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Properties

Label 18.10.2810414880...6032.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{18}\cdot 3^{18}\cdot 7^{13}\cdot 13^{4}$
Root discriminant $43.26$
Ramified primes $2, 3, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-97, -1812, -13008, -47316, -95673, -107592, -55584, 10296, 32466, 17366, 996, -2910, -1112, 66, 111, -2, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^16 - 2*x^15 + 111*x^14 + 66*x^13 - 1112*x^12 - 2910*x^11 + 996*x^10 + 17366*x^9 + 32466*x^8 + 10296*x^7 - 55584*x^6 - 107592*x^5 - 95673*x^4 - 47316*x^3 - 13008*x^2 - 1812*x - 97)
 
gp: K = bnfinit(x^18 - 12*x^16 - 2*x^15 + 111*x^14 + 66*x^13 - 1112*x^12 - 2910*x^11 + 996*x^10 + 17366*x^9 + 32466*x^8 + 10296*x^7 - 55584*x^6 - 107592*x^5 - 95673*x^4 - 47316*x^3 - 13008*x^2 - 1812*x - 97, 1)
 

Normalized defining polynomial

\( x^{18} - 12 x^{16} - 2 x^{15} + 111 x^{14} + 66 x^{13} - 1112 x^{12} - 2910 x^{11} + 996 x^{10} + 17366 x^{9} + 32466 x^{8} + 10296 x^{7} - 55584 x^{6} - 107592 x^{5} - 95673 x^{4} - 47316 x^{3} - 13008 x^{2} - 1812 x - 97 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(281041488075161284097055916032=2^{18}\cdot 3^{18}\cdot 7^{13}\cdot 13^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.26$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{14} a^{15} + \frac{3}{14} a^{14} - \frac{3}{7} a^{13} + \frac{3}{7} a^{11} - \frac{3}{7} a^{10} - \frac{1}{7} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{14} a - \frac{1}{14}$, $\frac{1}{28} a^{16} - \frac{1}{28} a^{14} - \frac{5}{14} a^{13} - \frac{2}{7} a^{12} + \frac{1}{7} a^{11} + \frac{1}{7} a^{10} - \frac{1}{14} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{7} - \frac{1}{2} a^{6} + \frac{2}{7} a^{5} - \frac{1}{2} a^{4} - \frac{11}{28} a^{2} + \frac{1}{7} a + \frac{3}{28}$, $\frac{1}{260378689023335432006044} a^{17} + \frac{1021941874247481941204}{65094672255833858001511} a^{16} - \frac{1915602165436488035767}{260378689023335432006044} a^{15} - \frac{19317537323788516998679}{130189344511667716003022} a^{14} - \frac{26331409502422931231}{73887255681990758231} a^{13} - \frac{11092421276809414933113}{65094672255833858001511} a^{12} + \frac{13742989813534738747423}{65094672255833858001511} a^{11} - \frac{59515768325367103149951}{130189344511667716003022} a^{10} + \frac{1561282262803678690714}{9299238893690551143073} a^{9} + \frac{3410735462921745054493}{65094672255833858001511} a^{8} + \frac{24703518941128813827799}{130189344511667716003022} a^{7} + \frac{262015994202941274074}{65094672255833858001511} a^{6} + \frac{3055793171029417795367}{130189344511667716003022} a^{5} + \frac{328297318880113315847}{9299238893690551143073} a^{4} + \frac{1820420711854381064557}{260378689023335432006044} a^{3} + \frac{14446000065036413189116}{65094672255833858001511} a^{2} + \frac{32678126412587713306489}{260378689023335432006044} a + \frac{141365731865223148043}{671079095420967608263}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 188591961.036 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T766:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 144 conjugacy class representatives for t18n766 are not computed
Character table for t18n766 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.25046451847872.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R $18$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.9.9.5$x^{9} + 3 x^{7} + 3 x^{6} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
3.9.9.5$x^{9} + 3 x^{7} + 3 x^{6} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.4.2$x^{6} - 13 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$