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Properties

Label 18.18.2096291596...4608.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{20}\cdot 7^{12}\cdot 13^{9}$
Root discriminant $70.99$
Ramified primes $2, 3, 7, 13$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $He_3:C_2$ (as 18T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1863, -14013, -33795, 147342, 38376, -257946, 7892, 193350, -29151, -76017, 16569, 16662, -4354, -2016, 588, 124, -39, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 39*x^16 + 124*x^15 + 588*x^14 - 2016*x^13 - 4354*x^12 + 16662*x^11 + 16569*x^10 - 76017*x^9 - 29151*x^8 + 193350*x^7 + 7892*x^6 - 257946*x^5 + 38376*x^4 + 147342*x^3 - 33795*x^2 - 14013*x + 1863)
 
gp: K = bnfinit(x^18 - 3*x^17 - 39*x^16 + 124*x^15 + 588*x^14 - 2016*x^13 - 4354*x^12 + 16662*x^11 + 16569*x^10 - 76017*x^9 - 29151*x^8 + 193350*x^7 + 7892*x^6 - 257946*x^5 + 38376*x^4 + 147342*x^3 - 33795*x^2 - 14013*x + 1863, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 39 x^{16} + 124 x^{15} + 588 x^{14} - 2016 x^{13} - 4354 x^{12} + 16662 x^{11} + 16569 x^{10} - 76017 x^{9} - 29151 x^{8} + 193350 x^{7} + 7892 x^{6} - 257946 x^{5} + 38376 x^{4} + 147342 x^{3} - 33795 x^{2} - 14013 x + 1863 \)

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:  \(2096291596176378856933557232324608=2^{12}\cdot 3^{20}\cdot 7^{12}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.99$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{12} a^{15} + \frac{1}{12} a^{12} + \frac{1}{6} a^{9} + \frac{5}{12} a^{3} + \frac{1}{4}$, $\frac{1}{72} a^{16} + \frac{1}{12} a^{14} - \frac{1}{36} a^{13} + \frac{1}{12} a^{12} + \frac{1}{36} a^{10} - \frac{1}{8} a^{8} - \frac{1}{6} a^{7} - \frac{1}{4} a^{6} + \frac{1}{6} a^{5} + \frac{13}{36} a^{4} - \frac{1}{2} a^{3} - \frac{1}{12} a + \frac{1}{8}$, $\frac{1}{1930833910671494184} a^{17} + \frac{1479895454144207}{643611303557164728} a^{16} - \frac{3150690870411721}{160902825889291182} a^{15} + \frac{37556470063850219}{965416955335747092} a^{14} - \frac{19099257190683149}{160902825889291182} a^{13} + \frac{5057261075605625}{107268550592860788} a^{12} - \frac{117386523455918723}{965416955335747092} a^{11} - \frac{8404607960948089}{321805651778582364} a^{10} - \frac{52578204287090149}{214537101185721576} a^{9} + \frac{119090170469812379}{643611303557164728} a^{8} + \frac{20363906712636371}{107268550592860788} a^{7} + \frac{43141632425948345}{321805651778582364} a^{6} - \frac{59046634068088937}{965416955335747092} a^{5} + \frac{26673997587667765}{160902825889291182} a^{4} - \frac{37428837380467687}{107268550592860788} a^{3} + \frac{80426596969805713}{160902825889291182} a^{2} - \frac{313763447725963}{1198531291540344} a - \frac{1548327435415737}{3109233350517704}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 127268889201 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$He_3:C_2$ (as 18T22):

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 $He_3:C_2$
Character table for $He_3:C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), 6.6.6836396112.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$7$7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$