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Properties

Label 18.0.11060909218...8583.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 13^{12}$
Root discriminant $27.98$
Ramified primes $7, 13$
Class number $13$
Class group $[13]$
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 15, 55, 201, 734, 2680, 1136, 1130, 704, 550, 366, 210, -76, 31, -10, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 5*x^16 - 10*x^15 + 31*x^14 - 76*x^13 + 210*x^12 + 366*x^11 + 550*x^10 + 704*x^9 + 1130*x^8 + 1136*x^7 + 2680*x^6 + 734*x^5 + 201*x^4 + 55*x^3 + 15*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^18 - x^17 + 5*x^16 - 10*x^15 + 31*x^14 - 76*x^13 + 210*x^12 + 366*x^11 + 550*x^10 + 704*x^9 + 1130*x^8 + 1136*x^7 + 2680*x^6 + 734*x^5 + 201*x^4 + 55*x^3 + 15*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 5 x^{16} - 10 x^{15} + 31 x^{14} - 76 x^{13} + 210 x^{12} + 366 x^{11} + 550 x^{10} + 704 x^{9} + 1130 x^{8} + 1136 x^{7} + 2680 x^{6} + 734 x^{5} + 201 x^{4} + 55 x^{3} + 15 x^{2} + 4 x + 1 \)

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:  \(-110609092182866440454328583=-\,7^{15}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.98$
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:  $7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(91=7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{91}(1,·)$, $\chi_{91}(66,·)$, $\chi_{91}(3,·)$, $\chi_{91}(68,·)$, $\chi_{91}(9,·)$, $\chi_{91}(74,·)$, $\chi_{91}(79,·)$, $\chi_{91}(16,·)$, $\chi_{91}(81,·)$, $\chi_{91}(22,·)$, $\chi_{91}(87,·)$, $\chi_{91}(27,·)$, $\chi_{91}(29,·)$, $\chi_{91}(40,·)$, $\chi_{91}(48,·)$, $\chi_{91}(53,·)$, $\chi_{91}(55,·)$, $\chi_{91}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{154646} a^{13} - \frac{30525}{154646} a^{12} + \frac{33475}{154646} a^{11} - \frac{465}{77323} a^{10} + \frac{10709}{154646} a^{9} + \frac{29419}{154646} a^{8} + \frac{14347}{154646} a^{7} - \frac{69219}{154646} a^{6} - \frac{18323}{154646} a^{5} - \frac{45007}{154646} a^{4} + \frac{20467}{77323} a^{3} - \frac{47993}{154646} a^{2} + \frac{24767}{154646} a + \frac{51619}{154646}$, $\frac{1}{309292} a^{14} + \frac{35065}{154646} a^{7} - \frac{18119}{309292}$, $\frac{1}{309292} a^{15} + \frac{35065}{154646} a^{8} - \frac{18119}{309292} a$, $\frac{1}{309292} a^{16} + \frac{35065}{154646} a^{9} - \frac{18119}{309292} a^{2}$, $\frac{1}{309292} a^{17} + \frac{35065}{154646} a^{10} - \frac{18119}{309292} a^{3}$

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

Class group and class number

$C_{13}$, which has order $13$

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:  \( \frac{6345}{309292} a^{17} - \frac{31725}{309292} a^{16} + \frac{31725}{154646} a^{15} - \frac{196695}{309292} a^{14} + \frac{120555}{77323} a^{13} - \frac{666225}{154646} a^{12} + \frac{864485}{77323} a^{11} - \frac{1744875}{154646} a^{10} - \frac{1116720}{77323} a^{9} - \frac{3584925}{154646} a^{8} - \frac{1801980}{77323} a^{7} - \frac{4251150}{77323} a^{6} - \frac{2328615}{154646} a^{5} - \frac{14059438}{77323} a^{4} - \frac{348975}{309292} a^{3} - \frac{95175}{309292} a^{2} - \frac{6345}{77323} a - \frac{6345}{309292} \) (order $14$)
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:  \( 205236.825908 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.3.169.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 3.3.8281.1, 6.0.9796423.1, 6.0.480024727.2, \(\Q(\zeta_{7})\), 6.0.480024727.1, 9.9.567869252041.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
7Data not computed
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$