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Properties

Label 18.0.88805207069...4187.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 1801^{12}$
Root discriminant $769.17$
Ramified primes $3, 1801$
Class number $14279586048$ (GRH)
Class group $[26, 26, 312, 67704]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![373714754427, 0, 0, 0, 0, 0, 418673070, 0, 0, 0, 0, 0, 118863, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 118863*x^12 + 418673070*x^6 + 373714754427)
 
gp: K = bnfinit(x^18 + 118863*x^12 + 418673070*x^6 + 373714754427, 1)
 

Normalized defining polynomial

\( x^{18} + 118863 x^{12} + 418673070 x^{6} + 373714754427 \)

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:  \(-8880520706942285236653498213029984177183105373284187=-\,3^{27}\cdot 1801^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $769.17$
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:  $3, 1801$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{9} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{441} a^{7} - \frac{1}{3} a^{5} - \frac{59}{147} a$, $\frac{1}{441} a^{8} - \frac{59}{147} a^{2}$, $\frac{1}{882} a^{9} - \frac{1}{18} a^{6} - \frac{1}{3} a^{4} + \frac{44}{147} a^{3} - \frac{1}{6}$, $\frac{1}{882} a^{10} - \frac{1}{882} a^{7} - \frac{1}{3} a^{5} + \frac{44}{147} a^{4} + \frac{59}{294} a$, $\frac{1}{2646} a^{11} - \frac{1}{882} a^{8} + \frac{44}{441} a^{5} - \frac{1}{3} a^{3} + \frac{59}{294} a^{2}$, $\frac{1}{233528022} a^{12} + \frac{1}{2646} a^{10} - \frac{1}{882} a^{7} + \frac{224129}{5987898} a^{6} - \frac{1}{3} a^{5} + \frac{191}{441} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} + \frac{59}{294} a - \frac{58757}{529542}$, $\frac{1}{233528022} a^{13} - \frac{6697}{5987898} a^{7} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{7468457}{25947558} a$, $\frac{1}{11442873078} a^{14} - \frac{33853}{293407002} a^{8} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{583433639}{1271430342} a^{2}$, $\frac{1}{560700780822} a^{15} + \frac{1812709}{7188471549} a^{9} - \frac{1}{3} a^{5} + \frac{22674227515}{62300086758} a^{3} - \frac{1}{2}$, $\frac{1}{82423014780834} a^{16} + \frac{476336699}{2113410635406} a^{10} + \frac{1}{1323} a^{8} - \frac{1}{882} a^{7} - \frac{1}{3} a^{5} - \frac{1554747016793}{9158112753426} a^{4} + \frac{88}{441} a^{2} - \frac{44}{147} a - \frac{1}{3}$, $\frac{1}{4038727724260866} a^{17} - \frac{161191181}{51778560567447} a^{11} - \frac{1}{2646} a^{9} - \frac{1}{18} a^{6} - \frac{194788849444523}{448747524917874} a^{5} - \frac{1}{3} a^{4} + \frac{103}{441} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{6}$

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

Class group and class number

$C_{26}\times C_{26}\times C_{312}\times C_{67704}$, which has order $14279586048$ (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:  \( \frac{1}{20766695586} a^{15} + \frac{4664}{798719061} a^{9} + \frac{213064305}{6922231862} a^{3} + \frac{1}{2} \) (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:  \( 1321946620.3369434 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.9730803.1 x3, 3.3.262731681.1, 6.0.284065581074427.2, 6.0.207083808603257283.2, 6.0.63843798483.2 x2, 9.3.54407477142216048038082723.1 x3

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

Sibling fields

Degree 6 sibling: data not computed
Degree 9 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.6.9.9$x^{6} + 6 x^{4} + 21$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.9$x^{6} + 6 x^{4} + 21$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.9$x^{6} + 6 x^{4} + 21$$6$$1$$9$$C_6$$[2]_{2}$
1801Data not computed