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Properties

Label 18.0.76262819902...3951.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,19^{16}\cdot 31^{9}$
Root discriminant $76.27$
Ramified primes $19, 31$
Class number $79812$ (GRH)
Class group $[2, 39906]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![707215681, -336518414, 536835424, -226538639, 193370792, -71859458, 43041793, -13992021, 6495797, -1828199, 687416, -164417, 50888, -9987, 2531, -378, 76, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 76*x^16 - 378*x^15 + 2531*x^14 - 9987*x^13 + 50888*x^12 - 164417*x^11 + 687416*x^10 - 1828199*x^9 + 6495797*x^8 - 13992021*x^7 + 43041793*x^6 - 71859458*x^5 + 193370792*x^4 - 226538639*x^3 + 536835424*x^2 - 336518414*x + 707215681)
 
gp: K = bnfinit(x^18 - 7*x^17 + 76*x^16 - 378*x^15 + 2531*x^14 - 9987*x^13 + 50888*x^12 - 164417*x^11 + 687416*x^10 - 1828199*x^9 + 6495797*x^8 - 13992021*x^7 + 43041793*x^6 - 71859458*x^5 + 193370792*x^4 - 226538639*x^3 + 536835424*x^2 - 336518414*x + 707215681, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 76 x^{16} - 378 x^{15} + 2531 x^{14} - 9987 x^{13} + 50888 x^{12} - 164417 x^{11} + 687416 x^{10} - 1828199 x^{9} + 6495797 x^{8} - 13992021 x^{7} + 43041793 x^{6} - 71859458 x^{5} + 193370792 x^{4} - 226538639 x^{3} + 536835424 x^{2} - 336518414 x + 707215681 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7626281990217745472007517214473951=-\,19^{16}\cdot 31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.27$
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:  $19, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(589=19\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{589}(1,·)$, $\chi_{589}(404,·)$, $\chi_{589}(216,·)$, $\chi_{589}(218,·)$, $\chi_{589}(92,·)$, $\chi_{589}(30,·)$, $\chi_{589}(187,·)$, $\chi_{589}(156,·)$, $\chi_{589}(125,·)$, $\chi_{589}(557,·)$, $\chi_{589}(495,·)$, $\chi_{589}(435,·)$, $\chi_{589}(309,·)$, $\chi_{589}(311,·)$, $\chi_{589}(340,·)$, $\chi_{589}(123,·)$, $\chi_{589}(61,·)$, $\chi_{589}(63,·)$$\rbrace$
This is 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}{37896151960943957609517688354002402475508815817956989731} a^{17} - \frac{16950195831095394527785182054256552404210953786346811464}{37896151960943957609517688354002402475508815817956989731} a^{16} + \frac{1606853799992990700271749592169222729257357704766981010}{37896151960943957609517688354002402475508815817956989731} a^{15} + \frac{13522905065716371029346844868808526966418586321481587903}{37896151960943957609517688354002402475508815817956989731} a^{14} - \frac{3398272787256622148288304875367371278252342046470121176}{37896151960943957609517688354002402475508815817956989731} a^{13} - \frac{7538851579146552991354046413364563477522882415833605222}{37896151960943957609517688354002402475508815817956989731} a^{12} + \frac{3350104237845733545834734091059734300871264148218693821}{37896151960943957609517688354002402475508815817956989731} a^{11} + \frac{8497577205132048082371106310640967248706530262282771674}{37896151960943957609517688354002402475508815817956989731} a^{10} + \frac{479273506119386798298836891627748385932785078628256944}{37896151960943957609517688354002402475508815817956989731} a^{9} - \frac{16066328916385969562956883271973607476811244709161411258}{37896151960943957609517688354002402475508815817956989731} a^{8} + \frac{5578232371249711499257806622827799168838028815677721714}{37896151960943957609517688354002402475508815817956989731} a^{7} - \frac{4854525947317002678487272620848750430365033581981235501}{37896151960943957609517688354002402475508815817956989731} a^{6} + \frac{13678125512323727928611180431339731383653456675049412822}{37896151960943957609517688354002402475508815817956989731} a^{5} + \frac{736710081092717032986651639705951738007742760909216687}{37896151960943957609517688354002402475508815817956989731} a^{4} + \frac{12035450576834859142387418653341366818334247367257129327}{37896151960943957609517688354002402475508815817956989731} a^{3} + \frac{819281503036621177336861157670294672068982619002708622}{37896151960943957609517688354002402475508815817956989731} a^{2} + \frac{16873132348516773336821200784216668431899109694447810712}{37896151960943957609517688354002402475508815817956989731} a + \frac{18189290285388980462843837967896739722632126449802372131}{37896151960943957609517688354002402475508815817956989731}$

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

Class group and class number

$C_{2}\times C_{39906}$, which has order $79812$ (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:  $8$
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:  \( 22305.8950792 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-31}) \), 3.3.361.1, 6.0.3882392911.2, \(\Q(\zeta_{19})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ $18$ R $18$ $18$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$19$19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
$31$31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$