↑

Properties

Label 18.2.20341215333...9184.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{21}\cdot 7^{15}$
Root discriminant $28.95$
Ramified primes $2, 3, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times C_3:S_3$ (as 18T12)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4913, 0, 0, -942, 0, 0, -313, 0, 0, 244, 0, 0, 25, 0, 0, -2, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^15 + 25*x^12 + 244*x^9 - 313*x^6 - 942*x^3 - 4913)
 
gp: K = bnfinit(x^18 - 2*x^15 + 25*x^12 + 244*x^9 - 313*x^6 - 942*x^3 - 4913, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{15} + 25 x^{12} + 244 x^{9} - 313 x^{6} - 942 x^{3} - 4913 \)

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:  \(203412153331596396123869184=2^{12}\cdot 3^{21}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.95$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{30} a^{9} - \frac{13}{30} a^{6} + \frac{3}{10} a^{3} - \frac{7}{30}$, $\frac{1}{30} a^{10} - \frac{13}{30} a^{7} + \frac{3}{10} a^{4} - \frac{7}{30} a$, $\frac{1}{30} a^{11} - \frac{13}{30} a^{8} + \frac{3}{10} a^{5} - \frac{7}{30} a^{2}$, $\frac{1}{150} a^{12} - \frac{1}{75} a^{9} - \frac{7}{75} a^{6} - \frac{14}{75} a^{3} - \frac{17}{150}$, $\frac{1}{150} a^{13} - \frac{1}{75} a^{10} - \frac{7}{75} a^{7} - \frac{14}{75} a^{4} - \frac{17}{150} a$, $\frac{1}{150} a^{14} - \frac{1}{75} a^{11} - \frac{7}{75} a^{8} - \frac{14}{75} a^{5} - \frac{17}{150} a^{2}$, $\frac{1}{1070100} a^{15} - \frac{1}{450} a^{14} - \frac{1}{450} a^{13} + \frac{827}{356700} a^{12} + \frac{1}{225} a^{11} + \frac{7}{450} a^{10} - \frac{7}{7134} a^{9} + \frac{7}{225} a^{8} + \frac{11}{50} a^{7} + \frac{26246}{53505} a^{6} - \frac{61}{225} a^{5} + \frac{73}{450} a^{4} - \frac{436171}{1070100} a^{3} + \frac{17}{450} a^{2} - \frac{28}{75} a - \frac{517021}{1070100}$, $\frac{1}{18191700} a^{16} - \frac{16543}{18191700} a^{13} + \frac{1}{450} a^{12} - \frac{1}{90} a^{11} - \frac{11558}{4547925} a^{10} - \frac{1}{225} a^{9} - \frac{17}{90} a^{8} - \frac{1381927}{9095850} a^{7} - \frac{7}{225} a^{6} - \frac{13}{30} a^{5} - \frac{217523}{6063900} a^{4} + \frac{61}{225} a^{3} + \frac{37}{90} a^{2} + \frac{1815797}{18191700} a - \frac{17}{450}$, $\frac{1}{309258900} a^{17} + \frac{589847}{309258900} a^{14} - \frac{1}{450} a^{13} - \frac{1}{450} a^{12} - \frac{617948}{77314725} a^{11} + \frac{1}{225} a^{10} + \frac{7}{450} a^{9} - \frac{70510387}{154629450} a^{8} + \frac{7}{225} a^{7} + \frac{11}{50} a^{6} - \frac{33771103}{103086300} a^{5} - \frac{61}{225} a^{4} + \frac{73}{450} a^{3} + \frac{127338527}{309258900} a^{2} + \frac{17}{450} a - \frac{28}{75}$

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:  $9$
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:  \( 1282241.5051674389 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3$ (as 18T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $C_2\times C_3:S_3$
Character table for $C_2\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{21}) \), 3.1.5292.1, 3.1.1323.1, 3.1.108.1, 3.1.588.1, 6.2.36756909.1, 6.2.588110544.1, 6.2.12002256.1, 6.2.7260624.1, 9.1.444611571264.1

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

Sibling fields

Galois closure: data not computed
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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$[\ ]_{3}^{2}$
$3$3.6.7.2$x^{6} + 3 x^{2} + 6$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
7Data not computed