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Properties

Label 18.8.19597718879...0608.2
Degree $18$
Signature $[8, 5]$
Discriminant $-\,2^{12}\cdot 3^{30}\cdot 7^{6}\cdot 12547^{3}$
Root discriminant $91.34$
Ramified primes $2, 3, 7, 12547$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T366

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-141570217, 637697091, -1160177862, 1159443999, -743501424, 322084668, -82704276, 1377153, 9075627, -4048559, 783678, -22953, -36342, 12393, -1377, 81, 39, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 39*x^16 + 81*x^15 - 1377*x^14 + 12393*x^13 - 36342*x^12 - 22953*x^11 + 783678*x^10 - 4048559*x^9 + 9075627*x^8 + 1377153*x^7 - 82704276*x^6 + 322084668*x^5 - 743501424*x^4 + 1159443999*x^3 - 1160177862*x^2 + 637697091*x - 141570217)
 
gp: K = bnfinit(x^18 - 3*x^17 + 39*x^16 + 81*x^15 - 1377*x^14 + 12393*x^13 - 36342*x^12 - 22953*x^11 + 783678*x^10 - 4048559*x^9 + 9075627*x^8 + 1377153*x^7 - 82704276*x^6 + 322084668*x^5 - 743501424*x^4 + 1159443999*x^3 - 1160177862*x^2 + 637697091*x - 141570217, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 39 x^{16} + 81 x^{15} - 1377 x^{14} + 12393 x^{13} - 36342 x^{12} - 22953 x^{11} + 783678 x^{10} - 4048559 x^{9} + 9075627 x^{8} + 1377153 x^{7} - 82704276 x^{6} + 322084668 x^{5} - 743501424 x^{4} + 1159443999 x^{3} - 1160177862 x^{2} + 637697091 x - 141570217 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-195977188797288322153846057926340608=-\,2^{12}\cdot 3^{30}\cdot 7^{6}\cdot 12547^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.34$
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:  $2, 3, 7, 12547$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{89} a^{16} + \frac{22}{89} a^{15} - \frac{1}{89} a^{14} - \frac{19}{89} a^{13} - \frac{16}{89} a^{12} - \frac{26}{89} a^{11} + \frac{38}{89} a^{10} + \frac{12}{89} a^{9} - \frac{15}{89} a^{8} - \frac{17}{89} a^{7} + \frac{29}{89} a^{5} - \frac{34}{89} a^{4} + \frac{5}{89} a^{3} + \frac{19}{89} a^{2} - \frac{13}{89} a + \frac{7}{89}$, $\frac{1}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{17} - \frac{372299696028322694019302358010070094030715999850951613832058899427239}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{16} + \frac{117530633781237582115574231587185882442690724844291853152121376934956172}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{15} - \frac{67052310234029555877625208291716370713646177654608257706520570860234376}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{14} - \frac{3965801405112933833095215202036064016472732206185133534635448603996115}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{13} + \frac{1109757777528637617462675851986452868705770070773955807324424316353756}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{12} + \frac{86619397189736092024064274449407955560405723687554181255559627571516380}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{11} - \frac{113289914876391260894602806809473748217172626918750046697422804492425693}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{10} - \frac{96722375552132960208170831353645323230363358685855066677673150983234111}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{9} + \frac{67344169503163051289249098462980022562432308521829506014016871689145798}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{8} + \frac{89558024371120301260895271994825472535920769185027278774242965061579838}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{7} - \frac{30088189071486786783581140280883188204955013129817200781726639732511633}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{6} + \frac{57576103572459108269865799540984295090240216141107961114124065031935804}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{5} + \frac{87267843649993863009828676167330601878132060602874057736860178182063917}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{4} + \frac{23705612492680673268694493639727892710347063356816623664847940519417288}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{3} + \frac{85655169667193730716023216019785098664829581345173250582074221349131715}{285727962069640093307827393894032875281021843851189577586934495904802831} a^{2} + \frac{4314549801855909478673146746425487617066921337881015731994838669643062}{285727962069640093307827393894032875281021843851189577586934495904802831} a - \frac{96111683339639216413089957717962544380661372295389117284517872144276475}{285727962069640093307827393894032875281021843851189577586934495904802831}$

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:  $12$
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:  \( 70064262840.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T366:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 48 conjugacy class representatives for t18n366
Character table for t18n366 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.756.1, 9.9.314987206464.1

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{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}$
12547Data not computed