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Properties

Label 18.6.86055014859...0000.2
Degree $18$
Signature $[6, 6]$
Discriminant $2^{24}\cdot 3^{32}\cdot 5^{6}\cdot 11^{6}$
Root discriminant $67.56$
Ramified primes $2, 3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T268

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![219707, 1622934, 4804836, 7835400, 8092887, 5703156, 2818683, 947232, 174777, -12420, -23556, -11448, -4416, -1548, -417, -54, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 - 54*x^15 - 417*x^14 - 1548*x^13 - 4416*x^12 - 11448*x^11 - 23556*x^10 - 12420*x^9 + 174777*x^8 + 947232*x^7 + 2818683*x^6 + 5703156*x^5 + 8092887*x^4 + 7835400*x^3 + 4804836*x^2 + 1622934*x + 219707)
 
gp: K = bnfinit(x^18 - 3*x^16 - 54*x^15 - 417*x^14 - 1548*x^13 - 4416*x^12 - 11448*x^11 - 23556*x^10 - 12420*x^9 + 174777*x^8 + 947232*x^7 + 2818683*x^6 + 5703156*x^5 + 8092887*x^4 + 7835400*x^3 + 4804836*x^2 + 1622934*x + 219707, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{16} - 54 x^{15} - 417 x^{14} - 1548 x^{13} - 4416 x^{12} - 11448 x^{11} - 23556 x^{10} - 12420 x^{9} + 174777 x^{8} + 947232 x^{7} + 2818683 x^{6} + 5703156 x^{5} + 8092887 x^{4} + 7835400 x^{3} + 4804836 x^{2} + 1622934 x + 219707 \)

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:  \(860550148596054437082169344000000=2^{24}\cdot 3^{32}\cdot 5^{6}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.56$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5893577452888113028414123565806476758248907851176593} a^{17} + \frac{961803832320029899348491197661870765652359577775018}{5893577452888113028414123565806476758248907851176593} a^{16} - \frac{2576808064873133932076713279324641693384174405912446}{5893577452888113028414123565806476758248907851176593} a^{15} - \frac{2218622417153596519917807375682869981628079924521902}{5893577452888113028414123565806476758248907851176593} a^{14} - \frac{1602760900913695829694340864081060789042946186045081}{5893577452888113028414123565806476758248907851176593} a^{13} + \frac{2824713149809946254631457398110703352499185939207898}{5893577452888113028414123565806476758248907851176593} a^{12} - \frac{2753006571493572248923163171311887250415399150092749}{5893577452888113028414123565806476758248907851176593} a^{11} - \frac{1175971226219509850113368768606821612405390643634233}{5893577452888113028414123565806476758248907851176593} a^{10} - \frac{82168559743751376508544323881282994570813910695924}{5893577452888113028414123565806476758248907851176593} a^{9} + \frac{2682554227226032140231634672726536648409822274670450}{5893577452888113028414123565806476758248907851176593} a^{8} + \frac{2257291493790740555069716414903737665807639081060336}{5893577452888113028414123565806476758248907851176593} a^{7} - \frac{820614663702730431372326786028718122201562899227070}{5893577452888113028414123565806476758248907851176593} a^{6} + \frac{737478832323974807242343263792320049673175026213657}{5893577452888113028414123565806476758248907851176593} a^{5} - \frac{24288515640086538858295517272085376798725959512883}{5893577452888113028414123565806476758248907851176593} a^{4} + \frac{53380189528867269532675828423739617628434527864892}{5893577452888113028414123565806476758248907851176593} a^{3} - \frac{2453036225764730743701504267829844834624142075615252}{5893577452888113028414123565806476758248907851176593} a^{2} + \frac{2633273394702899545602795321865358676289039946079130}{5893577452888113028414123565806476758248907851176593} a - \frac{1734063536349166345332095202109477017427839844088208}{5893577452888113028414123565806476758248907851176593}$

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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 7546900032.69 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T268:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 24 conjugacy class representatives for t18n268
Character table for t18n268 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.1620.1, 9.9.344373768000.1

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

Sibling fields

Degree 12 siblings: data not computed
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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$3$3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$