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Properties

Label 18.18.4064689890...8208.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{26}\cdot 3^{36}\cdot 7^{9}$
Root discriminant $64.81$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times D_9:C_3$ (as 18T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![898, -8928, 20736, 14880, -69120, -6696, 88704, 1116, -57024, -62, 20592, 0, -4368, 0, 540, 0, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 62*x^9 - 57024*x^8 + 1116*x^7 + 88704*x^6 - 6696*x^5 - 69120*x^4 + 14880*x^3 + 20736*x^2 - 8928*x + 898)
 
gp: K = bnfinit(x^18 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 62*x^9 - 57024*x^8 + 1116*x^7 + 88704*x^6 - 6696*x^5 - 69120*x^4 + 14880*x^3 + 20736*x^2 - 8928*x + 898, 1)
 

Normalized defining polynomial

\( x^{18} - 36 x^{16} + 540 x^{14} - 4368 x^{12} + 20592 x^{10} - 62 x^{9} - 57024 x^{8} + 1116 x^{7} + 88704 x^{6} - 6696 x^{5} - 69120 x^{4} + 14880 x^{3} + 20736 x^{2} - 8928 x + 898 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(406468989013028578759524596318208=2^{26}\cdot 3^{36}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.81$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{4}$, $\frac{1}{1347} a^{14} - \frac{10}{449} a^{13} - \frac{28}{1347} a^{12} - \frac{118}{1347} a^{11} - \frac{47}{449} a^{10} - \frac{167}{1347} a^{9} - \frac{111}{449} a^{8} - \frac{92}{449} a^{7} + \frac{221}{449} a^{6} + \frac{164}{1347} a^{5} + \frac{165}{449} a^{4} + \frac{115}{1347} a^{3} - \frac{263}{1347} a^{2} + \frac{154}{449} a + \frac{1}{3}$, $\frac{1}{1347} a^{15} - \frac{10}{449} a^{13} - \frac{20}{449} a^{12} - \frac{89}{1347} a^{11} + \frac{31}{449} a^{10} + \frac{15}{449} a^{9} + \frac{170}{449} a^{8} + \frac{155}{449} a^{7} - \frac{151}{1347} a^{6} + \frac{9}{449} a^{5} + \frac{199}{449} a^{4} - \frac{135}{449} a^{3} - \frac{244}{1347} a^{2} + \frac{130}{449} a$, $\frac{1}{1347} a^{16} - \frac{62}{1347} a^{13} - \frac{31}{1347} a^{12} + \frac{145}{1347} a^{11} - \frac{48}{449} a^{10} - \frac{10}{1347} a^{9} - \frac{32}{449} a^{8} - \frac{349}{1347} a^{7} - \frac{96}{449} a^{6} + \frac{43}{449} a^{5} + \frac{77}{1347} a^{4} - \frac{386}{1347} a^{3} - \frac{316}{1347} a^{2} + \frac{130}{449} a - \frac{1}{3}$, $\frac{1}{1347} a^{17} - \frac{95}{1347} a^{13} + \frac{205}{1347} a^{12} + \frac{173}{1347} a^{11} - \frac{221}{1347} a^{10} - \frac{41}{449} a^{9} + \frac{557}{1347} a^{8} + \frac{37}{449} a^{7} - \frac{174}{449} a^{6} - \frac{177}{449} a^{5} + \frac{221}{1347} a^{4} - \frac{370}{1347} a^{3} - \frac{650}{1347} a^{2} - \frac{541}{1347} a$

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:  $17$
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:  \( 138328022131 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{7}) \), 3.3.756.1, 6.6.64012032.1, 9.9.238130328086784.1

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 $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
2Data not computed
$3$3.9.18.28$x^{9} + 24 x^{3} + 9 x + 6$$9$$1$$18$$(C_9:C_3):C_2$$[3/2, 2, 5/2]_{2}$
3.9.18.28$x^{9} + 24 x^{3} + 9 x + 6$$9$$1$$18$$(C_9:C_3):C_2$$[3/2, 2, 5/2]_{2}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$