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Properties

Label 18.6.64973982181...0193.4
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 97^{3}\cdot 22679^{2}$
Root discriminant $23.90$
Ramified primes $7, 97, 22679$
Class number $1$
Class group Trivial
Galois group 18T879

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 20, 129, 337, 654, 686, 712, 139, 14, -383, -176, -140, -54, 2, -40, -10, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 - 10*x^15 - 40*x^14 + 2*x^13 - 54*x^12 - 140*x^11 - 176*x^10 - 383*x^9 + 14*x^8 + 139*x^7 + 712*x^6 + 686*x^5 + 654*x^4 + 337*x^3 + 129*x^2 + 20*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^16 - 10*x^15 - 40*x^14 + 2*x^13 - 54*x^12 - 140*x^11 - 176*x^10 - 383*x^9 + 14*x^8 + 139*x^7 + 712*x^6 + 686*x^5 + 654*x^4 + 337*x^3 + 129*x^2 + 20*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{16} - 10 x^{15} - 40 x^{14} + 2 x^{13} - 54 x^{12} - 140 x^{11} - 176 x^{10} - 383 x^{9} + 14 x^{8} + 139 x^{7} + 712 x^{6} + 686 x^{5} + 654 x^{4} + 337 x^{3} + 129 x^{2} + 20 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6497398218116128597530193=7^{12}\cdot 97^{3}\cdot 22679^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.90$
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:  $7, 97, 22679$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{6206346052567542071681} a^{17} + \frac{108914085577433508065}{6206346052567542071681} a^{16} - \frac{2806142346335021617674}{6206346052567542071681} a^{15} + \frac{555682675644945032798}{6206346052567542071681} a^{14} - \frac{1411466219734437630744}{6206346052567542071681} a^{13} + \frac{438980868938035271799}{6206346052567542071681} a^{12} - \frac{2055988400898150582897}{6206346052567542071681} a^{11} + \frac{2137536424869657864649}{6206346052567542071681} a^{10} - \frac{195275163822981005319}{6206346052567542071681} a^{9} - \frac{2232033914078222387181}{6206346052567542071681} a^{8} - \frac{1902679107989532119173}{6206346052567542071681} a^{7} - \frac{2430863778055248358440}{6206346052567542071681} a^{6} + \frac{1745679743561890526580}{6206346052567542071681} a^{5} + \frac{1286324593534975895510}{6206346052567542071681} a^{4} + \frac{1727057606670451776045}{6206346052567542071681} a^{3} + \frac{1600164354809968020613}{6206346052567542071681} a^{2} - \frac{500565565617610978219}{6206346052567542071681} a - \frac{2536438953382158924055}{6206346052567542071681}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 190383.105273 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T879:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 331776
The 360 conjugacy class representatives for t18n879 are not computed
Character table for t18n879 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.7.2668161671.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

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
7Data not computed
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.6.0.1$x^{6} - x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
22679Data not computed