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Properties

Label 18.6.17141791619...6477.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 1399^{3}\cdot 4523$
Root discriminant $19.53$
Ramified primes $7, 1399, 4523$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T879

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-71, 257, 1341, 1854, 1531, 230, -1362, -688, 40, 86, 201, -36, -42, 20, -7, 8, -1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - x^16 + 8*x^15 - 7*x^14 + 20*x^13 - 42*x^12 - 36*x^11 + 201*x^10 + 86*x^9 + 40*x^8 - 688*x^7 - 1362*x^6 + 230*x^5 + 1531*x^4 + 1854*x^3 + 1341*x^2 + 257*x - 71)
 
gp: K = bnfinit(x^18 - 3*x^17 - x^16 + 8*x^15 - 7*x^14 + 20*x^13 - 42*x^12 - 36*x^11 + 201*x^10 + 86*x^9 + 40*x^8 - 688*x^7 - 1362*x^6 + 230*x^5 + 1531*x^4 + 1854*x^3 + 1341*x^2 + 257*x - 71, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - x^{16} + 8 x^{15} - 7 x^{14} + 20 x^{13} - 42 x^{12} - 36 x^{11} + 201 x^{10} + 86 x^{9} + 40 x^{8} - 688 x^{7} - 1362 x^{6} + 230 x^{5} + 1531 x^{4} + 1854 x^{3} + 1341 x^{2} + 257 x - 71 \)

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:  \(171417916195550289266477=7^{12}\cdot 1399^{3}\cdot 4523\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.53$
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:  $7, 1399, 4523$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{29} a^{14} - \frac{7}{29} a^{13} - \frac{3}{29} a^{12} - \frac{11}{29} a^{11} - \frac{5}{29} a^{10} - \frac{7}{29} a^{9} + \frac{8}{29} a^{8} + \frac{12}{29} a^{7} - \frac{6}{29} a^{6} - \frac{1}{29} a^{5} + \frac{6}{29} a^{4} - \frac{10}{29} a^{3} - \frac{4}{29} a^{2} - \frac{9}{29} a - \frac{2}{29}$, $\frac{1}{29} a^{15} + \frac{6}{29} a^{13} - \frac{3}{29} a^{12} + \frac{5}{29} a^{11} - \frac{13}{29} a^{10} - \frac{12}{29} a^{9} + \frac{10}{29} a^{8} - \frac{9}{29} a^{7} - \frac{14}{29} a^{6} - \frac{1}{29} a^{5} + \frac{3}{29} a^{4} + \frac{13}{29} a^{3} - \frac{8}{29} a^{2} - \frac{7}{29} a - \frac{14}{29}$, $\frac{1}{116899} a^{16} + \frac{72}{116899} a^{15} + \frac{1448}{116899} a^{14} + \frac{9562}{116899} a^{13} - \frac{45775}{116899} a^{12} + \frac{291}{4031} a^{11} - \frac{54239}{116899} a^{10} + \frac{41571}{116899} a^{9} + \frac{20454}{116899} a^{8} + \frac{18585}{116899} a^{7} - \frac{25263}{116899} a^{6} + \frac{38828}{116899} a^{5} + \frac{6909}{116899} a^{4} + \frac{12869}{116899} a^{3} - \frac{9}{4031} a^{2} + \frac{44359}{116899} a - \frac{6096}{116899}$, $\frac{1}{48316314300122819} a^{17} + \frac{97856236759}{48316314300122819} a^{16} - \frac{256183366832050}{48316314300122819} a^{15} - \frac{619765645136319}{48316314300122819} a^{14} - \frac{22285820785060223}{48316314300122819} a^{13} - \frac{14708002420862645}{48316314300122819} a^{12} - \frac{13860377225822237}{48316314300122819} a^{11} - \frac{13572411404706579}{48316314300122819} a^{10} - \frac{19769473866755260}{48316314300122819} a^{9} - \frac{2933859353866101}{48316314300122819} a^{8} + \frac{14189617764085607}{48316314300122819} a^{7} + \frac{22297150256615975}{48316314300122819} a^{6} - \frac{22677946919635179}{48316314300122819} a^{5} - \frac{21486980065642474}{48316314300122819} a^{4} - \frac{8739400231784770}{48316314300122819} a^{3} + \frac{16113230974335227}{48316314300122819} a^{2} - \frac{3236387727244181}{48316314300122819} a + \frac{1595442943880151}{48316314300122819}$

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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 41082.8771552 \) (assuming GRH)
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

\(\Q(\zeta_{7})^+\), 9.3.164590951.1

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 $18$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
7Data not computed
1399Data not computed
4523Data not computed