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Properties

Label 18.2.84452981700...3001.1
Degree $18$
Signature $[2, 8]$
Discriminant $7^{12}\cdot 41^{3}\cdot 97^{4}$
Root discriminant $18.78$
Ramified primes $7, 41, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-113, -136, 210, 42, 188, -364, 130, -394, 466, -355, 128, 10, -65, 26, -12, -2, 4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 4*x^16 - 2*x^15 - 12*x^14 + 26*x^13 - 65*x^12 + 10*x^11 + 128*x^10 - 355*x^9 + 466*x^8 - 394*x^7 + 130*x^6 - 364*x^5 + 188*x^4 + 42*x^3 + 210*x^2 - 136*x - 113)
 
gp: K = bnfinit(x^18 - 3*x^17 + 4*x^16 - 2*x^15 - 12*x^14 + 26*x^13 - 65*x^12 + 10*x^11 + 128*x^10 - 355*x^9 + 466*x^8 - 394*x^7 + 130*x^6 - 364*x^5 + 188*x^4 + 42*x^3 + 210*x^2 - 136*x - 113, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 4 x^{16} - 2 x^{15} - 12 x^{14} + 26 x^{13} - 65 x^{12} + 10 x^{11} + 128 x^{10} - 355 x^{9} + 466 x^{8} - 394 x^{7} + 130 x^{6} - 364 x^{5} + 188 x^{4} + 42 x^{3} + 210 x^{2} - 136 x - 113 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(84452981700195737623001=7^{12}\cdot 41^{3}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.78$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{12134177503810010628469349819} a^{17} + \frac{492982679173954791906446957}{12134177503810010628469349819} a^{16} - \frac{1979994308524085138318988530}{12134177503810010628469349819} a^{15} + \frac{4207395210025891404860779375}{12134177503810010628469349819} a^{14} - \frac{201745677877460602122803920}{12134177503810010628469349819} a^{13} + \frac{1497946232848338787987604449}{12134177503810010628469349819} a^{12} - \frac{1780160256464964221919793713}{12134177503810010628469349819} a^{11} - \frac{3801438500471319452737890257}{12134177503810010628469349819} a^{10} - \frac{2004678625143476396887671566}{12134177503810010628469349819} a^{9} + \frac{5061198254893560001744501443}{12134177503810010628469349819} a^{8} + \frac{755782022259636913610563199}{12134177503810010628469349819} a^{7} - \frac{2073682921844658410641318107}{12134177503810010628469349819} a^{6} + \frac{4272976318750889923001635626}{12134177503810010628469349819} a^{5} - \frac{3003588771834173689695071674}{12134177503810010628469349819} a^{4} - \frac{2741416099144291178271001838}{12134177503810010628469349819} a^{3} + \frac{1768424896816953252028630243}{12134177503810010628469349819} a^{2} - \frac{4307806770105822278073985106}{12134177503810010628469349819} a + \frac{4393973880399575748802638336}{12134177503810010628469349819}$

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:  $9$
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:  \( 11681.4446341 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n765 are not computed
Character table for t18n765 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
7Data not computed
41Data not computed
97Data not computed