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Properties

Label 18.10.1353065020...5424.2
Degree $18$
Signature $[10, 4]$
Discriminant $2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{9}$
Root discriminant $53.64$
Ramified primes $2, 3, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53248, -89088, 25856, -116032, 238688, 5264, -240584, 89376, 85024, -50728, -12466, 12484, 155, -1648, 189, 120, -24, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 24*x^16 + 120*x^15 + 189*x^14 - 1648*x^13 + 155*x^12 + 12484*x^11 - 12466*x^10 - 50728*x^9 + 85024*x^8 + 89376*x^7 - 240584*x^6 + 5264*x^5 + 238688*x^4 - 116032*x^3 + 25856*x^2 - 89088*x + 53248)
 
gp: K = bnfinit(x^18 - 4*x^17 - 24*x^16 + 120*x^15 + 189*x^14 - 1648*x^13 + 155*x^12 + 12484*x^11 - 12466*x^10 - 50728*x^9 + 85024*x^8 + 89376*x^7 - 240584*x^6 + 5264*x^5 + 238688*x^4 - 116032*x^3 + 25856*x^2 - 89088*x + 53248, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 24 x^{16} + 120 x^{15} + 189 x^{14} - 1648 x^{13} + 155 x^{12} + 12484 x^{11} - 12466 x^{10} - 50728 x^{9} + 85024 x^{8} + 89376 x^{7} - 240584 x^{6} + 5264 x^{5} + 238688 x^{4} - 116032 x^{3} + 25856 x^{2} - 89088 x + 53248 \)

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:  \(13530650209602087833780282855424=2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.64$
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:  $2, 3, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{56} a^{12} - \frac{3}{28} a^{11} - \frac{1}{14} a^{9} - \frac{11}{56} a^{8} - \frac{13}{28} a^{7} - \frac{3}{8} a^{6} + \frac{9}{28} a^{5} - \frac{9}{28} a^{4} + \frac{2}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{112} a^{13} - \frac{1}{14} a^{11} - \frac{1}{28} a^{10} + \frac{3}{16} a^{9} + \frac{5}{28} a^{8} - \frac{37}{112} a^{7} + \frac{1}{28} a^{6} + \frac{3}{56} a^{5} - \frac{9}{28} a^{4} - \frac{1}{14} a^{3} - \frac{5}{14} a^{2} + \frac{3}{14} a - \frac{3}{7}$, $\frac{1}{448} a^{14} - \frac{1}{112} a^{12} - \frac{1}{8} a^{11} - \frac{5}{64} a^{10} + \frac{1}{112} a^{9} - \frac{81}{448} a^{8} - \frac{1}{28} a^{7} + \frac{101}{224} a^{6} + \frac{1}{7} a^{5} + \frac{11}{56} a^{4} - \frac{1}{14} a^{3} + \frac{1}{8} a^{2} + \frac{3}{28} a + \frac{3}{7}$, $\frac{1}{1792} a^{15} - \frac{1}{896} a^{14} + \frac{1}{448} a^{13} + \frac{61}{1792} a^{11} - \frac{13}{128} a^{10} + \frac{335}{1792} a^{9} + \frac{113}{896} a^{8} - \frac{263}{896} a^{7} - \frac{23}{64} a^{6} - \frac{101}{224} a^{5} - \frac{15}{112} a^{4} - \frac{37}{224} a^{3} - \frac{1}{4} a^{2} - \frac{25}{56} a + \frac{5}{14}$, $\frac{1}{3584} a^{16} + \frac{1}{448} a^{13} - \frac{3}{3584} a^{12} - \frac{31}{896} a^{11} - \frac{29}{3584} a^{10} + \frac{1}{14} a^{9} + \frac{45}{256} a^{8} + \frac{41}{112} a^{7} - \frac{47}{224} a^{6} + \frac{1}{28} a^{5} - \frac{177}{448} a^{4} - \frac{17}{224} a^{3} - \frac{13}{112} a^{2} + \frac{17}{56} a - \frac{1}{2}$, $\frac{1}{172567525615326507008} a^{17} + \frac{323090837916409}{2696367587739476672} a^{16} - \frac{53414534141145}{21570940701915813376} a^{15} + \frac{2581037247393393}{3081562957416544768} a^{14} - \frac{391815511507654819}{172567525615326507008} a^{13} + \frac{49303509150081959}{6163125914833089536} a^{12} - \frac{14317786731277153349}{172567525615326507008} a^{11} - \frac{194960552879912551}{10785470350957906688} a^{10} - \frac{18329985683274557857}{86283762807663253504} a^{9} - \frac{1652859083371577905}{10785470350957906688} a^{8} + \frac{186117640767385771}{674091896934869168} a^{7} - \frac{15296267257903879}{770390739354136192} a^{6} - \frac{8627541963054712441}{21570940701915813376} a^{5} + \frac{4695720289721894471}{10785470350957906688} a^{4} - \frac{660691149249075187}{5392735175478953344} a^{3} - \frac{634346665298641579}{2696367587739476672} a^{2} - \frac{79048800497066009}{168522974233717292} a - \frac{9643779680212687}{168522974233717292}$

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:  \( 2117395204.62 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.574470067776192.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$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}$
41Data not computed