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Properties

Label 18.0.54949782158...7808.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 7^{15}\cdot 41^{4}$
Root discriminant $18.34$
Ramified primes $2, 7, 41$
Class number $3$
Class group $[3]$
Galois group 18T201

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, 51, -153, 379, -576, 928, -1044, 1036, -786, 672, -492, 310, -168, 89, -34, 13, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 13*x^16 - 34*x^15 + 89*x^14 - 168*x^13 + 310*x^12 - 492*x^11 + 672*x^10 - 786*x^9 + 1036*x^8 - 1044*x^7 + 928*x^6 - 576*x^5 + 379*x^4 - 153*x^3 + 51*x^2 + 8*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 13*x^16 - 34*x^15 + 89*x^14 - 168*x^13 + 310*x^12 - 492*x^11 + 672*x^10 - 786*x^9 + 1036*x^8 - 1044*x^7 + 928*x^6 - 576*x^5 + 379*x^4 - 153*x^3 + 51*x^2 + 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 13 x^{16} - 34 x^{15} + 89 x^{14} - 168 x^{13} + 310 x^{12} - 492 x^{11} + 672 x^{10} - 786 x^{9} + 1036 x^{8} - 1044 x^{7} + 928 x^{6} - 576 x^{5} + 379 x^{4} - 153 x^{3} + 51 x^{2} + 8 x + 1 \)

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:  \(-54949782158942378487808=-\,2^{12}\cdot 7^{15}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.34$
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:  $2, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{2}$, $\frac{1}{3962545397341179428} a^{17} + \frac{224676366319577569}{990636349335294857} a^{16} + \frac{325386661386596625}{3962545397341179428} a^{15} - \frac{346171231981278129}{3962545397341179428} a^{14} - \frac{18371707875569934}{76202796102714989} a^{13} + \frac{430461141220438910}{990636349335294857} a^{12} + \frac{7730766444902509}{1981272698670589714} a^{11} - \frac{51786104146954903}{1981272698670589714} a^{10} - \frac{87562608741605691}{1981272698670589714} a^{9} - \frac{24954313206507621}{990636349335294857} a^{8} - \frac{145277704603750426}{990636349335294857} a^{7} + \frac{355861165161204243}{990636349335294857} a^{6} - \frac{386477765995737222}{990636349335294857} a^{5} - \frac{426503461277405167}{990636349335294857} a^{4} + \frac{1949296842234785835}{3962545397341179428} a^{3} - \frac{293452408278376783}{990636349335294857} a^{2} - \frac{1172189343572948049}{3962545397341179428} a + \frac{821975386547063819}{3962545397341179428}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( \frac{112075391173469115}{3962545397341179428} a^{17} - \frac{21354109484247591}{990636349335294857} a^{16} + \frac{780472308007355555}{3962545397341179428} a^{15} - \frac{797850338283581317}{3962545397341179428} a^{14} + \frac{92771434848219847}{152405592205429978} a^{13} + \frac{414741315494043175}{1981272698670589714} a^{12} - \frac{203916519172319166}{990636349335294857} a^{11} + \frac{2500480673523997270}{990636349335294857} a^{10} - \frac{6552161574179585486}{990636349335294857} a^{9} + \frac{22448979892988357913}{1981272698670589714} a^{8} - \frac{17418338800083091141}{1981272698670589714} a^{7} + \frac{45449647787107599899}{1981272698670589714} a^{6} - \frac{44260134734514755099}{1981272698670589714} a^{5} + \frac{46718658448653355569}{1981272698670589714} a^{4} - \frac{52777163693311870165}{3962545397341179428} a^{3} + \frac{21630654539792214629}{1981272698670589714} a^{2} - \frac{12542917482723644961}{3962545397341179428} a + \frac{2884140022879764165}{3962545397341179428} \) (order $14$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 23717.7684925 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T201:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 26 conjugacy class representatives for t18n201
Character table for t18n201 is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 9.5.12657150016.1

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 $18$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ $18$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
7Data not computed
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.6.0.1$x^{6} - x + 7$$1$$6$$0$$C_6$$[\ ]^{6}$
41.6.4.1$x^{6} + 1435 x^{3} + 2904768$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$