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Properties

Label 18.0.14845701048...9424.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 7^{15}\cdot 13^{12}$
Root discriminant $79.14$
Ramified primes $2, 7, 13$
Class number $39312$ (GRH)
Class group $[6, 6, 1092]$ (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![19683833, 11905764, 21350862, 4622682, 5324905, -960518, 892387, -443116, 276223, -87252, 50908, -16970, 6342, -1870, 633, -130, 33, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 33*x^16 - 130*x^15 + 633*x^14 - 1870*x^13 + 6342*x^12 - 16970*x^11 + 50908*x^10 - 87252*x^9 + 276223*x^8 - 443116*x^7 + 892387*x^6 - 960518*x^5 + 5324905*x^4 + 4622682*x^3 + 21350862*x^2 + 11905764*x + 19683833)
 
gp: K = bnfinit(x^18 - 6*x^17 + 33*x^16 - 130*x^15 + 633*x^14 - 1870*x^13 + 6342*x^12 - 16970*x^11 + 50908*x^10 - 87252*x^9 + 276223*x^8 - 443116*x^7 + 892387*x^6 - 960518*x^5 + 5324905*x^4 + 4622682*x^3 + 21350862*x^2 + 11905764*x + 19683833, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 33 x^{16} - 130 x^{15} + 633 x^{14} - 1870 x^{13} + 6342 x^{12} - 16970 x^{11} + 50908 x^{10} - 87252 x^{9} + 276223 x^{8} - 443116 x^{7} + 892387 x^{6} - 960518 x^{5} + 5324905 x^{4} + 4622682 x^{3} + 21350862 x^{2} + 11905764 x + 19683833 \)

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:  \(-14845701048926894165227270175719424=-\,2^{27}\cdot 7^{15}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.14$
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$$\chi_{728}(1,·)$, $\chi_{728}(417,·)$, $\chi_{728}(9,·)$, $\chi_{728}(269,·)$, $\chi_{728}(237,·)$, $\chi_{728}(81,·)$, $\chi_{728}(341,·)$, $\chi_{728}(625,·)$, $\chi_{728}(157,·)$, $\chi_{728}(677,·)$, $\chi_{728}(289,·)$, $\chi_{728}(549,·)$, $\chi_{728}(529,·)$, $\chi_{728}(685,·)$, $\chi_{728}(573,·)$, $\chi_{728}(113,·)$, $\chi_{728}(393,·)$, $\chi_{728}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4}$, $\frac{1}{36} a^{13} + \frac{1}{36} a^{12} - \frac{1}{6} a^{11} - \frac{1}{36} a^{9} + \frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{1}{18} a^{6} + \frac{1}{3} a^{5} - \frac{4}{9} a^{4} - \frac{1}{18} a^{3} + \frac{1}{9} a^{2} - \frac{1}{4} a + \frac{17}{36}$, $\frac{1}{36} a^{14} + \frac{1}{18} a^{12} + \frac{1}{6} a^{11} - \frac{1}{36} a^{10} + \frac{1}{9} a^{9} - \frac{1}{6} a^{8} - \frac{1}{9} a^{7} - \frac{2}{9} a^{6} + \frac{2}{9} a^{5} + \frac{7}{18} a^{4} + \frac{1}{6} a^{3} - \frac{13}{36} a^{2} - \frac{5}{18} a - \frac{2}{9}$, $\frac{1}{36} a^{15} + \frac{1}{9} a^{12} - \frac{7}{36} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{2}{9} a^{8} - \frac{1}{18} a^{7} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} - \frac{4}{9} a^{4} + \frac{1}{4} a^{3} + \frac{5}{18} a - \frac{4}{9}$, $\frac{1}{9946826061818589252} a^{16} - \frac{13772165683013753}{1105202895757621028} a^{15} - \frac{22029081548207725}{9946826061818589252} a^{14} + \frac{61882239752351387}{9946826061818589252} a^{13} - \frac{362120972724194587}{4973413030909294626} a^{12} - \frac{2061608280583981931}{9946826061818589252} a^{11} + \frac{292685041696166603}{3315608687272863084} a^{10} - \frac{15859574287361041}{1105202895757621028} a^{9} + \frac{742336899511342435}{9946826061818589252} a^{8} - \frac{177352035716352844}{2486706515454647313} a^{7} + \frac{119477455595708501}{828902171818215771} a^{6} + \frac{682945859754419071}{1657804343636431542} a^{5} + \frac{535546000385294749}{3315608687272863084} a^{4} + \frac{4228338209505153871}{9946826061818589252} a^{3} - \frac{162251102700862409}{3315608687272863084} a^{2} - \frac{603530922815145557}{3315608687272863084} a - \frac{4403795784474605477}{9946826061818589252}$, $\frac{1}{252813150738224539362457937844516} a^{17} + \frac{2865892893659}{126406575369112269681228968922258} a^{16} - \frac{694731492250881053966480832175}{126406575369112269681228968922258} a^{15} - \frac{74223228511289759704280233399}{84271050246074846454152645948172} a^{14} - \frac{45001053741014726427577874549}{7022587520506237204512720495681} a^{13} + \frac{3937538299392905281863368210173}{126406575369112269681228968922258} a^{12} + \frac{15738426197087651135851962811463}{63203287684556134840614484461129} a^{11} + \frac{19095302447797219369962602477411}{84271050246074846454152645948172} a^{10} + \frac{50612066796221257734285267767347}{252813150738224539362457937844516} a^{9} - \frac{29277967049226639582465662104195}{126406575369112269681228968922258} a^{8} + \frac{10522499604427964965898603094569}{63203287684556134840614484461129} a^{7} - \frac{22899085557753429700555435171591}{126406575369112269681228968922258} a^{6} - \frac{39438990972106757346945307751365}{252813150738224539362457937844516} a^{5} + \frac{3427495232607457237098006336484}{7022587520506237204512720495681} a^{4} - \frac{37032935260173323549745118896067}{126406575369112269681228968922258} a^{3} - \frac{55937632901002899913186887025433}{252813150738224539362457937844516} a^{2} + \frac{84770540801167036770475379127631}{252813150738224539362457937844516} a - \frac{39084864898688490202578035018311}{126406575369112269681228968922258}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{1092}$, which has order $39312$ (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:  \( 205236.825908 \) (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

\(\Q(\sqrt{-14}) \), 3.3.8281.1, 3.3.8281.2, 3.3.169.1, \(\Q(\zeta_{7})^+\), 6.0.245772660224.4, 6.0.245772660224.3, 6.0.5015768576.6, 6.0.8605184.1, 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$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$$[3]^{3}$
7Data not computed
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$