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Properties

Label 18.4.50056617315...7911.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,7^{12}\cdot 71\cdot 83^{4}\cdot 181^{4}$
Root discriminant $39.30$
Ramified primes $7, 71, 83, 181$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -24, 51, -55, 282, -314, -402, -86, 54, 181, 262, 272, 214, 37, 22, 15, -3, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 - 3*x^16 + 15*x^15 + 22*x^14 + 37*x^13 + 214*x^12 + 272*x^11 + 262*x^10 + 181*x^9 + 54*x^8 - 86*x^7 - 402*x^6 - 314*x^5 + 282*x^4 - 55*x^3 + 51*x^2 - 24*x + 1)
 
gp: K = bnfinit(x^18 - 5*x^17 - 3*x^16 + 15*x^15 + 22*x^14 + 37*x^13 + 214*x^12 + 272*x^11 + 262*x^10 + 181*x^9 + 54*x^8 - 86*x^7 - 402*x^6 - 314*x^5 + 282*x^4 - 55*x^3 + 51*x^2 - 24*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} - 3 x^{16} + 15 x^{15} + 22 x^{14} + 37 x^{13} + 214 x^{12} + 272 x^{11} + 262 x^{10} + 181 x^{9} + 54 x^{8} - 86 x^{7} - 402 x^{6} - 314 x^{5} + 282 x^{4} - 55 x^{3} + 51 x^{2} - 24 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-50056617315395675475570087911=-\,7^{12}\cdot 71\cdot 83^{4}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.30$
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:  $7, 71, 83, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{14} - \frac{2}{7} a^{13} - \frac{1}{7} a^{12} - \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{2}{7} a^{9} + \frac{1}{7} a^{8} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{16} - \frac{3}{7} a^{14} + \frac{1}{7} a^{13} + \frac{3}{7} a^{10} - \frac{1}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{28258046257438707741451} a^{17} - \frac{1358471837109209499762}{28258046257438707741451} a^{16} - \frac{165959136885984440234}{4036863751062672534493} a^{15} - \frac{1638873300500310831990}{4036863751062672534493} a^{14} - \frac{538038775963616825332}{4036863751062672534493} a^{13} - \frac{4741435163397098678193}{28258046257438707741451} a^{12} - \frac{654132978309365065982}{4036863751062672534493} a^{11} - \frac{1386017203710803682255}{4036863751062672534493} a^{10} + \frac{13827089240334085848486}{28258046257438707741451} a^{9} - \frac{6457989897233252596765}{28258046257438707741451} a^{8} + \frac{607296970618449107992}{4036863751062672534493} a^{7} - \frac{1758156523494976448005}{4036863751062672534493} a^{6} + \frac{5113632728353608982158}{28258046257438707741451} a^{5} + \frac{1691847084942600412748}{4036863751062672534493} a^{4} + \frac{232673492534296556126}{4036863751062672534493} a^{3} - \frac{666612315409395267459}{4036863751062672534493} a^{2} + \frac{5900944679341553165668}{28258046257438707741451} a - \frac{2252856313321745598870}{28258046257438707741451}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $10$
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:  \( 10374562.0209 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 192 conjugacy class representatives for t18n839 are not computed
Character table for t18n839 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.26552265046321.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R $18$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $18$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ $18$

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
$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$$[\ ]_{3}^{2}$
71Data not computed
83Data not computed
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$