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Properties

Label 19.19.7326960021...2481.1
Degree $19$
Signature $[19, 0]$
Discriminant $761^{18}$
Root discriminant $536.70$
Ramified prime $761$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{19}$ (as 19T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3483379661, -12659238391, 18814754704, 27241052007, -21796436285, -21259099080, 8595059019, 6678374954, -1501612590, -973700212, 120121397, 70534890, -4227138, -2558302, 58762, 45376, -173, -360, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - x^18 - 360*x^17 - 173*x^16 + 45376*x^15 + 58762*x^14 - 2558302*x^13 - 4227138*x^12 + 70534890*x^11 + 120121397*x^10 - 973700212*x^9 - 1501612590*x^8 + 6678374954*x^7 + 8595059019*x^6 - 21259099080*x^5 - 21796436285*x^4 + 27241052007*x^3 + 18814754704*x^2 - 12659238391*x - 3483379661)
 
gp: K = bnfinit(x^19 - x^18 - 360*x^17 - 173*x^16 + 45376*x^15 + 58762*x^14 - 2558302*x^13 - 4227138*x^12 + 70534890*x^11 + 120121397*x^10 - 973700212*x^9 - 1501612590*x^8 + 6678374954*x^7 + 8595059019*x^6 - 21259099080*x^5 - 21796436285*x^4 + 27241052007*x^3 + 18814754704*x^2 - 12659238391*x - 3483379661, 1)
 

Normalized defining polynomial

\( x^{19} - x^{18} - 360 x^{17} - 173 x^{16} + 45376 x^{15} + 58762 x^{14} - 2558302 x^{13} - 4227138 x^{12} + 70534890 x^{11} + 120121397 x^{10} - 973700212 x^{9} - 1501612590 x^{8} + 6678374954 x^{7} + 8595059019 x^{6} - 21259099080 x^{5} - 21796436285 x^{4} + 27241052007 x^{3} + 18814754704 x^{2} - 12659238391 x - 3483379661 \)

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

Invariants

Degree:  $19$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[19, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7326960021331421245780688833073465007173042939202481=761^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $536.70$
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:  $761$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(761\)
Dirichlet character group:    $\lbrace$$\chi_{761}(1,·)$, $\chi_{761}(258,·)$, $\chi_{761}(25,·)$, $\chi_{761}(274,·)$, $\chi_{761}(405,·)$, $\chi_{761}(625,·)$, $\chi_{761}(152,·)$, $\chi_{761}(473,·)$, $\chi_{761}(410,·)$, $\chi_{761}(357,·)$, $\chi_{761}(679,·)$, $\chi_{761}(680,·)$, $\chi_{761}(233,·)$, $\chi_{761}(554,·)$, $\chi_{761}(232,·)$, $\chi_{761}(498,·)$, $\chi_{761}(756,·)$, $\chi_{761}(636,·)$, $\chi_{761}(362,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{157} a^{16} + \frac{67}{157} a^{15} + \frac{52}{157} a^{14} + \frac{10}{157} a^{13} - \frac{56}{157} a^{12} + \frac{59}{157} a^{11} - \frac{36}{157} a^{10} - \frac{7}{157} a^{9} + \frac{2}{157} a^{8} + \frac{63}{157} a^{7} - \frac{62}{157} a^{6} + \frac{11}{157} a^{5} - \frac{38}{157} a^{4} - \frac{43}{157} a^{3} + \frac{17}{157} a^{2} + \frac{17}{157} a - \frac{53}{157}$, $\frac{1}{10519} a^{17} - \frac{32}{10519} a^{16} - \frac{3598}{10519} a^{15} - \frac{2783}{10519} a^{14} + \frac{1623}{10519} a^{13} - \frac{5230}{10519} a^{12} + \frac{3700}{10519} a^{11} - \frac{3351}{10519} a^{10} + \frac{1}{157} a^{9} - \frac{2961}{10519} a^{8} - \frac{333}{10519} a^{7} - \frac{1701}{10519} a^{6} + \frac{4682}{10519} a^{5} + \frac{3562}{10519} a^{4} - \frac{2163}{10519} a^{3} - \frac{2294}{10519} a^{2} + \frac{3131}{10519} a - \frac{1818}{10519}$, $\frac{1}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{18} - \frac{112480927600848173427572286566641942509293827529284685303377101624305220547358950443130}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{17} - \frac{1980006658650159103389020321380596718161073213238656896929050243257946911824913702232957}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{16} - \frac{4031736710462742011518308731887738353161508585253361630935519486831605525183691215619179215}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{15} + \frac{2093632334155429692768489884759319572285107133295647671392392629099477273613654508725447536}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{14} + \frac{1339520512437990486569996569493380154352435202961785774284035731330937355739970644250055335}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{13} + \frac{3705868490608764098232523456155600723565021372593475312079258362069980505540791749943304691}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{12} + \frac{4993461878152931187789755621295394243159330250928544126911743962993812552573749345374798873}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{11} - \frac{5062147014423863368236857529102268827045452369536798425221215066406623299034359070776264187}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{10} + \frac{573386293370649748905004431425821306482070903017751290608089458221098336082305984610087139}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{9} + \frac{1824348758135733512374350065945891851396962916038187214958885683056857634389649235259216282}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{8} + \frac{824637325124721916465643740769244575786359272896142564153647762693218166000332800330780683}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{7} - \frac{5071343630096268630599909431567420948262482905265142179481597449606686959661004829744792067}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{6} - \frac{783075306822531794179713630859923563460627530256526098283363694793874701154323379866345959}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{5} + \frac{2261750216336770522827539267358901080529757329750398845390609502458252805078878916866752184}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{4} - \frac{196537495551380174709227030257148782025102845622829174818811176538812830983731497422284490}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{3} - \frac{1167031691965930866654204999729125948119786467900695646417754499291673590391863130590766187}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a^{2} - \frac{589714758548603648265305576415644140170601625468176257536726051769693635832417109950435786}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971} a - \frac{3533574894593310772119105895751834403463660490992599035544847933319925319941426773984551162}{10325124348661486292605003346731115757710824425761196091106924479282231882475596246810254971}$

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:  $18$
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:  \( 94833868482425830000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{19}$ (as 19T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 19
The 19 conjugacy class representatives for $C_{19}$
Character table for $C_{19}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{19}$ $19$ $19$ $19$ $19$

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
761Data not computed