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Properties

Label 18.10.2799928238...0217.5
Degree $18$
Signature $[10, 4]$
Discriminant $7^{12}\cdot 53^{6}\cdot 97^{3}$
Root discriminant $29.46$
Ramified primes $7, 53, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T366

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, 253, -1089, -4614, -1295, 3000, 3171, -1788, -52, 1063, -685, -11, 162, -26, 2, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 5*x^16 + 2*x^15 - 26*x^14 + 162*x^13 - 11*x^12 - 685*x^11 + 1063*x^10 - 52*x^9 - 1788*x^8 + 3171*x^7 + 3000*x^6 - 1295*x^5 - 4614*x^4 - 1089*x^3 + 253*x^2 + 4*x - 1)
 
gp: K = bnfinit(x^18 - 2*x^17 - 5*x^16 + 2*x^15 - 26*x^14 + 162*x^13 - 11*x^12 - 685*x^11 + 1063*x^10 - 52*x^9 - 1788*x^8 + 3171*x^7 + 3000*x^6 - 1295*x^5 - 4614*x^4 - 1089*x^3 + 253*x^2 + 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 5 x^{16} + 2 x^{15} - 26 x^{14} + 162 x^{13} - 11 x^{12} - 685 x^{11} + 1063 x^{10} - 52 x^{9} - 1788 x^{8} + 3171 x^{7} + 3000 x^{6} - 1295 x^{5} - 4614 x^{4} - 1089 x^{3} + 253 x^{2} + 4 x - 1 \)

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:  \(279992823820843547402730217=7^{12}\cdot 53^{6}\cdot 97^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.46$
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:  $7, 53, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{5967102057548373881711176149884} a^{17} - \frac{172279310705048670468881957893}{5967102057548373881711176149884} a^{16} + \frac{555390376227064231464306080941}{2983551028774186940855588074942} a^{15} - \frac{550768365504823845795571590711}{2983551028774186940855588074942} a^{14} + \frac{628474427206739372406474257871}{2983551028774186940855588074942} a^{13} + \frac{556292041070777919155604456130}{1491775514387093470427794037471} a^{12} - \frac{2226212636539858091652059894617}{5967102057548373881711176149884} a^{11} - \frac{504631504674592495508421292365}{2983551028774186940855588074942} a^{10} - \frac{57291297627793508881934344955}{5967102057548373881711176149884} a^{9} - \frac{2212244318877698358678744791339}{5967102057548373881711176149884} a^{8} - \frac{1426563021762483795851728111245}{5967102057548373881711176149884} a^{7} - \frac{1287850960761377122707706071819}{2983551028774186940855588074942} a^{6} + \frac{1266643996505312778649694789851}{2983551028774186940855588074942} a^{5} + \frac{2588963105148377380399366888941}{5967102057548373881711176149884} a^{4} - \frac{58976421954904888076785238797}{5967102057548373881711176149884} a^{3} - \frac{441873850856305515843792100915}{2983551028774186940855588074942} a^{2} + \frac{2412721289501902917213048279719}{5967102057548373881711176149884} a + \frac{1622252325642023552318811967707}{5967102057548373881711176149884}$

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:  \( 4742583.82996 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T366:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 48 conjugacy class representatives for t18n366
Character table for t18n366 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 9.9.17515230173.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$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$$0$Trivial$[\ ]$
$\Q_{97}$$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}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$