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Properties

Label 18.18.9540369687...2997.1
Degree $18$
Signature $[18, 0]$
Discriminant $37^{9}\cdot 151^{4}\cdot 613^{4}$
Root discriminant $77.22$
Ramified primes $37, 151, 613$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T888

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25, 7505, -145537, 624213, -159932, -1066391, 423035, 700492, -272786, -233941, 82703, 42867, -13496, -4267, 1215, 212, -56, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 56*x^16 + 212*x^15 + 1215*x^14 - 4267*x^13 - 13496*x^12 + 42867*x^11 + 82703*x^10 - 233941*x^9 - 272786*x^8 + 700492*x^7 + 423035*x^6 - 1066391*x^5 - 159932*x^4 + 624213*x^3 - 145537*x^2 + 7505*x - 25)
 
gp: K = bnfinit(x^18 - 4*x^17 - 56*x^16 + 212*x^15 + 1215*x^14 - 4267*x^13 - 13496*x^12 + 42867*x^11 + 82703*x^10 - 233941*x^9 - 272786*x^8 + 700492*x^7 + 423035*x^6 - 1066391*x^5 - 159932*x^4 + 624213*x^3 - 145537*x^2 + 7505*x - 25, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 56 x^{16} + 212 x^{15} + 1215 x^{14} - 4267 x^{13} - 13496 x^{12} + 42867 x^{11} + 82703 x^{10} - 233941 x^{9} - 272786 x^{8} + 700492 x^{7} + 423035 x^{6} - 1066391 x^{5} - 159932 x^{4} + 624213 x^{3} - 145537 x^{2} + 7505 x - 25 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9540369687303470979346612772502997=37^{9}\cdot 151^{4}\cdot 613^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.22$
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:  $37, 151, 613$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3805683437877220924578374650367916080} a^{17} - \frac{1076116150287809747772279920191334359}{3805683437877220924578374650367916080} a^{16} + \frac{1399212398564405233700472728473456829}{3805683437877220924578374650367916080} a^{15} + \frac{1456319720194418278038534967516236637}{3805683437877220924578374650367916080} a^{14} + \frac{8764669006192745785629406408726055}{95142085946930523114459366259197902} a^{13} - \frac{1416692126424772748857606502130255027}{3805683437877220924578374650367916080} a^{12} - \frac{998341618782472177806987711806122831}{3805683437877220924578374650367916080} a^{11} + \frac{37381429645161639335926893521457467}{237855214867326307786148415647994755} a^{10} - \frac{459007950467189908664058793152743617}{3805683437877220924578374650367916080} a^{9} + \frac{707562651164288945744405844223678367}{1902841718938610462289187325183958040} a^{8} - \frac{104368960347159655045115757199046519}{951420859469305231144593662591979020} a^{7} - \frac{82733282758950159203425000538642743}{237855214867326307786148415647994755} a^{6} - \frac{246178671064169103128143501422186801}{761136687575444184915674930073583216} a^{5} - \frac{236616494368890967777025195920611297}{475710429734652615572296831295989510} a^{4} + \frac{411276324214994795695332783443744227}{951420859469305231144593662591979020} a^{3} - \frac{558900610389848831868145109672918847}{3805683437877220924578374650367916080} a^{2} - \frac{454958216178007923216718894166797753}{951420859469305231144593662591979020} a - \frac{378017627952353368828353480355709111}{761136687575444184915674930073583216}$

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:  $17$
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:  \( 184865536690 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T888:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 362880
The 36 conjugacy class representatives for t18n888
Character table for t18n888 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 9.9.11729467378561.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ $18$ ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $18$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$37$37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$151$$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
151.4.2.2$x^{4} - 151 x^{2} + 273612$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
151.4.2.2$x^{4} - 151 x^{2} + 273612$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
151.4.0.1$x^{4} - x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
151.4.0.1$x^{4} - x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
613Data not computed