac{6326}{290895723} a^{16} + \frac{2596}{290895723} a^{15} + \frac{31702319}{290895723} a^{14} + \frac{4220262}{32321747} a^{13} - \frac{25761383}{290895723} a^{12} + \frac{29616713}{290895723} a^{11} + \frac{38686040}{290895723} a^{10} + \frac{5634133}{290895723} a^{9} - \frac{69943381}{290895723} a^{8} - \frac{97301218}{290895723} a^{7} + \frac{55013864}{290895723} a^{6} - \frac{3511574}{96965241} a^{5} - \frac{39360335}{290895723} a^{4} + \frac{31217047}{96965241} a^{3} - \frac{4485985}{96965241} a^{2} - \frac{37403}{1042637} a + \frac{96053600}{290895723}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{21928142}{96965241} a^{17} - \frac{3741457}{96965241} a^{16} - \frac{198489655}{96965241} a^{15} + \frac{3289712}{32321747} a^{14} - \frac{351849787}{32321747} a^{13} + \frac{181144893}{32321747} a^{12} - \frac{5502460603}{96965241} a^{11} + \frac{2409753023}{32321747} a^{10} - \frac{6386654785}{32321747} a^{9} + \frac{6221726801}{32321747} a^{8} - \frac{28790534099}{96965241} a^{7} + \frac{23428772026}{96965241} a^{6} - \frac{28148672042}{96965241} a^{5} + \frac{5859249682}{32321747} a^{4} - \frac{3311530456}{32321747} a^{3} + \frac{3080825285}{96965241} a^{2} - \frac{981911767}{96965241} a + \frac{164578034}{96965241} \) (order $6$)
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:  \( 535161.981474463 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.34992.1, 9.3.890970363072.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 9 siblings: data not computed
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
3Data not computed
$29$29.6.4.2$x^{6} - 29 x^{3} + 2523$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
29.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
29.6.4.2$x^{6} - 29 x^{3} + 2523$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$