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Properties

Label 18.0.60299121381...0563.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{12}\cdot 19^{12}$
Root discriminant $45.13$
Ramified primes $3, 7, 19$
Class number $108$ (GRH)
Class group $[6, 18]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3136, -4704, 19600, -14224, 66332, -50344, 104749, -33026, 69607, -25326, 27163, -5250, 4312, -518, 495, -28, 26, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 26*x^16 - 28*x^15 + 495*x^14 - 518*x^13 + 4312*x^12 - 5250*x^11 + 27163*x^10 - 25326*x^9 + 69607*x^8 - 33026*x^7 + 104749*x^6 - 50344*x^5 + 66332*x^4 - 14224*x^3 + 19600*x^2 - 4704*x + 3136)
 
gp: K = bnfinit(x^18 + 26*x^16 - 28*x^15 + 495*x^14 - 518*x^13 + 4312*x^12 - 5250*x^11 + 27163*x^10 - 25326*x^9 + 69607*x^8 - 33026*x^7 + 104749*x^6 - 50344*x^5 + 66332*x^4 - 14224*x^3 + 19600*x^2 - 4704*x + 3136, 1)
 

Normalized defining polynomial

\( x^{18} + 26 x^{16} - 28 x^{15} + 495 x^{14} - 518 x^{13} + 4312 x^{12} - 5250 x^{11} + 27163 x^{10} - 25326 x^{9} + 69607 x^{8} - 33026 x^{7} + 104749 x^{6} - 50344 x^{5} + 66332 x^{4} - 14224 x^{3} + 19600 x^{2} - 4704 x + 3136 \)

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:  \(-602991213815902363206590020563=-\,3^{9}\cdot 7^{12}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.13$
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:  $3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(399=3\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{399}(64,·)$, $\chi_{399}(1,·)$, $\chi_{399}(197,·)$, $\chi_{399}(134,·)$, $\chi_{399}(11,·)$, $\chi_{399}(277,·)$, $\chi_{399}(163,·)$, $\chi_{399}(296,·)$, $\chi_{399}(106,·)$, $\chi_{399}(235,·)$, $\chi_{399}(172,·)$, $\chi_{399}(239,·)$, $\chi_{399}(368,·)$, $\chi_{399}(305,·)$, $\chi_{399}(121,·)$, $\chi_{399}(58,·)$, $\chi_{399}(254,·)$, $\chi_{399}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{13} + \frac{1}{12} a^{11} - \frac{1}{12} a^{10} + \frac{1}{6} a^{6} + \frac{1}{12} a^{5} + \frac{1}{4} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{2428776} a^{15} + \frac{25303}{303597} a^{13} + \frac{8225}{173484} a^{12} - \frac{67301}{809592} a^{11} - \frac{1495}{86742} a^{10} - \frac{4465}{57828} a^{9} - \frac{407}{14457} a^{8} - \frac{3337}{39816} a^{7} - \frac{5969}{57828} a^{6} - \frac{844733}{2428776} a^{5} - \frac{12373}{86742} a^{4} - \frac{83621}{2428776} a^{3} + \frac{2483}{173484} a^{2} + \frac{2023}{9638} a + \frac{3569}{43371}$, $\frac{1}{37757751696} a^{16} + \frac{533}{2696982264} a^{15} + \frac{13}{18878875848} a^{14} + \frac{12116237}{449497044} a^{13} + \frac{3798473551}{37757751696} a^{12} - \frac{16587523}{337122783} a^{11} + \frac{21423071}{337122783} a^{10} - \frac{11253535}{99888232} a^{9} + \frac{3908732663}{37757751696} a^{8} - \frac{18486832}{337122783} a^{7} - \frac{16644988025}{37757751696} a^{6} - \frac{164290043}{449497044} a^{5} - \frac{317677027}{4195305744} a^{4} - \frac{568041023}{2696982264} a^{3} - \frac{511531145}{1348491132} a^{2} + \frac{169523225}{674245566} a + \frac{41955256}{337122783}$, $\frac{1}{21385872846285685605671328} a^{17} - \frac{11104280416283}{1527562346163263257547952} a^{16} + \frac{125217663926614933}{3564312141047614267611888} a^{15} - \frac{54575912749609542349}{190945293270407907193494} a^{14} + \frac{1252899371668866862807375}{21385872846285685605671328} a^{13} + \frac{499166475335610344131}{4041170227945140892984} a^{12} + \frac{40007561124994754462489}{381890586540815814386988} a^{11} - \frac{5051847818053269246011}{1527562346163263257547952} a^{10} + \frac{2344980347798631179356415}{21385872846285685605671328} a^{9} + \frac{70279974747175773666217}{763781173081631628773976} a^{8} - \frac{722878150545173466827947}{7128624282095228535223776} a^{7} + \frac{732675834295706108135}{381890586540815814386988} a^{6} + \frac{1149441405268250841199093}{2376208094031742845074592} a^{5} + \frac{36626904391173461098531}{218223192309037608221136} a^{4} + \frac{8859330278619137635873}{18185266025753134018428} a^{3} + \frac{18716420861410101122509}{381890586540815814386988} a^{2} - \frac{4738426225397042021971}{27277899038629701027642} a + \frac{3612159074074353207953}{13638949519314850513821}$

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

Class group and class number

$C_{6}\times C_{18}$, which has order $108$ (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:  \( -\frac{38724148526401575}{305700256533094409504} a^{17} - \frac{728381908217615}{16376799457130057652} a^{16} - \frac{1496128809748957177}{458550384799641614256} a^{15} + \frac{11170285176335245}{4679085559180016472} a^{14} - \frac{55648086374743938683}{917100769599283228512} a^{13} + \frac{2794496768606740637}{65507197828520230608} a^{12} - \frac{8325413098995655573}{16376799457130057652} a^{11} + \frac{29894948547853751263}{65507197828520230608} a^{10} - \frac{942390385174484590621}{305700256533094409504} a^{9} + \frac{17179735950168823559}{9358171118360032944} a^{8} - \frac{6345668448739098263683}{917100769599283228512} a^{7} + \frac{21259695672813150187}{65507197828520230608} a^{6} - \frac{9241596130446617188801}{917100769599283228512} a^{5} + \frac{15321696623295374999}{16376799457130057652} a^{4} - \frac{42367580428887836471}{10917866304753371768} a^{3} - \frac{48292226997470044753}{16376799457130057652} a^{2} - \frac{1183038253162527997}{1169771389795004118} a + \frac{1305820652555963}{9588290080286919} \) (order $6$)
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

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{7})^+\), 3.3.17689.2, 3.3.17689.1, 3.3.361.1, 6.0.64827.1, 6.0.8448319467.1, 6.0.8448319467.2, 6.0.3518667.1, 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$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$$[\ ]_{3}$
$19$19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$