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Properties

Label 18.10.5219341921...5488.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{18}\cdot 3^{18}\cdot 7^{12}\cdot 13^{5}$
Root discriminant $44.77$
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![-13, 0, 780, 0, -2262, 0, -24581, 0, 8391, 0, 3855, 0, -305, 0, -120, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 120*x^14 - 305*x^12 + 3855*x^10 + 8391*x^8 - 24581*x^6 - 2262*x^4 + 780*x^2 - 13)
 
gp: K = bnfinit(x^18 + 3*x^16 - 120*x^14 - 305*x^12 + 3855*x^10 + 8391*x^8 - 24581*x^6 - 2262*x^4 + 780*x^2 - 13, 1)
 

Normalized defining polynomial

\( x^{18} + 3 x^{16} - 120 x^{14} - 305 x^{12} + 3855 x^{10} + 8391 x^{8} - 24581 x^{6} - 2262 x^{4} + 780 x^{2} - 13 \)

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:  \(521934192139585241894532415488=2^{18}\cdot 3^{18}\cdot 7^{12}\cdot 13^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.77$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{86} a^{14} + \frac{9}{43} a^{12} - \frac{3}{86} a^{10} + \frac{35}{86} a^{8} - \frac{1}{2} a^{7} - \frac{10}{43} a^{6} - \frac{8}{43} a^{4} - \frac{1}{2} a^{3} + \frac{20}{43} a^{2} - \frac{1}{2} a + \frac{6}{43}$, $\frac{1}{172} a^{15} + \frac{9}{86} a^{13} - \frac{1}{4} a^{12} + \frac{10}{43} a^{11} - \frac{1}{4} a^{10} - \frac{2}{43} a^{9} - \frac{1}{4} a^{8} + \frac{33}{86} a^{7} - \frac{1}{4} a^{6} + \frac{35}{86} a^{5} + \frac{1}{4} a^{4} - \frac{23}{86} a^{3} + \frac{1}{4} a^{2} - \frac{31}{172} a - \frac{1}{4}$, $\frac{1}{100422697900612} a^{16} - \frac{137890895465}{25105674475153} a^{14} - \frac{1}{4} a^{13} + \frac{10355966298323}{50211348950306} a^{12} - \frac{1}{4} a^{11} + \frac{4117389652451}{50211348950306} a^{10} - \frac{1}{4} a^{9} + \frac{7151141707095}{25105674475153} a^{8} - \frac{1}{4} a^{7} + \frac{9512848688923}{25105674475153} a^{6} + \frac{1}{4} a^{5} + \frac{8784122825522}{25105674475153} a^{4} + \frac{1}{4} a^{3} - \frac{5873420062565}{100422697900612} a^{2} - \frac{1}{4} a + \frac{9066304736341}{25105674475153}$, $\frac{1}{100422697900612} a^{17} + \frac{32289312911}{100422697900612} a^{15} - \frac{1}{172} a^{14} - \frac{9495032123891}{50211348950306} a^{13} - \frac{9}{86} a^{12} - \frac{4655613463641}{25105674475153} a^{11} - \frac{10}{43} a^{10} + \frac{5983435917553}{25105674475153} a^{9} - \frac{39}{86} a^{8} - \frac{11918506045017}{50211348950306} a^{7} - \frac{33}{86} a^{6} - \frac{12208251982277}{50211348950306} a^{5} + \frac{4}{43} a^{4} + \frac{17480695728275}{100422697900612} a^{3} - \frac{10}{43} a^{2} + \frac{18165779207463}{100422697900612} a + \frac{31}{172}$

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:  \( 242249876.124 \) (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 ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ $18$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ $18$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.18.64$x^{12} + 14 x^{11} + 12 x^{10} + 4 x^{9} + 10 x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 16 x - 8$$4$$3$$18$12T51$[2, 2, 2, 2]^{6}$
$3$3.9.9.5$x^{9} + 3 x^{7} + 3 x^{6} + 54$$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$$[3/2, 3/2, 3/2]_{2}^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$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}$
13.6.5.6$x^{6} + 416$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$