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Properties

Label 18.18.2246956500...4288.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 7^{15}\cdot 13^{4}\cdot 43^{6}$
Root discriminant $62.71$
Ramified primes $2, 7, 13, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T552

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-634207, 0, 7448245, 0, -10709832, 0, 6099030, 0, -1817410, 0, 315462, 0, -33054, 0, 2058, 0, -70, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 70*x^16 + 2058*x^14 - 33054*x^12 + 315462*x^10 - 1817410*x^8 + 6099030*x^6 - 10709832*x^4 + 7448245*x^2 - 634207)
 
gp: K = bnfinit(x^18 - 70*x^16 + 2058*x^14 - 33054*x^12 + 315462*x^10 - 1817410*x^8 + 6099030*x^6 - 10709832*x^4 + 7448245*x^2 - 634207, 1)
 

Normalized defining polynomial

\( x^{18} - 70 x^{16} + 2058 x^{14} - 33054 x^{12} + 315462 x^{10} - 1817410 x^{8} + 6099030 x^{6} - 10709832 x^{4} + 7448245 x^{2} - 634207 \)

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:  \(224695650053140671980101468684288=2^{18}\cdot 7^{15}\cdot 13^{4}\cdot 43^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.71$
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:  $2, 7, 13, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{7} a^{8}$, $\frac{1}{7} a^{9}$, $\frac{1}{7} a^{10}$, $\frac{1}{7} a^{11}$, $\frac{1}{147} a^{12} - \frac{1}{21} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{147} a^{13} - \frac{1}{21} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{147} a^{14} - \frac{1}{21} a^{8} - \frac{1}{21} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{147} a^{15} - \frac{1}{21} a^{9} - \frac{1}{21} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{42203953882671} a^{16} + \frac{12943424616}{14067984627557} a^{14} - \frac{6696559255}{6029136268953} a^{12} + \frac{401258880719}{6029136268953} a^{10} + \frac{44916167579}{861305181279} a^{8} - \frac{215607068449}{6029136268953} a^{6} - \frac{119387878174}{287101727093} a^{4} - \frac{333328382248}{861305181279} a^{2} + \frac{325362580}{20030353053}$, $\frac{1}{42203953882671} a^{17} + \frac{12943424616}{14067984627557} a^{15} - \frac{6696559255}{6029136268953} a^{13} + \frac{401258880719}{6029136268953} a^{11} + \frac{44916167579}{861305181279} a^{9} - \frac{215607068449}{6029136268953} a^{7} - \frac{119387878174}{287101727093} a^{5} - \frac{333328382248}{861305181279} a^{3} + \frac{325362580}{20030353053} a$

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:  \( 17180134719.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T552:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10368
The 64 conjugacy class representatives for t18n552 are not computed
Character table for t18n552 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.36763077169.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ $18$ R $18$ R ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
7Data not computed
$13$13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.4.1$x^{6} + 39 x^{3} + 676$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.6.4.1$x^{6} + 344 x^{3} + 49923$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
43.6.0.1$x^{6} - x + 26$$1$$6$$0$$C_6$$[\ ]^{6}$