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Properties

Label 18.18.3924616230...0000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 5^{14}\cdot 13^{9}\cdot 23^{6}$
Root discriminant $56.91$
Ramified primes $2, 5, 13, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^3:C_2^2$ (as 18T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-289, 1785, 4358, -27711, 16235, 64265, -78445, -34860, 87225, -15335, -31392, 12350, 4444, -2693, -160, 244, -15, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - 15*x^16 + 244*x^15 - 160*x^14 - 2693*x^13 + 4444*x^12 + 12350*x^11 - 31392*x^10 - 15335*x^9 + 87225*x^8 - 34860*x^7 - 78445*x^6 + 64265*x^5 + 16235*x^4 - 27711*x^3 + 4358*x^2 + 1785*x - 289)
 
gp: K = bnfinit(x^18 - 8*x^17 - 15*x^16 + 244*x^15 - 160*x^14 - 2693*x^13 + 4444*x^12 + 12350*x^11 - 31392*x^10 - 15335*x^9 + 87225*x^8 - 34860*x^7 - 78445*x^6 + 64265*x^5 + 16235*x^4 - 27711*x^3 + 4358*x^2 + 1785*x - 289, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} - 15 x^{16} + 244 x^{15} - 160 x^{14} - 2693 x^{13} + 4444 x^{12} + 12350 x^{11} - 31392 x^{10} - 15335 x^{9} + 87225 x^{8} - 34860 x^{7} - 78445 x^{6} + 64265 x^{5} + 16235 x^{4} - 27711 x^{3} + 4358 x^{2} + 1785 x - 289 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(39246162302049939925000000000000=2^{12}\cdot 5^{14}\cdot 13^{9}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.91$
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:  $2, 5, 13, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{8} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{85} a^{15} + \frac{8}{85} a^{14} - \frac{2}{85} a^{13} + \frac{6}{85} a^{12} - \frac{4}{85} a^{11} + \frac{4}{85} a^{10} + \frac{4}{85} a^{9} - \frac{31}{85} a^{8} + \frac{3}{85} a^{7} - \frac{26}{85} a^{6} + \frac{19}{85} a^{5} + \frac{4}{17} a^{4} + \frac{3}{17} a^{3} - \frac{3}{17} a^{2} - \frac{27}{85} a - \frac{1}{5}$, $\frac{1}{5525} a^{16} - \frac{4}{5525} a^{15} + \frac{9}{221} a^{14} - \frac{28}{1105} a^{13} - \frac{18}{221} a^{12} - \frac{33}{5525} a^{11} + \frac{262}{5525} a^{10} + \frac{93}{1105} a^{9} + \frac{7}{1105} a^{8} + \frac{90}{221} a^{7} - \frac{67}{221} a^{6} - \frac{453}{1105} a^{5} - \frac{94}{221} a^{4} - \frac{69}{221} a^{3} + \frac{17}{65} a^{2} + \frac{1089}{5525} a - \frac{118}{325}$, $\frac{1}{65882846482069175} a^{17} + \frac{160354260069}{3875461557768775} a^{16} + \frac{97362189965682}{65882846482069175} a^{15} - \frac{404438381412062}{13176569296413835} a^{14} + \frac{387359802764579}{13176569296413835} a^{13} - \frac{721612471307278}{65882846482069175} a^{12} - \frac{2573963073082329}{65882846482069175} a^{11} + \frac{4195554288023439}{65882846482069175} a^{10} + \frac{5950905838088}{1013582253570295} a^{9} - \frac{922734331298275}{2635313859282767} a^{8} + \frac{113387962878}{2635313859282767} a^{7} + \frac{4110912067741399}{13176569296413835} a^{6} - \frac{843616680943290}{2635313859282767} a^{5} + \frac{416772002785903}{1013582253570295} a^{4} + \frac{6556304870353857}{13176569296413835} a^{3} + \frac{24433823766545534}{65882846482069175} a^{2} + \frac{60626387292262}{298112427520675} a - \frac{26973948258378}{227968326927575}$

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:  $17$
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:  \( 12399191314.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^3:C_2^2$ (as 18T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 15 conjugacy class representatives for $C_3^3:C_2^2$
Character table for $C_3^3:C_2^2$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.1300.1 x3, 6.6.464885200.2, 6.6.21970000.1

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

Sibling fields

Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$23$23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
23.9.6.1$x^{9} - 529 x^{3} + 48668$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$