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Properties

Label 18.0.82756087703...4224.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 7^{12}\cdot 97^{6}\cdot 1399^{3}$
Root discriminant $112.46$
Ramified primes $2, 7, 97, 1399$
Class number $3744$ (GRH)
Class group $[2, 2, 2, 468]$ (GRH)
Galois group 18T401

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1165327834895671, 0, 27840157113194, 0, -56039034873, 0, 69465367376, 0, -38209949, 0, -20867028, 0, 612762, 0, -1931, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 - 1931*x^14 + 612762*x^12 - 20867028*x^10 - 38209949*x^8 + 69465367376*x^6 - 56039034873*x^4 + 27840157113194*x^2 + 1165327834895671)
 
gp: K = bnfinit(x^18 - 6*x^16 - 1931*x^14 + 612762*x^12 - 20867028*x^10 - 38209949*x^8 + 69465367376*x^6 - 56039034873*x^4 + 27840157113194*x^2 + 1165327834895671, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{16} - 1931 x^{14} + 612762 x^{12} - 20867028 x^{10} - 38209949 x^{8} + 69465367376 x^{6} - 56039034873 x^{4} + 27840157113194 x^{2} + 1165327834895671 \)

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:  \(-8275608770373144126499153648830644224=-\,2^{18}\cdot 7^{12}\cdot 97^{6}\cdot 1399^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.46$
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:  $2, 7, 97, 1399$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{97} a^{8} - \frac{6}{97} a^{6} + \frac{9}{97} a^{4} + \frac{13}{97} a^{2}$, $\frac{1}{97} a^{9} - \frac{6}{97} a^{7} + \frac{9}{97} a^{5} + \frac{13}{97} a^{3}$, $\frac{1}{9409} a^{10} - \frac{6}{9409} a^{8} - \frac{1931}{9409} a^{6} + \frac{1177}{9409} a^{4} + \frac{22}{97} a^{2}$, $\frac{1}{9409} a^{11} - \frac{6}{9409} a^{9} - \frac{1931}{9409} a^{7} + \frac{1177}{9409} a^{5} + \frac{22}{97} a^{3}$, $\frac{1}{912673} a^{12} - \frac{6}{912673} a^{10} - \frac{1931}{912673} a^{8} - \frac{299911}{912673} a^{6} + \frac{1283}{9409} a^{4} + \frac{13}{97} a^{2}$, $\frac{1}{912673} a^{13} - \frac{6}{912673} a^{11} - \frac{1931}{912673} a^{9} - \frac{299911}{912673} a^{7} + \frac{1283}{9409} a^{5} + \frac{13}{97} a^{3}$, $\frac{1}{88529281} a^{14} - \frac{6}{88529281} a^{12} - \frac{1931}{88529281} a^{10} - \frac{299911}{88529281} a^{8} - \frac{158670}{912673} a^{6} + \frac{4475}{9409} a^{4} - \frac{46}{97} a^{2}$, $\frac{1}{88529281} a^{15} - \frac{6}{88529281} a^{13} - \frac{1931}{88529281} a^{11} - \frac{299911}{88529281} a^{9} - \frac{158670}{912673} a^{7} + \frac{4475}{9409} a^{5} - \frac{46}{97} a^{3}$, $\frac{1}{9374077312896325604881779139023142810220729} a^{16} + \frac{3898693256274913602096470919854914}{721082870222794277298598395309472523863133} a^{14} - \frac{4306004257511249305375803157584759874}{9374077312896325604881779139023142810220729} a^{12} + \frac{359659126440811296640000811729476550521}{9374077312896325604881779139023142810220729} a^{10} - \frac{397174150419072718542439551684499359370}{96639972297900263967853393185805595981657} a^{8} - \frac{83193816492789861180807895931579369429}{996288374205157360493333950369129855481} a^{6} + \frac{4326925985976364163311231339198210398}{10271014167063477943230246911021957273} a^{4} + \frac{53064682611337285742103640207105}{8145134153103471802720259247440093} a^{2} + \frac{414551062592152889100160577005793}{1091615917426238489024364641409497}$, $\frac{1}{9374077312896325604881779139023142810220729} a^{17} + \frac{3898693256274913602096470919854914}{721082870222794277298598395309472523863133} a^{15} - \frac{4306004257511249305375803157584759874}{9374077312896325604881779139023142810220729} a^{13} + \frac{359659126440811296640000811729476550521}{9374077312896325604881779139023142810220729} a^{11} - \frac{397174150419072718542439551684499359370}{96639972297900263967853393185805595981657} a^{9} - \frac{83193816492789861180807895931579369429}{996288374205157360493333950369129855481} a^{7} + \frac{4326925985976364163311231339198210398}{10271014167063477943230246911021957273} a^{5} + \frac{53064682611337285742103640207105}{8145134153103471802720259247440093} a^{3} + \frac{414551062592152889100160577005793}{1091615917426238489024364641409497} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{468}$, which has order $3744$ (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:  $8$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 17653328.7584 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T401:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2592
The 44 conjugacy class representatives for t18n401
Character table for t18n401 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.0.2022708581824.1, 9.3.164590951.1

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

Sibling fields

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
2Data not computed
$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}$
$97$97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.6.0.1$x^{6} - x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
1399Data not computed