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Properties

Label 18.6.34027126022...7133.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 29077^{3}$
Root discriminant $20.29$
Ramified primes $7, 29077$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T362

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, -2, -6, 29, -45, 81, 105, -101, -11, -42, 10, -16, 27, -8, 0, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 3*x^16 - 8*x^14 + 27*x^13 - 16*x^12 + 10*x^11 - 42*x^10 - 11*x^9 - 101*x^8 + 105*x^7 + 81*x^6 - 45*x^5 + 29*x^4 - 6*x^3 - 2*x^2 + 4*x - 1)
 
gp: K = bnfinit(x^18 - x^17 + 3*x^16 - 8*x^14 + 27*x^13 - 16*x^12 + 10*x^11 - 42*x^10 - 11*x^9 - 101*x^8 + 105*x^7 + 81*x^6 - 45*x^5 + 29*x^4 - 6*x^3 - 2*x^2 + 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 3 x^{16} - 8 x^{14} + 27 x^{13} - 16 x^{12} + 10 x^{11} - 42 x^{10} - 11 x^{9} - 101 x^{8} + 105 x^{7} + 81 x^{6} - 45 x^{5} + 29 x^{4} - 6 x^{3} - 2 x^{2} + 4 x - 1 \)

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:  \(340271260224306722157133=7^{12}\cdot 29077^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.29$
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:  $7, 29077$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{6} a^{15} + \frac{1}{6} a^{12} - \frac{1}{6} a^{11} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} + \frac{1}{6} a^{6} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{161848573309602} a^{17} + \frac{713165687715}{17983174812178} a^{16} - \frac{8757506487}{17983174812178} a^{15} + \frac{454072945037}{17983174812178} a^{14} - \frac{2143330218779}{161848573309602} a^{13} - \frac{13116902674561}{80924286654801} a^{12} + \frac{19595100667409}{53949524436534} a^{11} - \frac{36740711124631}{80924286654801} a^{10} - \frac{25394003597723}{161848573309602} a^{9} + \frac{30895074902188}{80924286654801} a^{8} + \frac{7205541131854}{26974762218267} a^{7} + \frac{911834398849}{17983174812178} a^{6} - \frac{6631462012873}{17983174812178} a^{5} + \frac{5224387881182}{26974762218267} a^{4} + \frac{6066453971149}{80924286654801} a^{3} + \frac{9680202790741}{80924286654801} a^{2} + \frac{23642509864127}{53949524436534} a + \frac{8149951005455}{80924286654801}$

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:  \( 71433.2998726 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T362:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2160
The 33 conjugacy class representatives for t18n362
Character table for t18n362 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.2.29077.1

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

Sibling fields

Degree 18 sibling: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ $15{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
29077Data not computed