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Properties

Label 18.18.7084786458...1157.1
Degree $18$
Signature $[18, 0]$
Discriminant $7^{12}\cdot 13^{15}$
Root discriminant $31.02$
Ramified primes $7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
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, -6, 45, 89, -466, -522, 1831, 1287, -3300, -1420, 2914, 711, -1259, -180, 269, 22, -27, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 27*x^16 + 22*x^15 + 269*x^14 - 180*x^13 - 1259*x^12 + 711*x^11 + 2914*x^10 - 1420*x^9 - 3300*x^8 + 1287*x^7 + 1831*x^6 - 522*x^5 - 466*x^4 + 89*x^3 + 45*x^2 - 6*x - 1)
 
gp: K = bnfinit(x^18 - x^17 - 27*x^16 + 22*x^15 + 269*x^14 - 180*x^13 - 1259*x^12 + 711*x^11 + 2914*x^10 - 1420*x^9 - 3300*x^8 + 1287*x^7 + 1831*x^6 - 522*x^5 - 466*x^4 + 89*x^3 + 45*x^2 - 6*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 27 x^{16} + 22 x^{15} + 269 x^{14} - 180 x^{13} - 1259 x^{12} + 711 x^{11} + 2914 x^{10} - 1420 x^{9} - 3300 x^{8} + 1287 x^{7} + 1831 x^{6} - 522 x^{5} - 466 x^{4} + 89 x^{3} + 45 x^{2} - 6 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(708478645847689707516501157=7^{12}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.02$
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$$\chi_{91}(64,·)$, $\chi_{91}(1,·)$, $\chi_{91}(4,·)$, $\chi_{91}(9,·)$, $\chi_{91}(74,·)$, $\chi_{91}(79,·)$, $\chi_{91}(16,·)$, $\chi_{91}(81,·)$, $\chi_{91}(22,·)$, $\chi_{91}(23,·)$, $\chi_{91}(88,·)$, $\chi_{91}(25,·)$, $\chi_{91}(29,·)$, $\chi_{91}(30,·)$, $\chi_{91}(36,·)$, $\chi_{91}(43,·)$, $\chi_{91}(51,·)$, $\chi_{91}(53,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{14} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{207958299548697} a^{17} - \frac{14810111393447}{207958299548697} a^{16} - \frac{6188279408926}{207958299548697} a^{15} - \frac{56756337643564}{207958299548697} a^{14} + \frac{1121297346959}{69319433182899} a^{13} + \frac{31335340800697}{69319433182899} a^{12} + \frac{28633158890638}{207958299548697} a^{11} + \frac{21802970469404}{207958299548697} a^{10} + \frac{76838089823483}{207958299548697} a^{9} - \frac{21728458089637}{69319433182899} a^{8} - \frac{2412430972069}{69319433182899} a^{7} - \frac{5070516683702}{69319433182899} a^{6} + \frac{76107206311390}{207958299548697} a^{5} + \frac{90962557399241}{207958299548697} a^{4} - \frac{2961557531285}{69319433182899} a^{3} - \frac{70998714843817}{207958299548697} a^{2} + \frac{12427514976847}{207958299548697} a + \frac{68749997353073}{207958299548697}$

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:  \( 37124865.5559 \) (assuming GRH)
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

\(\Q(\sqrt{13}) \), 3.3.8281.1, 3.3.169.1, \(\Q(\zeta_{7})^+\), 3.3.8281.2, 6.6.891474493.1, \(\Q(\zeta_{13})^+\), 6.6.5274997.1, 6.6.891474493.2, 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$