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Properties

Label 18.0.11639651445...5367.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{9}\cdot 19^{16}$
Root discriminant $36.24$
Ramified primes $7, 19$
Class number $148$ (GRH)
Class group $[2, 74]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![512, -1280, 10240, 0, 32480, -4496, 41176, -4536, 26418, -2825, 9881, -914, 2258, -179, 319, -19, 26, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 26*x^16 - 19*x^15 + 319*x^14 - 179*x^13 + 2258*x^12 - 914*x^11 + 9881*x^10 - 2825*x^9 + 26418*x^8 - 4536*x^7 + 41176*x^6 - 4496*x^5 + 32480*x^4 + 10240*x^2 - 1280*x + 512)
 
gp: K = bnfinit(x^18 - x^17 + 26*x^16 - 19*x^15 + 319*x^14 - 179*x^13 + 2258*x^12 - 914*x^11 + 9881*x^10 - 2825*x^9 + 26418*x^8 - 4536*x^7 + 41176*x^6 - 4496*x^5 + 32480*x^4 + 10240*x^2 - 1280*x + 512, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 26 x^{16} - 19 x^{15} + 319 x^{14} - 179 x^{13} + 2258 x^{12} - 914 x^{11} + 9881 x^{10} - 2825 x^{9} + 26418 x^{8} - 4536 x^{7} + 41176 x^{6} - 4496 x^{5} + 32480 x^{4} + 10240 x^{2} - 1280 x + 512 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-11639651445632252525480175367=-\,7^{9}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.24$
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:  $7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(133=7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{133}(64,·)$, $\chi_{133}(1,·)$, $\chi_{133}(6,·)$, $\chi_{133}(83,·)$, $\chi_{133}(20,·)$, $\chi_{133}(85,·)$, $\chi_{133}(92,·)$, $\chi_{133}(99,·)$, $\chi_{133}(36,·)$, $\chi_{133}(104,·)$, $\chi_{133}(106,·)$, $\chi_{133}(43,·)$, $\chi_{133}(111,·)$, $\chi_{133}(118,·)$, $\chi_{133}(55,·)$, $\chi_{133}(120,·)$, $\chi_{133}(125,·)$, $\chi_{133}(62,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{11} + \frac{1}{16} a^{10} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} + \frac{3}{8} a^{7} - \frac{3}{8} a^{6} - \frac{7}{16} a^{5} - \frac{1}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{13} - \frac{1}{16} a^{12} + \frac{1}{32} a^{11} - \frac{1}{32} a^{10} + \frac{1}{32} a^{9} + \frac{3}{16} a^{8} - \frac{3}{16} a^{7} + \frac{9}{32} a^{6} + \frac{15}{32} a^{5} + \frac{3}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{15} - \frac{1}{64} a^{14} - \frac{1}{32} a^{13} + \frac{1}{64} a^{12} - \frac{1}{64} a^{11} + \frac{1}{64} a^{10} - \frac{13}{32} a^{9} + \frac{13}{32} a^{8} - \frac{23}{64} a^{7} - \frac{17}{64} a^{6} - \frac{13}{32} a^{5} + \frac{7}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4736} a^{16} - \frac{5}{4736} a^{15} - \frac{21}{2368} a^{14} - \frac{3}{4736} a^{13} - \frac{117}{4736} a^{12} + \frac{553}{4736} a^{11} - \frac{453}{2368} a^{10} + \frac{27}{64} a^{9} - \frac{671}{4736} a^{8} - \frac{733}{4736} a^{7} - \frac{793}{2368} a^{6} - \frac{73}{592} a^{5} + \frac{167}{592} a^{4} - \frac{67}{296} a^{3} + \frac{7}{37} a^{2} + \frac{6}{37} a - \frac{5}{37}$, $\frac{1}{3286709792573598033802496} a^{17} - \frac{215770965449050865501}{3286709792573598033802496} a^{16} - \frac{2445858885533688158913}{1643354896286799016901248} a^{15} - \frac{23067760764204942904747}{3286709792573598033802496} a^{14} + \frac{491020196381887664779}{29085927367907947201792} a^{13} + \frac{42309349823304998863041}{3286709792573598033802496} a^{12} + \frac{23146089489841244826995}{1643354896286799016901248} a^{11} + \frac{72851807720290550079211}{1643354896286799016901248} a^{10} + \frac{1517413864507216216671401}{3286709792573598033802496} a^{9} - \frac{859923161310066372971813}{3286709792573598033802496} a^{8} + \frac{499984370557850566690115}{1643354896286799016901248} a^{7} + \frac{83675598202111999661129}{205419362035849877112656} a^{6} + \frac{12265361935070684567021}{25677420254481234639082} a^{5} + \frac{2336123684059370245461}{12838710127240617319541} a^{4} - \frac{15235432006187974529779}{51354840508962469278164} a^{3} - \frac{16876346955721589598461}{51354840508962469278164} a^{2} - \frac{8039987531013250719967}{25677420254481234639082} a - \frac{1897745210837244605108}{12838710127240617319541}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{74}$, which has order $148$ (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:  $8$
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:  \( 22305.8950792 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.3.361.1, 6.0.44700103.1, \(\Q(\zeta_{19})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ $18$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ $18$ $18$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
$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}$
19Data not computed