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Properties

Label 18.6.60006020121...5793.2
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 41^{6}\cdot 97^{3}$
Root discriminant $27.05$
Ramified primes $7, 41, 97$
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![-2309, -3345, 1560, 1658, -2903, 212, 2799, -1299, -1040, 1462, -186, -395, 368, -118, -67, 41, -7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 7*x^16 + 41*x^15 - 67*x^14 - 118*x^13 + 368*x^12 - 395*x^11 - 186*x^10 + 1462*x^9 - 1040*x^8 - 1299*x^7 + 2799*x^6 + 212*x^5 - 2903*x^4 + 1658*x^3 + 1560*x^2 - 3345*x - 2309)
 
gp: K = bnfinit(x^18 - x^17 - 7*x^16 + 41*x^15 - 67*x^14 - 118*x^13 + 368*x^12 - 395*x^11 - 186*x^10 + 1462*x^9 - 1040*x^8 - 1299*x^7 + 2799*x^6 + 212*x^5 - 2903*x^4 + 1658*x^3 + 1560*x^2 - 3345*x - 2309, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 7 x^{16} + 41 x^{15} - 67 x^{14} - 118 x^{13} + 368 x^{12} - 395 x^{11} - 186 x^{10} + 1462 x^{9} - 1040 x^{8} - 1299 x^{7} + 2799 x^{6} + 212 x^{5} - 2903 x^{4} + 1658 x^{3} + 1560 x^{2} - 3345 x - 2309 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(60006020121228767347575793=7^{12}\cdot 41^{6}\cdot 97^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.05$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{41} a^{16} - \frac{5}{41} a^{15} + \frac{9}{41} a^{14} - \frac{16}{41} a^{13} + \frac{2}{41} a^{12} + \frac{20}{41} a^{11} - \frac{7}{41} a^{10} + \frac{4}{41} a^{9} - \frac{10}{41} a^{8} + \frac{10}{41} a^{7} - \frac{15}{41} a^{6} - \frac{8}{41} a^{5} - \frac{20}{41} a^{4} - \frac{4}{41} a^{3} - \frac{19}{41} a^{2} - \frac{13}{41} a + \frac{7}{41}$, $\frac{1}{12505921508797005169984744926811} a^{17} + \frac{42077762487056890814658843326}{12505921508797005169984744926811} a^{16} - \frac{1516917834515183555452165780077}{12505921508797005169984744926811} a^{15} - \frac{5717645085529502842193779548323}{12505921508797005169984744926811} a^{14} + \frac{2122272245638247781317887463388}{12505921508797005169984744926811} a^{13} + \frac{1458573882481702929047687793416}{12505921508797005169984744926811} a^{12} - \frac{443856172068533437927571849112}{12505921508797005169984744926811} a^{11} + \frac{3127376701171554906217660665956}{12505921508797005169984744926811} a^{10} + \frac{4976418980723192345250938919935}{12505921508797005169984744926811} a^{9} + \frac{3986541349682324884395230005156}{12505921508797005169984744926811} a^{8} - \frac{1577371785190151546614643632950}{12505921508797005169984744926811} a^{7} + \frac{4028417818818550576134035786703}{12505921508797005169984744926811} a^{6} + \frac{2745115841131330032369110675969}{12505921508797005169984744926811} a^{5} + \frac{5899334249150337289598837159430}{12505921508797005169984744926811} a^{4} + \frac{400304809363977444752233352362}{12505921508797005169984744926811} a^{3} + \frac{1386394591125479987073976105610}{12505921508797005169984744926811} a^{2} - \frac{4469736781260492904615798438815}{12505921508797005169984744926811} a + \frac{2426680828684036600307527277687}{12505921508797005169984744926811}$

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:  $11$
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:  \( 711722.007002 \) (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

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

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
7Data not computed
41Data not computed
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$