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Properties

Label 18.2.30576476160...0000.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{37}\cdot 3^{6}\cdot 5^{15}$
Root discriminant $22.92$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times S_3^2$ (as 18T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -21, 76, -180, 312, -456, 548, -498, 400, -346, 348, -354, 284, -150, 36, 7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 7*x^16 + 36*x^15 - 150*x^14 + 284*x^13 - 354*x^12 + 348*x^11 - 346*x^10 + 400*x^9 - 498*x^8 + 548*x^7 - 456*x^6 + 312*x^5 - 180*x^4 + 76*x^3 - 21*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 7*x^16 + 36*x^15 - 150*x^14 + 284*x^13 - 354*x^12 + 348*x^11 - 346*x^10 + 400*x^9 - 498*x^8 + 548*x^7 - 456*x^6 + 312*x^5 - 180*x^4 + 76*x^3 - 21*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 7 x^{16} + 36 x^{15} - 150 x^{14} + 284 x^{13} - 354 x^{12} + 348 x^{11} - 346 x^{10} + 400 x^{9} - 498 x^{8} + 548 x^{7} - 456 x^{6} + 312 x^{5} - 180 x^{4} + 76 x^{3} - 21 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3057647616000000000000000=2^{37}\cdot 3^{6}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.92$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{9} - \frac{1}{2} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{10} - \frac{1}{3} a^{8} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{11} - \frac{1}{3} a^{9} + \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{18} a^{16} + \frac{1}{18} a^{15} + \frac{1}{18} a^{14} - \frac{1}{18} a^{12} + \frac{1}{18} a^{11} - \frac{1}{18} a^{10} - \frac{1}{9} a^{9} + \frac{1}{18} a^{8} + \frac{7}{18} a^{7} - \frac{1}{6} a^{6} - \frac{4}{9} a^{5} - \frac{5}{18} a^{4} + \frac{1}{6} a^{3} - \frac{1}{18} a^{2} - \frac{1}{3} a - \frac{4}{9}$, $\frac{1}{37132380774} a^{17} - \frac{663396931}{37132380774} a^{16} + \frac{710686937}{37132380774} a^{15} + \frac{35818939}{807225669} a^{14} - \frac{1168761704}{18566190387} a^{13} + \frac{17178887}{229212227} a^{12} - \frac{58076281}{12377460258} a^{11} - \frac{281306161}{2062910043} a^{10} - \frac{9230401658}{18566190387} a^{9} + \frac{6373436977}{18566190387} a^{8} - \frac{8088476429}{37132380774} a^{7} + \frac{1213410122}{18566190387} a^{6} + \frac{9122397157}{18566190387} a^{5} - \frac{144011207}{314681193} a^{4} - \frac{1104969349}{37132380774} a^{3} + \frac{6108483442}{18566190387} a^{2} - \frac{3325159007}{37132380774} a - \frac{12675243611}{37132380774}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 165711.7100392055 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3^2$ (as 18T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 18 conjugacy class representatives for $C_2\times S_3^2$
Character table for $C_2\times S_3^2$

Intermediate fields

\(\Q(\sqrt{10}) \), 3.1.200.1, 3.1.300.1, 6.2.57600000.1, 6.2.6400000.1, 9.1.3456000000.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 18 siblings: 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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.11.5$x^{6} + 6$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.12.26.99$x^{12} + 4 x^{11} + 6 x^{10} + 6 x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} + 6$$12$$1$$26$$S_3 \times C_2^2$$[2, 3]_{3}^{2}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.6.5.2$x^{6} + 10$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$