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Properties

Label 18.18.4328192725...2768.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{27}\cdot 7^{12}\cdot 13^{12}$
Root discriminant $57.22$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-727, 10964, -40450, 11322, 152385, -144998, -168253, 220500, 55551, -121028, -740, 31254, -2802, -4062, 537, 254, -39, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 39*x^16 + 254*x^15 + 537*x^14 - 4062*x^13 - 2802*x^12 + 31254*x^11 - 740*x^10 - 121028*x^9 + 55551*x^8 + 220500*x^7 - 168253*x^6 - 144998*x^5 + 152385*x^4 + 11322*x^3 - 40450*x^2 + 10964*x - 727)
 
gp: K = bnfinit(x^18 - 6*x^17 - 39*x^16 + 254*x^15 + 537*x^14 - 4062*x^13 - 2802*x^12 + 31254*x^11 - 740*x^10 - 121028*x^9 + 55551*x^8 + 220500*x^7 - 168253*x^6 - 144998*x^5 + 152385*x^4 + 11322*x^3 - 40450*x^2 + 10964*x - 727, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 39 x^{16} + 254 x^{15} + 537 x^{14} - 4062 x^{13} - 2802 x^{12} + 31254 x^{11} - 740 x^{10} - 121028 x^{9} + 55551 x^{8} + 220500 x^{7} - 168253 x^{6} - 144998 x^{5} + 152385 x^{4} + 11322 x^{3} - 40450 x^{2} + 10964 x - 727 \)

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:  \(43281927256346630219321487392768=2^{27}\cdot 7^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.22$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(728=2^{3}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{728}(1,·)$, $\chi_{728}(261,·)$, $\chi_{728}(417,·)$, $\chi_{728}(393,·)$, $\chi_{728}(653,·)$, $\chi_{728}(81,·)$, $\chi_{728}(625,·)$, $\chi_{728}(477,·)$, $\chi_{728}(289,·)$, $\chi_{728}(165,·)$, $\chi_{728}(529,·)$, $\chi_{728}(365,·)$, $\chi_{728}(29,·)$, $\chi_{728}(113,·)$, $\chi_{728}(373,·)$, $\chi_{728}(9,·)$, $\chi_{728}(445,·)$, $\chi_{728}(53,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{1559401678852} a^{16} - \frac{71926507223}{1559401678852} a^{15} - \frac{91440749995}{779700839426} a^{14} + \frac{138056269617}{1559401678852} a^{13} + \frac{109839630371}{1559401678852} a^{12} + \frac{40617447573}{1559401678852} a^{11} + \frac{112549909615}{779700839426} a^{10} + \frac{370224281395}{1559401678852} a^{9} + \frac{17376958906}{389850419713} a^{8} + \frac{45020033200}{389850419713} a^{7} - \frac{136481985449}{779700839426} a^{6} - \frac{96270729122}{389850419713} a^{5} + \frac{370895820707}{1559401678852} a^{4} + \frac{418500007433}{1559401678852} a^{3} + \frac{34748157499}{389850419713} a^{2} + \frac{770775462647}{1559401678852} a - \frac{104106022122}{389850419713}$, $\frac{1}{40572913646602800964} a^{17} + \frac{4408905}{40572913646602800964} a^{16} - \frac{3135388383875203217}{40572913646602800964} a^{15} + \frac{1021518958757019983}{20286456823301400482} a^{14} - \frac{1808810496335204923}{40572913646602800964} a^{13} - \frac{1330370708827277091}{20286456823301400482} a^{12} - \frac{4042227765320036225}{40572913646602800964} a^{11} + \frac{2220769412345448755}{20286456823301400482} a^{10} + \frac{3956226556918952331}{20286456823301400482} a^{9} - \frac{6045868647173662349}{40572913646602800964} a^{8} + \frac{1146954139384072557}{20286456823301400482} a^{7} - \frac{226796097248983017}{10143228411650700241} a^{6} + \frac{1362379499764676685}{40572913646602800964} a^{5} + \frac{4312676669974254285}{40572913646602800964} a^{4} + \frac{9133593467678263621}{40572913646602800964} a^{3} - \frac{5611531735215324087}{20286456823301400482} a^{2} + \frac{3571391916674065267}{20286456823301400482} a + \frac{7562116359049399101}{40572913646602800964}$

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:  \( 8169166181.92 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{2}) \), 3.3.8281.1, 3.3.169.1, \(\Q(\zeta_{7})^+\), 3.3.8281.2, 6.6.35110380032.2, 6.6.14623232.1, 6.6.1229312.1, 6.6.35110380032.1, 9.9.567869252041.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$