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Properties

Label 19.19.3947090208...5889.1
Degree $19$
Signature $[19, 0]$
Discriminant $647^{18}$
Root discriminant $460.22$
Ramified prime $647$
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![50781641759, -18736713884, -71781895812, 33454527221, 31790175776, -16762277726, -6509889712, 3902105097, 678792838, -488507308, -34840236, 34401066, 704483, -1361758, 130, 29370, -79, -306, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - x^18 - 306*x^17 - 79*x^16 + 29370*x^15 + 130*x^14 - 1361758*x^13 + 704483*x^12 + 34401066*x^11 - 34840236*x^10 - 488507308*x^9 + 678792838*x^8 + 3902105097*x^7 - 6509889712*x^6 - 16762277726*x^5 + 31790175776*x^4 + 33454527221*x^3 - 71781895812*x^2 - 18736713884*x + 50781641759)
 
gp: K = bnfinit(x^19 - x^18 - 306*x^17 - 79*x^16 + 29370*x^15 + 130*x^14 - 1361758*x^13 + 704483*x^12 + 34401066*x^11 - 34840236*x^10 - 488507308*x^9 + 678792838*x^8 + 3902105097*x^7 - 6509889712*x^6 - 16762277726*x^5 + 31790175776*x^4 + 33454527221*x^3 - 71781895812*x^2 - 18736713884*x + 50781641759, 1)
 

Normalized defining polynomial

\( x^{19} - x^{18} - 306 x^{17} - 79 x^{16} + 29370 x^{15} + 130 x^{14} - 1361758 x^{13} + 704483 x^{12} + 34401066 x^{11} - 34840236 x^{10} - 488507308 x^{9} + 678792838 x^{8} + 3902105097 x^{7} - 6509889712 x^{6} - 16762277726 x^{5} + 31790175776 x^{4} + 33454527221 x^{3} - 71781895812 x^{2} - 18736713884 x + 50781641759 \)

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:  \(394709020826813768130190448054073575320458976755889=647^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $460.22$
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:  $647$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(647\)
Dirichlet character group:    $\lbrace$$\chi_{647}(1,·)$, $\chi_{647}(257,·)$, $\chi_{647}(200,·)$, $\chi_{647}(201,·)$, $\chi_{647}(279,·)$, $\chi_{647}(464,·)$, $\chi_{647}(533,·)$, $\chi_{647}(86,·)$, $\chi_{647}(544,·)$, $\chi_{647}(158,·)$, $\chi_{647}(287,·)$, $\chi_{647}(96,·)$, $\chi_{647}(548,·)$, $\chi_{647}(104,·)$, $\chi_{647}(492,·)$, $\chi_{647}(437,·)$, $\chi_{647}(55,·)$, $\chi_{647}(56,·)$, $\chi_{647}(378,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3551} a^{16} + \frac{925}{3551} a^{15} + \frac{1738}{3551} a^{14} + \frac{1301}{3551} a^{13} - \frac{1681}{3551} a^{12} - \frac{1595}{3551} a^{11} - \frac{20}{3551} a^{10} - \frac{447}{3551} a^{9} + \frac{1604}{3551} a^{8} + \frac{1337}{3551} a^{7} + \frac{1445}{3551} a^{6} + \frac{1555}{3551} a^{5} - \frac{452}{3551} a^{4} + \frac{964}{3551} a^{3} + \frac{313}{3551} a^{2} - \frac{696}{3551} a - \frac{1611}{3551}$, $\frac{1}{44739049} a^{17} - \frac{4398}{44739049} a^{16} - \frac{4598896}{44739049} a^{15} + \frac{12549516}{44739049} a^{14} + \frac{2543613}{44739049} a^{13} - \frac{12973955}{44739049} a^{12} + \frac{962045}{44739049} a^{11} + \frac{8340782}{44739049} a^{10} - \frac{9965842}{44739049} a^{9} - \frac{11725553}{44739049} a^{8} - \frac{20232800}{44739049} a^{7} - \frac{9558006}{44739049} a^{6} - \frac{10099380}{44739049} a^{5} - \frac{10518680}{44739049} a^{4} - \frac{11494451}{44739049} a^{3} + \frac{13471118}{44739049} a^{2} - \frac{20947845}{44739049} a - \frac{10908}{1040443}$, $\frac{1}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{18} + \frac{530505971002626225593906207968651773732614567494498940863424859}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{17} - \frac{11702626846555300910894490713479645533051648920325592805416634225196}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{16} - \frac{47493068417316141807175443413748508809750610166286072144308719315040351}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{15} + \frac{38384453329226613953782861938256280346094897940480397000679245841266854}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{14} + \frac{29474153276620523515010782386321134242436277751254390205212718828365920}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{13} - \frac{45797272046162531499821549792300739377282464494546751320371664401011371}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{12} - \frac{3999859762885648914530526889993753567152455593098403981484224955389263}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{11} - \frac{954754117224238331209990606512427706724963487704076017003635006569523}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{10} + \frac{28721420030929331575811656942936340496241137175910750005429364325860120}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{9} - \frac{27637496923057524088522176757955377367146140545640617599157167674358992}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{8} - \frac{20775720322827710887842456725804385530733676547959835910920194921510432}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{7} + \frac{23419434999626292134438840427799368785885357064785164875021723187264391}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{6} + \frac{54120291000471665560389504055371961609197628966347769786825558715587805}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{5} + \frac{46996522584115040456354974874361320417505443292685337635113327151306249}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{4} + \frac{47255924530949364554175129587260226275040576418754341864751230857398082}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{3} - \frac{24518707673682169432644222886832699655024176288637106340507356657634223}{123361847490546084562745526376800363181390569154240294664716818055988993} a^{2} - \frac{55230131313593605893398510361720703961570759937527921439937787671880416}{123361847490546084562745526376800363181390569154240294664716818055988993} a - \frac{503193698139948773965343352222087281317471915591462837873092540277290}{2868880174198746152621988985506985190264896957075355689877135303627651}$

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:  \( 24719888574728266000 \) (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 $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{19}$ $19$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{19}$ $19$

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
647Data not computed