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Properties

Label 18.18.3640833096...4121.1
Degree $18$
Signature $[18, 0]$
Discriminant $7^{12}\cdot 138041^{3}$
Root discriminant $26.31$
Ramified primes $7, 138041$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T207

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -7, 44, 222, -779, -864, 3019, 1489, -4871, -1473, 3837, 872, -1523, -276, 301, 40, -28, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 28*x^16 + 40*x^15 + 301*x^14 - 276*x^13 - 1523*x^12 + 872*x^11 + 3837*x^10 - 1473*x^9 - 4871*x^8 + 1489*x^7 + 3019*x^6 - 864*x^5 - 779*x^4 + 222*x^3 + 44*x^2 - 7*x - 1)
 
gp: K = bnfinit(x^18 - 2*x^17 - 28*x^16 + 40*x^15 + 301*x^14 - 276*x^13 - 1523*x^12 + 872*x^11 + 3837*x^10 - 1473*x^9 - 4871*x^8 + 1489*x^7 + 3019*x^6 - 864*x^5 - 779*x^4 + 222*x^3 + 44*x^2 - 7*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 28 x^{16} + 40 x^{15} + 301 x^{14} - 276 x^{13} - 1523 x^{12} + 872 x^{11} + 3837 x^{10} - 1473 x^{9} - 4871 x^{8} + 1489 x^{7} + 3019 x^{6} - 864 x^{5} - 779 x^{4} + 222 x^{3} + 44 x^{2} - 7 x - 1 \)

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:  \(36408330967717863107914121=7^{12}\cdot 138041^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.31$
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:  $7, 138041$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1020435438061423} a^{17} + \frac{130945069671443}{1020435438061423} a^{16} - \frac{298129488463027}{1020435438061423} a^{15} + \frac{356377475919382}{1020435438061423} a^{14} - \frac{284632878079495}{1020435438061423} a^{13} + \frac{189637727616475}{1020435438061423} a^{12} + \frac{473997103876885}{1020435438061423} a^{11} - \frac{19722450114131}{1020435438061423} a^{10} + \frac{409147114392915}{1020435438061423} a^{9} + \frac{461584276608922}{1020435438061423} a^{8} + \frac{379021307216700}{1020435438061423} a^{7} - \frac{44587831118051}{1020435438061423} a^{6} + \frac{167832362561480}{1020435438061423} a^{5} + \frac{86855358804499}{1020435438061423} a^{4} + \frac{117727854102923}{1020435438061423} a^{3} + \frac{325156484733787}{1020435438061423} a^{2} + \frac{92835708409057}{1020435438061423} a + \frac{442463787897563}{1020435438061423}$

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:  \( 6072713.5102 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T207:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 17 conjugacy class representatives for t18n207
Character table for t18n207

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.6.331436441.1, 9.9.16240385609.1 x3

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

Sibling fields

Degree 9 sibling: data not computed
Degree 12 sibling: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
138041Data not computed