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Properties

Label 18.4.30367959915...1603.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,3^{27}\cdot 73^{5}\cdot 577^{3}$
Root discriminant $49.37$
Ramified primes $3, 73, 577$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 18T765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37981, 23652, 7833, 93928, -76062, 149169, -115747, 134766, -99219, 78591, -46083, 23430, -9267, 2544, -480, 5, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 21*x^16 + 5*x^15 - 480*x^14 + 2544*x^13 - 9267*x^12 + 23430*x^11 - 46083*x^10 + 78591*x^9 - 99219*x^8 + 134766*x^7 - 115747*x^6 + 149169*x^5 - 76062*x^4 + 93928*x^3 + 7833*x^2 + 23652*x + 37981)
 
gp: K = bnfinit(x^18 - 6*x^17 + 21*x^16 + 5*x^15 - 480*x^14 + 2544*x^13 - 9267*x^12 + 23430*x^11 - 46083*x^10 + 78591*x^9 - 99219*x^8 + 134766*x^7 - 115747*x^6 + 149169*x^5 - 76062*x^4 + 93928*x^3 + 7833*x^2 + 23652*x + 37981, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 21 x^{16} + 5 x^{15} - 480 x^{14} + 2544 x^{13} - 9267 x^{12} + 23430 x^{11} - 46083 x^{10} + 78591 x^{9} - 99219 x^{8} + 134766 x^{7} - 115747 x^{6} + 149169 x^{5} - 76062 x^{4} + 93928 x^{3} + 7833 x^{2} + 23652 x + 37981 \)

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:  \(-3036795991579620615531492351603=-\,3^{27}\cdot 73^{5}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.37$
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:  $3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{86147113579821441617924820106068907321171785437} a^{17} + \frac{13170537161516834629308826356267236605100672995}{86147113579821441617924820106068907321171785437} a^{16} - \frac{1940671341369281266647622961555200964222124615}{86147113579821441617924820106068907321171785437} a^{15} + \frac{37906693838686658522362506105218289996246660683}{86147113579821441617924820106068907321171785437} a^{14} - \frac{8219643456341000132392959626767282828823306951}{86147113579821441617924820106068907321171785437} a^{13} + \frac{14301367105514379225496086755202502067471517356}{86147113579821441617924820106068907321171785437} a^{12} + \frac{24011956768153235179041669443703500658159061912}{86147113579821441617924820106068907321171785437} a^{11} + \frac{31214613176716814822887617126088573772758801776}{86147113579821441617924820106068907321171785437} a^{10} - \frac{16164105297676493389015212294419748797695194756}{86147113579821441617924820106068907321171785437} a^{9} - \frac{27199104888884565947694923616787993743472126178}{86147113579821441617924820106068907321171785437} a^{8} - \frac{3822628952135973971606407089303354990587552733}{86147113579821441617924820106068907321171785437} a^{7} + \frac{7642630527887101252815850612552432911428093927}{86147113579821441617924820106068907321171785437} a^{6} - \frac{21564764910763594455117124467431030009881691515}{86147113579821441617924820106068907321171785437} a^{5} - \frac{139786473558769787847582385211764232259079786}{1213339627884809036872180564874209962270025147} a^{4} - \frac{2342476705916381471485806734306547023170280493}{86147113579821441617924820106068907321171785437} a^{3} - \frac{40024409500029224293014842920763779305384103030}{86147113579821441617924820106068907321171785437} a^{2} + \frac{20746070045563038873397585813330093035928281124}{86147113579821441617924820106068907321171785437} a + \frac{29744387660717385401430306137705459629144488799}{86147113579821441617924820106068907321171785437}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 31302538.0806 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T765:

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 t18n765 are not computed
Character table for t18n765 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

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 $18$ R $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $18$ $18$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
3Data not computed
73Data not computed
577Data not computed