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Properties

Label 18.4.75754328775...0567.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,7^{13}\cdot 52879^{4}$
Root discriminant $45.71$
Ramified primes $7, 52879$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T767

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![307943, -1269256, 2270524, -2607975, 2341399, -1614434, 931848, -431269, 157050, -40794, 3197, 3432, -2569, 949, -285, 26, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 26*x^15 - 285*x^14 + 949*x^13 - 2569*x^12 + 3432*x^11 + 3197*x^10 - 40794*x^9 + 157050*x^8 - 431269*x^7 + 931848*x^6 - 1614434*x^5 + 2341399*x^4 - 2607975*x^3 + 2270524*x^2 - 1269256*x + 307943)
 
gp: K = bnfinit(x^18 - 4*x^17 + 26*x^15 - 285*x^14 + 949*x^13 - 2569*x^12 + 3432*x^11 + 3197*x^10 - 40794*x^9 + 157050*x^8 - 431269*x^7 + 931848*x^6 - 1614434*x^5 + 2341399*x^4 - 2607975*x^3 + 2270524*x^2 - 1269256*x + 307943, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 26 x^{15} - 285 x^{14} + 949 x^{13} - 2569 x^{12} + 3432 x^{11} + 3197 x^{10} - 40794 x^{9} + 157050 x^{8} - 431269 x^{7} + 931848 x^{6} - 1614434 x^{5} + 2341399 x^{4} - 2607975 x^{3} + 2270524 x^{2} - 1269256 x + 307943 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-757543287754796256482850550567=-\,7^{13}\cdot 52879^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.71$
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:  $7, 52879$
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}{181399042953870335814897318793513907243774149061987} a^{17} + \frac{87033365705388788621407271027492989096236664626254}{181399042953870335814897318793513907243774149061987} a^{16} + \frac{33876693532965496019494804284014494366523894926881}{181399042953870335814897318793513907243774149061987} a^{15} + \frac{48471473917612868201172269449345192538575152187710}{181399042953870335814897318793513907243774149061987} a^{14} - \frac{33624514577127541975783346507092649685815446859713}{181399042953870335814897318793513907243774149061987} a^{13} + \frac{4051987505279802221401070572617698284038476912742}{181399042953870335814897318793513907243774149061987} a^{12} - \frac{37712185458723314165951315922036914694289544655310}{181399042953870335814897318793513907243774149061987} a^{11} - \frac{20723856632070403498883715617004971997236924521476}{181399042953870335814897318793513907243774149061987} a^{10} + \frac{46558310478919113994599699656911288858971369370331}{181399042953870335814897318793513907243774149061987} a^{9} + \frac{32231810588240315543454331690039372349891651757205}{181399042953870335814897318793513907243774149061987} a^{8} + \frac{72836982381766097917907591200632612640153617233425}{181399042953870335814897318793513907243774149061987} a^{7} + \frac{17442310771587170638368225432109084044790481348555}{181399042953870335814897318793513907243774149061987} a^{6} + \frac{63822160018181291995557403015286425137285606298383}{181399042953870335814897318793513907243774149061987} a^{5} - \frac{48501040767030683925346537497176943319755643060881}{181399042953870335814897318793513907243774149061987} a^{4} + \frac{88830167655563356013420898816650249126844220189404}{181399042953870335814897318793513907243774149061987} a^{3} + \frac{26215808351479603961212934275434211512639801959530}{181399042953870335814897318793513907243774149061987} a^{2} + \frac{29963037102467843406326966214781567895585640946248}{181399042953870335814897318793513907243774149061987} a - \frac{20654045093385861703798287339841893048127554325723}{181399042953870335814897318793513907243774149061987}$

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:  $10$
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:  \( 49924075.6539 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T767:

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 t18n767 are not computed
Character table for t18n767 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.7.6221161471.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $18$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
52879Data not computed