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Properties

Label 18.0.63744201554...4375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,5^{9}\cdot 7^{12}\cdot 11^{9}$
Root discriminant $27.14$
Ramified primes $5, 7, 11$
Class number $12$
Class group $[12]$
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![64, -288, 688, -1784, 5500, -13718, 25499, -36489, 41417, -37881, 28338, -17428, 8805, -3630, 1216, -318, 66, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 66*x^16 - 318*x^15 + 1216*x^14 - 3630*x^13 + 8805*x^12 - 17428*x^11 + 28338*x^10 - 37881*x^9 + 41417*x^8 - 36489*x^7 + 25499*x^6 - 13718*x^5 + 5500*x^4 - 1784*x^3 + 688*x^2 - 288*x + 64)
 
gp: K = bnfinit(x^18 - 9*x^17 + 66*x^16 - 318*x^15 + 1216*x^14 - 3630*x^13 + 8805*x^12 - 17428*x^11 + 28338*x^10 - 37881*x^9 + 41417*x^8 - 36489*x^7 + 25499*x^6 - 13718*x^5 + 5500*x^4 - 1784*x^3 + 688*x^2 - 288*x + 64, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 66 x^{16} - 318 x^{15} + 1216 x^{14} - 3630 x^{13} + 8805 x^{12} - 17428 x^{11} + 28338 x^{10} - 37881 x^{9} + 41417 x^{8} - 36489 x^{7} + 25499 x^{6} - 13718 x^{5} + 5500 x^{4} - 1784 x^{3} + 688 x^{2} - 288 x + 64 \)

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:  \(-63744201554816021271484375=-\,5^{9}\cdot 7^{12}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.14$
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:  $5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{3}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{3}{8} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{1736} a^{15} - \frac{3}{868} a^{14} + \frac{17}{1736} a^{13} - \frac{3}{217} a^{12} - \frac{159}{1736} a^{11} + \frac{15}{124} a^{10} - \frac{241}{1736} a^{9} - \frac{153}{868} a^{8} - \frac{11}{1736} a^{7} - \frac{359}{868} a^{6} - \frac{27}{248} a^{5} + \frac{425}{868} a^{4} + \frac{577}{1736} a^{3} + \frac{100}{217} a^{2} + \frac{61}{217} a + \frac{76}{217}$, $\frac{1}{24304} a^{16} - \frac{1}{3472} a^{15} - \frac{187}{3038} a^{14} - \frac{129}{12152} a^{13} - \frac{22}{1519} a^{12} + \frac{727}{12152} a^{11} + \frac{2153}{24304} a^{10} + \frac{293}{12152} a^{9} - \frac{2131}{12152} a^{8} - \frac{163}{3472} a^{7} - \frac{10321}{24304} a^{6} - \frac{9811}{24304} a^{5} - \frac{597}{3472} a^{4} + \frac{55}{3038} a^{3} - \frac{295}{6076} a^{2} - \frac{752}{1519} a - \frac{38}{1519}$, $\frac{1}{4359785240608} a^{17} - \frac{7125887}{4359785240608} a^{16} - \frac{23469071}{1089946310152} a^{15} + \frac{18918536103}{2179892620304} a^{14} - \frac{14080293301}{272486577538} a^{13} - \frac{21376112179}{2179892620304} a^{12} - \frac{173665739427}{4359785240608} a^{11} - \frac{320945518501}{2179892620304} a^{10} + \frac{381810128115}{2179892620304} a^{9} - \frac{93776370477}{4359785240608} a^{8} - \frac{357175805717}{4359785240608} a^{7} - \frac{6387824647}{88975208992} a^{6} + \frac{2111682685101}{4359785240608} a^{5} + \frac{231707339793}{544973155076} a^{4} - \frac{65970260093}{1089946310152} a^{3} - \frac{197874539313}{544973155076} a^{2} - \frac{102621535197}{272486577538} a - \frac{12101662151}{136243288769}$

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

Class group and class number

$C_{12}$, which has order $12$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 64768.63578 \)
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

\(\Q(\sqrt{-55}) \), 3.1.2695.1 x3, \(\Q(\zeta_{7})^+\), 6.0.399466375.1, 6.0.399466375.2, 6.0.8152375.1 x2, 9.3.19573852375.2 x3

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

Sibling fields

Degree 6 sibling: 6.0.8152375.1
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R R R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$