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Properties

Label 18.2.93367623828...1661.1
Degree $18$
Signature $[2, 8]$
Discriminant $101^{9}\cdot 292181^{2}$
Root discriminant $40.69$
Ramified primes $101, 292181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T913

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-408451709, 194016950, 209940349, -40115066, -84556770, 2085791, 19195542, 1454633, -2888104, -369935, 309106, 38942, -23537, -2349, 1320, 84, -49, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 49*x^16 + 84*x^15 + 1320*x^14 - 2349*x^13 - 23537*x^12 + 38942*x^11 + 309106*x^10 - 369935*x^9 - 2888104*x^8 + 1454633*x^7 + 19195542*x^6 + 2085791*x^5 - 84556770*x^4 - 40115066*x^3 + 209940349*x^2 + 194016950*x - 408451709)
 
gp: K = bnfinit(x^18 - 2*x^17 - 49*x^16 + 84*x^15 + 1320*x^14 - 2349*x^13 - 23537*x^12 + 38942*x^11 + 309106*x^10 - 369935*x^9 - 2888104*x^8 + 1454633*x^7 + 19195542*x^6 + 2085791*x^5 - 84556770*x^4 - 40115066*x^3 + 209940349*x^2 + 194016950*x - 408451709, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 49 x^{16} + 84 x^{15} + 1320 x^{14} - 2349 x^{13} - 23537 x^{12} + 38942 x^{11} + 309106 x^{10} - 369935 x^{9} - 2888104 x^{8} + 1454633 x^{7} + 19195542 x^{6} + 2085791 x^{5} - 84556770 x^{4} - 40115066 x^{3} + 209940349 x^{2} + 194016950 x - 408451709 \)

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:  \(93367623828446393959890781661=101^{9}\cdot 292181^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.69$
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:  $101, 292181$
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}{9092186297214936479221129499173652039238455301691507578471616754819} a^{17} - \frac{104766751743487572262142602883549706556705225061572299710616921530}{9092186297214936479221129499173652039238455301691507578471616754819} a^{16} - \frac{2874714153057725024433798246945575443108491799398416869419778871717}{9092186297214936479221129499173652039238455301691507578471616754819} a^{15} - \frac{127493351775908354903047553100711192524311209548434415739189697479}{9092186297214936479221129499173652039238455301691507578471616754819} a^{14} + \frac{1833204552199618693963152927152511856813765886444869549688357800515}{9092186297214936479221129499173652039238455301691507578471616754819} a^{13} + \frac{1791029188839401498914436073212537315513354845572086203305247060191}{9092186297214936479221129499173652039238455301691507578471616754819} a^{12} - \frac{1778042764344903660485431167181322675645953742501263902851088188210}{9092186297214936479221129499173652039238455301691507578471616754819} a^{11} + \frac{337279281563369873572588948947445664579187548596273678654062126183}{9092186297214936479221129499173652039238455301691507578471616754819} a^{10} + \frac{2984144788506613035265708401188436783164390996166465062541988246509}{9092186297214936479221129499173652039238455301691507578471616754819} a^{9} + \frac{4304586238381428933450199183678859042342514141064177443083701284572}{9092186297214936479221129499173652039238455301691507578471616754819} a^{8} - \frac{1092176832736202563021749146453591981818679020023359139189472890208}{9092186297214936479221129499173652039238455301691507578471616754819} a^{7} - \frac{4176696852585880303296300727959965566910117286044579168682507531928}{9092186297214936479221129499173652039238455301691507578471616754819} a^{6} - \frac{1209477977417270961997514345925954472227349747518817244867671049355}{9092186297214936479221129499173652039238455301691507578471616754819} a^{5} - \frac{2776957353433688369891220440478366945294432243521072055950829363832}{9092186297214936479221129499173652039238455301691507578471616754819} a^{4} + \frac{2496645253683280338788487216206222159114477752676218963269843119943}{9092186297214936479221129499173652039238455301691507578471616754819} a^{3} - \frac{41432178705825778840035440865993139449583476760418012725522184597}{9092186297214936479221129499173652039238455301691507578471616754819} a^{2} + \frac{2875937874297297877927503370723821252166434269319972010299082344972}{9092186297214936479221129499173652039238455301691507578471616754819} a - \frac{557625268221740656604281826761191259149720354801352111420541258086}{9092186297214936479221129499173652039238455301691507578471616754819}$

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:  \( 12396446.7025 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T913:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 725760
The 60 conjugacy class representatives for t18n913 are not computed
Character table for t18n913 is not computed

Intermediate fields

\(\Q(\sqrt{101}) \), 9.1.29510281.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 sibling: 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 $18$ $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ $18$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
101Data not computed
292181Data not computed