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Properties

Label 18.0.26735514947...9952.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{27}\cdot 7^{12}\cdot 13^{12}\cdot 61^{12}$
Root discriminant $2585.93$
Ramified primes $2, 3, 7, 13, 61$
Class number $17458368291$ (GRH)
Class group $[3, 3, 3, 3, 3, 3, 3, 3, 3, 57, 15561]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![699092182315008, 0, 0, 0, 0, 0, 63795390480, 0, 0, 0, 0, 0, 1465272, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 1465272*x^12 + 63795390480*x^6 + 699092182315008)
 
gp: K = bnfinit(x^18 + 1465272*x^12 + 63795390480*x^6 + 699092182315008, 1)
 

Normalized defining polynomial

\( x^{18} + 1465272 x^{12} + 63795390480 x^{6} + 699092182315008 \)

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:  \(-26735514947708767000623460382175089698901347865969225007869952=-\,2^{12}\cdot 3^{27}\cdot 7^{12}\cdot 13^{12}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2585.93$
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:  $2, 3, 7, 13, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{72} a^{6} + \frac{1}{12} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{3096} a^{7} + \frac{1}{12} a^{5} - \frac{1}{2} a^{2} + \frac{79}{258} a$, $\frac{1}{6192} a^{8} - \frac{179}{516} a^{2}$, $\frac{1}{24768} a^{9} + \frac{337}{2064} a^{3}$, $\frac{1}{24768} a^{10} - \frac{179}{2064} a^{4} - \frac{1}{2} a$, $\frac{1}{74304} a^{11} - \frac{695}{6192} a^{5} - \frac{1}{6} a^{3}$, $\frac{1}{80880498432} a^{12} + \frac{1}{74304} a^{10} - \frac{1}{49536} a^{9} - \frac{38910535}{6740041536} a^{6} - \frac{179}{6192} a^{4} - \frac{337}{4128} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{567751}{1632762}$, $\frac{1}{80880498432} a^{13} - \frac{1}{49536} a^{10} + \frac{275753}{6740041536} a^{7} + \frac{1}{12} a^{5} - \frac{337}{4128} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{11507471}{70208766} a$, $\frac{1}{3477861432576} a^{14} - \frac{1}{148608} a^{11} + \frac{14426357}{289821786048} a^{8} + \frac{695}{12384} a^{5} + \frac{1}{12} a^{4} - \frac{1}{6} a^{3} + \frac{2268334819}{6037953876} a^{2} - \frac{1}{2} a$, $\frac{1}{299096083201536} a^{15} + \frac{223690207}{12462336800064} a^{9} - \frac{1}{12} a^{5} + \frac{420648827029}{2077056133344} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{77166789465996288} a^{16} - \frac{9839566253}{3215282894416512} a^{10} - \frac{1}{18576} a^{8} - \frac{1}{6192} a^{7} + \frac{1}{12} a^{5} - \frac{13112418460379}{535880482402752} a^{4} - \frac{79}{1548} a^{2} + \frac{179}{516} a + \frac{1}{3}$, $\frac{1}{6636343894075680768} a^{17} - \frac{572375602367}{276514328919820032} a^{11} + \frac{1}{74304} a^{9} - \frac{1}{12384} a^{8} + \frac{1036752912939457}{46085721486636672} a^{5} + \frac{1369}{6192} a^{3} - \frac{337}{1032} a^{2} - \frac{1}{3} a$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{57}\times C_{15561}$, which has order $17458368291$ (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:  \( \frac{1}{11077632711168} a^{15} + \frac{186857}{1384704088896} a^{9} + \frac{2026772295}{230784014816} a^{3} + \frac{1}{2} \) (order $6$)
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:  \( 973307437394863.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), Deg 3 x3, Deg 3, 6.0.410174498845670832.1, 6.0.18688575603655877283.1, Deg 6 x2, Deg 9 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 sibling: data not computed
Degree 9 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$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}$
$61$61.3.2.2$x^{3} + 122$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.2.2$x^{3} + 122$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.2.2$x^{3} + 122$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.2.2$x^{3} + 122$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.2.2$x^{3} + 122$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.2.2$x^{3} + 122$$3$$1$$2$$C_3$$[\ ]_{3}$