↑

Properties

Label 18.0.21012633925...1099.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 19^{15}$
Root discriminant $42.56$
Ramified primes $7, 19$
Class number $27$ (GRH)
Class group $[3, 3, 3]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3241, -2534, 22127, -23297, 37362, -55511, 28219, 5713, -6056, 741, -380, -206, 296, -141, 91, -5, -5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 5*x^16 - 5*x^15 + 91*x^14 - 141*x^13 + 296*x^12 - 206*x^11 - 380*x^10 + 741*x^9 - 6056*x^8 + 5713*x^7 + 28219*x^6 - 55511*x^5 + 37362*x^4 - 23297*x^3 + 22127*x^2 - 2534*x + 3241)
 
gp: K = bnfinit(x^18 - 3*x^17 - 5*x^16 - 5*x^15 + 91*x^14 - 141*x^13 + 296*x^12 - 206*x^11 - 380*x^10 + 741*x^9 - 6056*x^8 + 5713*x^7 + 28219*x^6 - 55511*x^5 + 37362*x^4 - 23297*x^3 + 22127*x^2 - 2534*x + 3241, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 5 x^{16} - 5 x^{15} + 91 x^{14} - 141 x^{13} + 296 x^{12} - 206 x^{11} - 380 x^{10} + 741 x^{9} - 6056 x^{8} + 5713 x^{7} + 28219 x^{6} - 55511 x^{5} + 37362 x^{4} - 23297 x^{3} + 22127 x^{2} - 2534 x + 3241 \)

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:  \(-210126339255361190328405271099=-\,7^{12}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.56$
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$$\chi_{133}(64,·)$, $\chi_{133}(1,·)$, $\chi_{133}(65,·)$, $\chi_{133}(8,·)$, $\chi_{133}(11,·)$, $\chi_{133}(18,·)$, $\chi_{133}(88,·)$, $\chi_{133}(30,·)$, $\chi_{133}(37,·)$, $\chi_{133}(102,·)$, $\chi_{133}(39,·)$, $\chi_{133}(106,·)$, $\chi_{133}(107,·)$, $\chi_{133}(46,·)$, $\chi_{133}(113,·)$, $\chi_{133}(50,·)$, $\chi_{133}(121,·)$, $\chi_{133}(58,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{385} a^{15} + \frac{1}{5} a^{14} + \frac{174}{385} a^{13} + \frac{17}{55} a^{12} + \frac{24}{385} a^{11} - \frac{18}{55} a^{10} + \frac{18}{55} a^{9} - \frac{8}{55} a^{8} - \frac{72}{385} a^{7} + \frac{4}{55} a^{6} + \frac{8}{77} a^{5} - \frac{1}{5} a^{4} + \frac{57}{385} a^{3} - \frac{27}{55} a^{2} + \frac{4}{55} a + \frac{17}{55}$, $\frac{1}{44065945} a^{16} + \frac{45114}{44065945} a^{15} - \frac{14571152}{44065945} a^{14} + \frac{9291857}{44065945} a^{13} + \frac{3148372}{44065945} a^{12} + \frac{4861847}{44065945} a^{11} - \frac{2732183}{6295135} a^{10} + \frac{546218}{6295135} a^{9} - \frac{3259804}{44065945} a^{8} - \frac{10666206}{44065945} a^{7} - \frac{12801049}{44065945} a^{6} + \frac{9780143}{44065945} a^{5} + \frac{12777668}{44065945} a^{4} + \frac{365082}{8813189} a^{3} + \frac{22263}{114457} a^{2} - \frac{556691}{1259027} a + \frac{1749854}{6295135}$, $\frac{1}{622519484474364578450802683138755} a^{17} + \frac{2256362007238795144676046}{622519484474364578450802683138755} a^{16} - \frac{599091542600716352369796579061}{622519484474364578450802683138755} a^{15} + \frac{77582138178726874375107517826339}{622519484474364578450802683138755} a^{14} - \frac{3154223620977532465777589592078}{12704479274987032213281687410995} a^{13} - \frac{45235527786438652124394589704857}{622519484474364578450802683138755} a^{12} - \frac{51890253222453306148546186606633}{124503896894872915690160536627751} a^{11} + \frac{10943059921483869493751584551213}{88931354924909225492971811876965} a^{10} + \frac{265441920153498035943371237080861}{622519484474364578450802683138755} a^{9} + \frac{165686476329819006992158115584933}{622519484474364578450802683138755} a^{8} - \frac{95065768087596188406644444437387}{622519484474364578450802683138755} a^{7} + \frac{43074098076226784068286018772959}{622519484474364578450802683138755} a^{6} + \frac{224843403835557556028357003484049}{622519484474364578450802683138755} a^{5} + \frac{35711453860701122518579957298895}{124503896894872915690160536627751} a^{4} - \frac{157407058426917529781675025541434}{622519484474364578450802683138755} a^{3} - \frac{31554002124590335212435271147371}{88931354924909225492971811876965} a^{2} - \frac{809923873020852902345220245234}{8084668629537202317542891988815} a - \frac{35726852831103139375661574021596}{88931354924909225492971811876965}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (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:  \( 833965.243856 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-19}) \), 3.3.361.1, 3.3.17689.2, \(\Q(\zeta_{7})^+\), 3.3.17689.1, 6.0.2476099.1, 6.0.5945113699.2, 6.0.16468459.1, 6.0.5945113699.1, 9.9.5534900853769.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
19Data not computed