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Properties

Label 18.0.72651683078...2688.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{44}\cdot 23^{9}$
Root discriminant $111.65$
Ramified primes $2, 3, 23$
Class number $36$ (GRH)
Class group $[36]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16877600, -33133680, 43262136, -42759972, 36128628, -23140440, 13325322, -6738858, 3159846, -1269783, 480933, -153774, 47256, -11700, 2844, -504, 90, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 90*x^16 - 504*x^15 + 2844*x^14 - 11700*x^13 + 47256*x^12 - 153774*x^11 + 480933*x^10 - 1269783*x^9 + 3159846*x^8 - 6738858*x^7 + 13325322*x^6 - 23140440*x^5 + 36128628*x^4 - 42759972*x^3 + 43262136*x^2 - 33133680*x + 16877600)
 
gp: K = bnfinit(x^18 - 9*x^17 + 90*x^16 - 504*x^15 + 2844*x^14 - 11700*x^13 + 47256*x^12 - 153774*x^11 + 480933*x^10 - 1269783*x^9 + 3159846*x^8 - 6738858*x^7 + 13325322*x^6 - 23140440*x^5 + 36128628*x^4 - 42759972*x^3 + 43262136*x^2 - 33133680*x + 16877600, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 90 x^{16} - 504 x^{15} + 2844 x^{14} - 11700 x^{13} + 47256 x^{12} - 153774 x^{11} + 480933 x^{10} - 1269783 x^{9} + 3159846 x^{8} - 6738858 x^{7} + 13325322 x^{6} - 23140440 x^{5} + 36128628 x^{4} - 42759972 x^{3} + 43262136 x^{2} - 33133680 x + 16877600 \)

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:  \(-7265168307811659818615258509987442688=-\,2^{12}\cdot 3^{44}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $111.65$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{8} + \frac{1}{6} a^{6} - \frac{1}{6} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} + \frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{1}{4} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{4} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{13} - \frac{1}{24} a^{12} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{1}{24} a^{9} + \frac{1}{8} a^{8} + \frac{1}{24} a^{7} + \frac{1}{6} a^{6} + \frac{5}{12} a^{5} + \frac{5}{12} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{15} + \frac{1}{6} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{816} a^{16} - \frac{1}{102} a^{15} + \frac{7}{408} a^{14} - \frac{7}{408} a^{13} - \frac{13}{408} a^{12} - \frac{5}{136} a^{11} + \frac{1}{24} a^{10} + \frac{19}{204} a^{9} + \frac{113}{816} a^{8} - \frac{55}{408} a^{7} - \frac{5}{204} a^{6} + \frac{21}{136} a^{5} + \frac{19}{51} a^{4} + \frac{25}{51} a^{3} - \frac{13}{204} a^{2} + \frac{35}{102} a - \frac{1}{3}$, $\frac{1}{27326880793465094285098618900431546226770011372400} a^{17} - \frac{7276452651058209638721669155658748842940325869}{27326880793465094285098618900431546226770011372400} a^{16} - \frac{1175713206033504266933594413155773893802936948}{113862003306104559521244245418464775944875047385} a^{15} + \frac{14924748312252011228813957860250476569722956983}{2277240066122091190424884908369295518897500947700} a^{14} + \frac{6316670401607560857574046559563037706590264674}{1707930049591568392818663681276971639173125710775} a^{13} + \frac{18190213999609688981147385267023654677990015221}{455448013224418238084976981673859103779500189540} a^{12} - \frac{341963860468735672136722398048447009730875293311}{6831720198366273571274654725107886556692502843100} a^{11} - \frac{1061981438278164961997992609026541276386016520167}{13663440396732547142549309450215773113385005686200} a^{10} + \frac{4280502665469330598873703502552304855365320948473}{27326880793465094285098618900431546226770011372400} a^{9} - \frac{6409285763632093280021184514499424069251088312463}{27326880793465094285098618900431546226770011372400} a^{8} - \frac{175234428893545787971987894983260701208140515327}{2277240066122091190424884908369295518897500947700} a^{7} - \frac{1708930221783085515613687300843275186011398854359}{13663440396732547142549309450215773113385005686200} a^{6} - \frac{2255852675839186653974739033484905916292484460233}{4554480132244182380849769816738591037795001895400} a^{5} - \frac{67267975432454655375386174746884592416003891611}{455448013224418238084976981673859103779500189540} a^{4} - \frac{3344354998337815393052165396695816910256407986993}{6831720198366273571274654725107886556692502843100} a^{3} + \frac{2302396204699410052993232539414449114715362945787}{6831720198366273571274654725107886556692502843100} a^{2} - \frac{553740084639482399733772260189260757749616667934}{1707930049591568392818663681276971639173125710775} a - \frac{1614609360336241941564451190691237441187548014}{20093294701077275209631337426787901637330890715}$

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

Class group and class number

$C_{36}$, which has order $36$ (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:  \( 36522830000.442375 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.1.243.1, 6.0.718449183.1, 9.1.2008387814976.4

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.9.22.46$x^{9} + 18 x^{5} + 3$$9$$1$$22$$C_3^2 : C_6$$[2, 5/2, 17/6]_{2}$
3.9.22.46$x^{9} + 18 x^{5} + 3$$9$$1$$22$$C_3^2 : C_6$$[2, 5/2, 17/6]_{2}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.12.6.1$x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$