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Properties

Label 18.0.39187042708...5504.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{32}\cdot 19^{9}$
Root discriminant $56.91$
Ramified primes $2, 3, 19$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times D_9:C_3$ (as 18T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1912285, -3954051, 6963867, -7264110, 7249680, -5527062, 4021662, -2387844, 1330443, -619917, 268533, -97128, 32646, -8946, 2286, -444, 81, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 81*x^16 - 444*x^15 + 2286*x^14 - 8946*x^13 + 32646*x^12 - 97128*x^11 + 268533*x^10 - 619917*x^9 + 1330443*x^8 - 2387844*x^7 + 4021662*x^6 - 5527062*x^5 + 7249680*x^4 - 7264110*x^3 + 6963867*x^2 - 3954051*x + 1912285)
 
gp: K = bnfinit(x^18 - 9*x^17 + 81*x^16 - 444*x^15 + 2286*x^14 - 8946*x^13 + 32646*x^12 - 97128*x^11 + 268533*x^10 - 619917*x^9 + 1330443*x^8 - 2387844*x^7 + 4021662*x^6 - 5527062*x^5 + 7249680*x^4 - 7264110*x^3 + 6963867*x^2 - 3954051*x + 1912285, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 81 x^{16} - 444 x^{15} + 2286 x^{14} - 8946 x^{13} + 32646 x^{12} - 97128 x^{11} + 268533 x^{10} - 619917 x^{9} + 1330443 x^{8} - 2387844 x^{7} + 4021662 x^{6} - 5527062 x^{5} + 7249680 x^{4} - 7264110 x^{3} + 6963867 x^{2} - 3954051 x + 1912285 \)

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:  \(-39187042708921278373282522005504=-\,2^{16}\cdot 3^{32}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.91$
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:  $2, 3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{31949523268048985373131551845640842512879472783} a^{17} - \frac{1585723622458263784630227406205146204962511970}{10649841089349661791043850615213614170959824261} a^{16} - \frac{1054136099346244391301379325981799291363707490}{10649841089349661791043850615213614170959824261} a^{15} - \frac{4447159574484870672165277750631880575342346043}{31949523268048985373131551845640842512879472783} a^{14} + \frac{1128387210667528558449505027708262691667918770}{10649841089349661791043850615213614170959824261} a^{13} + \frac{449800164331134989600520268419573208104119912}{10649841089349661791043850615213614170959824261} a^{12} - \frac{1906638590793592470051548665133651316125783743}{31949523268048985373131551845640842512879472783} a^{11} - \frac{665231898159046575230904733960886253825550020}{10649841089349661791043850615213614170959824261} a^{10} + \frac{3127827253742219919902646804107701236870711756}{10649841089349661791043850615213614170959824261} a^{9} + \frac{4044959097317235818638953685552734817883502730}{31949523268048985373131551845640842512879472783} a^{8} + \frac{83456793077660531205771907580379082414491361}{10649841089349661791043850615213614170959824261} a^{7} + \frac{266213390910036647599379272539114561615015385}{10649841089349661791043850615213614170959824261} a^{6} - \frac{7219231885343772689456977290226669665631908089}{31949523268048985373131551845640842512879472783} a^{5} - \frac{838334360316829839977878913281047176304761721}{10649841089349661791043850615213614170959824261} a^{4} - \frac{110174995540758719545556688435782160531450583}{10649841089349661791043850615213614170959824261} a^{3} + \frac{15265182520316577933213874470183671078307626120}{31949523268048985373131551845640842512879472783} a^{2} - \frac{795184464886790805653465453993597442174215484}{10649841089349661791043850615213614170959824261} a + \frac{76606659726120886191389334617306721349525966}{10649841089349661791043850615213614170959824261}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 158639448.38554028 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-19}) \), 3.1.108.1, 6.0.80003376.4, 9.1.11019960576.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
2Data not computed
3Data not computed
19Data not computed