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Properties

Label 18.6.39487629517...5289.2
Degree $18$
Signature $[6, 6]$
Discriminant $19^{16}\cdot 37^{2}$
Root discriminant $20.46$
Ramified primes $19, 37$
Class number $1$
Class group Trivial
Galois group 18T368

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 15, -64, 44, 154, -145, -322, 577, -305, 125, -167, 93, 12, -13, 4, -10, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 4*x^16 - 10*x^15 + 4*x^14 - 13*x^13 + 12*x^12 + 93*x^11 - 167*x^10 + 125*x^9 - 305*x^8 + 577*x^7 - 322*x^6 - 145*x^5 + 154*x^4 + 44*x^3 - 64*x^2 + 15*x - 1)
 
gp: K = bnfinit(x^18 - x^17 + 4*x^16 - 10*x^15 + 4*x^14 - 13*x^13 + 12*x^12 + 93*x^11 - 167*x^10 + 125*x^9 - 305*x^8 + 577*x^7 - 322*x^6 - 145*x^5 + 154*x^4 + 44*x^3 - 64*x^2 + 15*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 4 x^{16} - 10 x^{15} + 4 x^{14} - 13 x^{13} + 12 x^{12} + 93 x^{11} - 167 x^{10} + 125 x^{9} - 305 x^{8} + 577 x^{7} - 322 x^{6} - 145 x^{5} + 154 x^{4} + 44 x^{3} - 64 x^{2} + 15 x - 1 \)

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:  \(394876295174073378555289=19^{16}\cdot 37^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.46$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{229} a^{16} - \frac{103}{229} a^{15} - \frac{105}{229} a^{14} + \frac{36}{229} a^{13} + \frac{28}{229} a^{12} - \frac{60}{229} a^{11} - \frac{29}{229} a^{10} - \frac{104}{229} a^{9} - \frac{34}{229} a^{8} + \frac{109}{229} a^{7} + \frac{33}{229} a^{6} + \frac{61}{229} a^{5} - \frac{57}{229} a^{4} + \frac{41}{229} a^{3} - \frac{98}{229} a^{2} + \frac{78}{229} a - \frac{82}{229}$, $\frac{1}{37171192863439} a^{17} - \frac{59974799760}{37171192863439} a^{16} - \frac{15933409609840}{37171192863439} a^{15} - \frac{11137204067115}{37171192863439} a^{14} - \frac{4681692393282}{37171192863439} a^{13} + \frac{2236028493133}{37171192863439} a^{12} - \frac{14835591393638}{37171192863439} a^{11} - \frac{5983928999774}{37171192863439} a^{10} + \frac{1003774295895}{37171192863439} a^{9} - \frac{2646513457410}{37171192863439} a^{8} - \frac{7090982936443}{37171192863439} a^{7} + \frac{12348818989239}{37171192863439} a^{6} + \frac{12015128846045}{37171192863439} a^{5} + \frac{11492401760068}{37171192863439} a^{4} + \frac{13575724505675}{37171192863439} a^{3} - \frac{1866594111876}{37171192863439} a^{2} - \frac{16010370355898}{37171192863439} a - \frac{5508154908350}{37171192863439}$

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:  \( 59372.2035473 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T368:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for t18n368
Character table for t18n368 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$19$19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
37Data not computed