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Properties

Label 18.4.21481164688...8944.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,2^{12}\cdot 7^{9}\cdot 37^{9}$
Root discriminant $25.55$
Ramified primes $2, 7, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3:S_3:S_4$ (as 18T155)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65, 254, -197, -693, -901, 1551, 763, 323, -1125, -190, -32, 309, 21, -27, 3, -24, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 7*x^16 - 24*x^15 + 3*x^14 - 27*x^13 + 21*x^12 + 309*x^11 - 32*x^10 - 190*x^9 - 1125*x^8 + 323*x^7 + 763*x^6 + 1551*x^5 - 901*x^4 - 693*x^3 - 197*x^2 + 254*x + 65)
 
gp: K = bnfinit(x^18 - x^17 + 7*x^16 - 24*x^15 + 3*x^14 - 27*x^13 + 21*x^12 + 309*x^11 - 32*x^10 - 190*x^9 - 1125*x^8 + 323*x^7 + 763*x^6 + 1551*x^5 - 901*x^4 - 693*x^3 - 197*x^2 + 254*x + 65, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 7 x^{16} - 24 x^{15} + 3 x^{14} - 27 x^{13} + 21 x^{12} + 309 x^{11} - 32 x^{10} - 190 x^{9} - 1125 x^{8} + 323 x^{7} + 763 x^{6} + 1551 x^{5} - 901 x^{4} - 693 x^{3} - 197 x^{2} + 254 x + 65 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-21481164688288963759058944=-\,2^{12}\cdot 7^{9}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.55$
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:  $2, 7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} - \frac{3}{7} a^{12} - \frac{2}{7} a^{11} + \frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{15} - \frac{3}{7} a^{13} - \frac{2}{7} a^{12} + \frac{1}{7} a^{11} + \frac{3}{7} a^{10} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{539} a^{16} + \frac{38}{539} a^{15} - \frac{27}{539} a^{14} + \frac{6}{49} a^{13} - \frac{150}{539} a^{12} + \frac{250}{539} a^{11} + \frac{1}{49} a^{10} + \frac{116}{539} a^{9} + \frac{213}{539} a^{8} - \frac{10}{49} a^{7} - \frac{72}{539} a^{6} - \frac{120}{539} a^{5} + \frac{130}{539} a^{4} - \frac{109}{539} a^{3} - \frac{107}{539} a^{2} - \frac{244}{539} a - \frac{38}{539}$, $\frac{1}{361836476522620134445487} a^{17} - \frac{330885807867897806610}{361836476522620134445487} a^{16} - \frac{462889162380454265031}{7384417888216737437663} a^{15} + \frac{20663454062900874644368}{361836476522620134445487} a^{14} + \frac{2450694299199585275203}{361836476522620134445487} a^{13} - \frac{116075328874782125389556}{361836476522620134445487} a^{12} + \frac{6786530981855769987321}{51690925217517162063641} a^{11} + \frac{4395817073910503767279}{361836476522620134445487} a^{10} + \frac{104841198520806385944611}{361836476522620134445487} a^{9} - \frac{105706635045779054377971}{361836476522620134445487} a^{8} + \frac{170000438222312519591478}{361836476522620134445487} a^{7} + \frac{7943616498426173424967}{361836476522620134445487} a^{6} - \frac{56965926253740914250307}{361836476522620134445487} a^{5} + \frac{45011772064140027806370}{361836476522620134445487} a^{4} - \frac{4728173963440289532442}{32894225138420012222317} a^{3} + \frac{13345474556426319516749}{32894225138420012222317} a^{2} + \frac{121840488560427157985498}{361836476522620134445487} a - \frac{165679906810157470864811}{361836476522620134445487}$

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:  $10$
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:  \( 465630.315564 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3:S_4$ (as 18T155):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_3:S_3:S_4$
Character table for $C_3:S_3:S_4$

Intermediate fields

3.3.148.1, 6.4.277983664.1, 9.3.1111934656.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
37.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
37.12.9.1$x^{12} - 74 x^{8} - 20535 x^{4} - 810448$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$