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Properties

Label 18.0.69712928281...4375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 5^{12}\cdot 29^{9}$
Root discriminant $27.27$
Ramified primes $3, 5, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_9$ (as 18T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![783, 0, -506, 0, 817, 0, -1574, 0, 1993, 0, -1094, 0, 358, 0, -56, 0, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 7*x^16 - 56*x^14 + 358*x^12 - 1094*x^10 + 1993*x^8 - 1574*x^6 + 817*x^4 - 506*x^2 + 783)
 
gp: K = bnfinit(x^18 + 7*x^16 - 56*x^14 + 358*x^12 - 1094*x^10 + 1993*x^8 - 1574*x^6 + 817*x^4 - 506*x^2 + 783, 1)
 

Normalized defining polynomial

\( x^{18} + 7 x^{16} - 56 x^{14} + 358 x^{12} - 1094 x^{10} + 1993 x^{8} - 1574 x^{6} + 817 x^{4} - 506 x^{2} + 783 \)

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:  \(-69712928281989630615234375=-\,3^{9}\cdot 5^{12}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.27$
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:  $3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{15} a^{6} + \frac{4}{15} a^{4} + \frac{7}{15} a^{2} - \frac{1}{5}$, $\frac{1}{15} a^{7} - \frac{1}{15} a^{5} + \frac{2}{15} a^{3} + \frac{7}{15} a$, $\frac{1}{45} a^{8} + \frac{7}{15} a^{4} + \frac{14}{45} a^{2} - \frac{2}{5}$, $\frac{1}{90} a^{9} - \frac{1}{90} a^{8} + \frac{1}{15} a^{5} - \frac{7}{30} a^{4} + \frac{22}{45} a^{3} + \frac{31}{90} a^{2} + \frac{2}{15} a - \frac{3}{10}$, $\frac{1}{90} a^{10} - \frac{1}{90} a^{8} - \frac{1}{6} a^{5} - \frac{1}{90} a^{4} - \frac{1}{6} a^{3} + \frac{1}{90} a^{2} - \frac{1}{6} a - \frac{1}{10}$, $\frac{1}{90} a^{11} - \frac{1}{90} a^{8} - \frac{1}{30} a^{6} + \frac{1}{18} a^{5} + \frac{2}{15} a^{4} - \frac{1}{2} a^{3} + \frac{1}{9} a^{2} + \frac{1}{30} a + \frac{3}{10}$, $\frac{1}{1350} a^{12} - \frac{1}{675} a^{10} - \frac{1}{270} a^{8} - \frac{1}{30} a^{7} + \frac{7}{270} a^{6} - \frac{2}{15} a^{5} + \frac{38}{135} a^{4} + \frac{4}{15} a^{3} - \frac{56}{675} a^{2} + \frac{1}{10} a - \frac{23}{50}$, $\frac{1}{1350} a^{13} - \frac{1}{675} a^{11} - \frac{1}{270} a^{9} - \frac{1}{90} a^{8} + \frac{7}{270} a^{7} - \frac{7}{135} a^{5} + \frac{4}{15} a^{4} - \frac{281}{675} a^{3} + \frac{31}{90} a^{2} + \frac{31}{150} a + \frac{1}{5}$, $\frac{1}{4050} a^{14} + \frac{1}{675} a^{10} - \frac{7}{810} a^{8} + \frac{1}{45} a^{6} - \frac{1}{6} a^{5} - \frac{292}{675} a^{4} + \frac{1}{3} a^{3} + \frac{149}{810} a^{2} - \frac{1}{6} a + \frac{22}{75}$, $\frac{1}{4050} a^{15} + \frac{1}{675} a^{11} + \frac{1}{405} a^{9} - \frac{1}{90} a^{8} + \frac{1}{45} a^{7} - \frac{1}{30} a^{6} - \frac{22}{675} a^{5} - \frac{11}{30} a^{4} + \frac{1}{162} a^{3} + \frac{1}{9} a^{2} - \frac{6}{25} a + \frac{3}{10}$, $\frac{1}{36450} a^{16} + \frac{1}{18225} a^{14} - \frac{2}{6075} a^{12} + \frac{13}{36450} a^{10} + \frac{38}{3645} a^{8} - \frac{1}{30} a^{7} + \frac{173}{6075} a^{6} - \frac{2}{15} a^{5} - \frac{11219}{36450} a^{4} - \frac{7}{30} a^{3} - \frac{623}{18225} a^{2} - \frac{2}{5} a + \frac{281}{675}$, $\frac{1}{36450} a^{17} + \frac{1}{18225} a^{15} - \frac{2}{6075} a^{13} + \frac{13}{36450} a^{11} - \frac{1}{1458} a^{9} + \frac{173}{6075} a^{7} - \frac{1499}{36450} a^{5} - \frac{3458}{18225} a^{3} - \frac{1}{2} a^{2} - \frac{259}{675} a - \frac{1}{2}$

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:  $8$
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:  \( 3534920.45401 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 18T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-87}) \), 3.1.87.1 x3, 6.0.658503.1, 9.1.895152515625.1 x9

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

Sibling fields

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R R ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$