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Properties

Label 18.10.8191939224...1097.1
Degree $18$
Signature $[10, 4]$
Discriminant $7^{12}\cdot 41^{3}\cdot 97^{5}$
Root discriminant $24.21$
Ramified primes $7, 41, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T401

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![127, 680, -535, -1700, 2527, 1458, -2713, -10, 405, -389, 542, 132, -187, -67, 12, 33, -5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 5*x^16 + 33*x^15 + 12*x^14 - 67*x^13 - 187*x^12 + 132*x^11 + 542*x^10 - 389*x^9 + 405*x^8 - 10*x^7 - 2713*x^6 + 1458*x^5 + 2527*x^4 - 1700*x^3 - 535*x^2 + 680*x + 127)
 
gp: K = bnfinit(x^18 - 4*x^17 - 5*x^16 + 33*x^15 + 12*x^14 - 67*x^13 - 187*x^12 + 132*x^11 + 542*x^10 - 389*x^9 + 405*x^8 - 10*x^7 - 2713*x^6 + 1458*x^5 + 2527*x^4 - 1700*x^3 - 535*x^2 + 680*x + 127, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 5 x^{16} + 33 x^{15} + 12 x^{14} - 67 x^{13} - 187 x^{12} + 132 x^{11} + 542 x^{10} - 389 x^{9} + 405 x^{8} - 10 x^{7} - 2713 x^{6} + 1458 x^{5} + 2527 x^{4} - 1700 x^{3} - 535 x^{2} + 680 x + 127 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8191939224918986549431097=7^{12}\cdot 41^{3}\cdot 97^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.21$
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:  $7, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1257020437514944583539379929} a^{17} - \frac{277198485242563520746095541}{1257020437514944583539379929} a^{16} + \frac{31485262075370897789629405}{1257020437514944583539379929} a^{15} + \frac{521028497251799019830707759}{1257020437514944583539379929} a^{14} - \frac{362832680947607008301366895}{1257020437514944583539379929} a^{13} + \frac{56383527911689229821923339}{1257020437514944583539379929} a^{12} + \frac{527571201878362820950405132}{1257020437514944583539379929} a^{11} - \frac{277840627345162128238141479}{1257020437514944583539379929} a^{10} + \frac{364216996224029912745556530}{1257020437514944583539379929} a^{9} + \frac{87232843342957078920179979}{1257020437514944583539379929} a^{8} + \frac{321375609555057552868428856}{1257020437514944583539379929} a^{7} + \frac{96475176750692909669453489}{1257020437514944583539379929} a^{6} + \frac{166028797519333624787067779}{1257020437514944583539379929} a^{5} - \frac{151216379614797677994648004}{1257020437514944583539379929} a^{4} + \frac{15890944933646451882943281}{1257020437514944583539379929} a^{3} - \frac{62998394089719759730431639}{1257020437514944583539379929} a^{2} - \frac{117533537070825238066863089}{1257020437514944583539379929} a + \frac{543831146773209987313565255}{1257020437514944583539379929}$

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:  $13$
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:  \( 599136.815697 \) (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

\(\Q(\zeta_{7})^+\), 6.6.9548777.1, 9.5.467890073.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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}$
41Data not computed
97Data not computed