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Properties

Label 18.10.1461042292...5693.1
Degree $18$
Signature $[10, 4]$
Discriminant $19^{16}\cdot 37^{3}$
Root discriminant $25.01$
Ramified primes $19, 37$
Class number $1$
Class group Trivial
Galois group 18T264

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -45, 217, -128, -871, 2007, -1802, 481, 924, -2010, 2256, -1401, 261, 287, -248, 73, 2, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 2*x^16 + 73*x^15 - 248*x^14 + 287*x^13 + 261*x^12 - 1401*x^11 + 2256*x^10 - 2010*x^9 + 924*x^8 + 481*x^7 - 1802*x^6 + 2007*x^5 - 871*x^4 - 128*x^3 + 217*x^2 - 45*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 2*x^16 + 73*x^15 - 248*x^14 + 287*x^13 + 261*x^12 - 1401*x^11 + 2256*x^10 - 2010*x^9 + 924*x^8 + 481*x^7 - 1802*x^6 + 2007*x^5 - 871*x^4 - 128*x^3 + 217*x^2 - 45*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 2 x^{16} + 73 x^{15} - 248 x^{14} + 287 x^{13} + 261 x^{12} - 1401 x^{11} + 2256 x^{10} - 2010 x^{9} + 924 x^{8} + 481 x^{7} - 1802 x^{6} + 2007 x^{5} - 871 x^{4} - 128 x^{3} + 217 x^{2} - 45 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14610422921440715006545693=19^{16}\cdot 37^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.01$
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:  $19, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{3}{11} a^{14} + \frac{5}{11} a^{13} - \frac{5}{11} a^{12} + \frac{1}{11} a^{11} - \frac{3}{11} a^{9} - \frac{4}{11} a^{8} - \frac{3}{11} a^{7} - \frac{4}{11} a^{6} + \frac{4}{11} a^{5} + \frac{3}{11} a^{3} - \frac{5}{11} a^{2} - \frac{3}{11} a - \frac{5}{11}$, $\frac{1}{11} a^{16} - \frac{4}{11} a^{14} - \frac{1}{11} a^{13} - \frac{3}{11} a^{12} + \frac{3}{11} a^{11} - \frac{3}{11} a^{10} - \frac{2}{11} a^{9} - \frac{4}{11} a^{8} - \frac{2}{11} a^{7} + \frac{3}{11} a^{6} + \frac{1}{11} a^{5} + \frac{3}{11} a^{4} + \frac{4}{11} a^{3} + \frac{4}{11} a^{2} - \frac{3}{11} a - \frac{4}{11}$, $\frac{1}{5021066049241} a^{17} - \frac{218767000845}{5021066049241} a^{16} + \frac{56719975667}{5021066049241} a^{15} + \frac{1142735617891}{5021066049241} a^{14} - \frac{48386134636}{5021066049241} a^{13} + \frac{160550834821}{5021066049241} a^{12} - \frac{1502281963686}{5021066049241} a^{11} - \frac{145977254731}{456460549931} a^{10} - \frac{1142779583680}{5021066049241} a^{9} + \frac{1472080905880}{5021066049241} a^{8} - \frac{1585507057928}{5021066049241} a^{7} + \frac{1948540086827}{5021066049241} a^{6} - \frac{746064601105}{5021066049241} a^{5} - \frac{564829820377}{5021066049241} a^{4} + \frac{1391492556260}{5021066049241} a^{3} + \frac{1185979808779}{5021066049241} a^{2} - \frac{1745282294966}{5021066049241} a + \frac{2356185278059}{5021066049241}$

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:  $13$
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:  \( 731326.099824 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T264:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 32 conjugacy class representatives for t18n264
Character table for t18n264 is not computed

Intermediate fields

3.3.361.1, \(\Q(\zeta_{19})^+\)

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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ $18$ $18$ R $18$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
19Data not computed
37Data not computed