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Properties

Label 19.19.4163417357...7961.1
Degree $19$
Signature $[19, 0]$
Discriminant $571^{18}$
Root discriminant $408.84$
Ramified prime $571$
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![1072348621073, -549551190705, -749021546949, 398834944226, 187074906809, -104945874488, -24108274876, 14258200659, 1826548777, -1133336624, -85340779, 55186847, 2453428, -1648168, -41181, 28987, 351, -270, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - x^18 - 270*x^17 + 351*x^16 + 28987*x^15 - 41181*x^14 - 1648168*x^13 + 2453428*x^12 + 55186847*x^11 - 85340779*x^10 - 1133336624*x^9 + 1826548777*x^8 + 14258200659*x^7 - 24108274876*x^6 - 104945874488*x^5 + 187074906809*x^4 + 398834944226*x^3 - 749021546949*x^2 - 549551190705*x + 1072348621073)
 
gp: K = bnfinit(x^19 - x^18 - 270*x^17 + 351*x^16 + 28987*x^15 - 41181*x^14 - 1648168*x^13 + 2453428*x^12 + 55186847*x^11 - 85340779*x^10 - 1133336624*x^9 + 1826548777*x^8 + 14258200659*x^7 - 24108274876*x^6 - 104945874488*x^5 + 187074906809*x^4 + 398834944226*x^3 - 749021546949*x^2 - 549551190705*x + 1072348621073, 1)
 

Normalized defining polynomial

\( x^{19} - x^{18} - 270 x^{17} + 351 x^{16} + 28987 x^{15} - 41181 x^{14} - 1648168 x^{13} + 2453428 x^{12} + 55186847 x^{11} - 85340779 x^{10} - 1133336624 x^{9} + 1826548777 x^{8} + 14258200659 x^{7} - 24108274876 x^{6} - 104945874488 x^{5} + 187074906809 x^{4} + 398834944226 x^{3} - 749021546949 x^{2} - 549551190705 x + 1072348621073 \)

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:  \(41634173570364661205169708858211372543325791407961=571^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $408.84$
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:  $571$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(571\)
Dirichlet character group:    $\lbrace$$\chi_{571}(64,·)$, $\chi_{571}(1,·)$, $\chi_{571}(323,·)$, $\chi_{571}(390,·)$, $\chi_{571}(271,·)$, $\chi_{571}(131,·)$, $\chi_{571}(214,·)$, $\chi_{571}(407,·)$, $\chi_{571}(94,·)$, $\chi_{571}(31,·)$, $\chi_{571}(353,·)$, $\chi_{571}(99,·)$, $\chi_{571}(170,·)$, $\chi_{571}(306,·)$, $\chi_{571}(563,·)$, $\chi_{571}(116,·)$, $\chi_{571}(350,·)$, $\chi_{571}(55,·)$, $\chi_{571}(59,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7313} a^{16} - \frac{2980}{7313} a^{15} + \frac{1898}{7313} a^{14} - \frac{1779}{7313} a^{13} + \frac{2580}{7313} a^{12} - \frac{137}{7313} a^{11} + \frac{198}{7313} a^{10} - \frac{953}{7313} a^{9} + \frac{3196}{7313} a^{8} - \frac{1761}{7313} a^{7} + \frac{1231}{7313} a^{6} + \frac{3290}{7313} a^{5} - \frac{1346}{7313} a^{4} - \frac{262}{7313} a^{3} - \frac{1588}{7313} a^{2} + \frac{1129}{7313} a - \frac{1458}{7313}$, $\frac{1}{16183669} a^{17} - \frac{34}{16183669} a^{16} + \frac{6477736}{16183669} a^{15} + \frac{397499}{16183669} a^{14} - \frac{6518129}{16183669} a^{13} - \frac{1284752}{16183669} a^{12} - \frac{2904450}{16183669} a^{11} + \frac{5621012}{16183669} a^{10} - \frac{5239571}{16183669} a^{9} + \frac{3548629}{16183669} a^{8} + \frac{5351358}{16183669} a^{7} - \frac{5511434}{16183669} a^{6} + \frac{2612010}{16183669} a^{5} + \frac{7969238}{16183669} a^{4} + \frac{7409807}{16183669} a^{3} - \frac{5013517}{16183669} a^{2} + \frac{5503850}{16183669} a + \frac{7632235}{16183669}$, $\frac{1}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{18} + \frac{28315567514945721617073307456960391981692235884892140895703848041}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{17} + \frac{90329817933674128687643570612962234841079190453749125066301141763062}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{16} - \frac{569171928162569258712457530448034659757766714060280489003248811116012682}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{15} - \frac{826532473760836787319063447430311396327464025188442810740573503144365504}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{14} + \frac{399846729692606367332866770188065697646149201025165423437849799019350092}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{13} - \frac{538530000763033124540272693107500783336696142266997616865253224747762667}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{12} - \frac{1080504061834062932574398021643790213337653159201445444944837103329473652}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{11} + \frac{176205553015894190883653810320393277421341395718727114994140177108634948}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{10} + \frac{1158889844153482781130980727052093375848001696882742359785918638830821479}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{9} + \frac{229154753102215322526029320353201586531069635814983097972174203871510433}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{8} - \frac{221722297175753565267040702228578665802579545855940593554283068419578951}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{7} + \frac{1124034139307779581847740746625086662488220921405689240859185564183311705}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{6} - \frac{698083134691320959505053686510042488132191716339928877048352373674711050}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{5} + \frac{51285763772835864888565177410174516905238343263773904041116199989284401}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{4} + \frac{435857341183523958675481977212552034897816546884541921034401824066529660}{2601127624625722985032968148360703355647974765822798880262361977322034857} a^{3} - \frac{6199059426631544077784355453179299803279225488570712696520336656911329}{23863556189226816376449249067529388583926373998374301653783137406624173} a^{2} + \frac{567333253262137522643300277274789503326175515054627798575046151129970432}{2601127624625722985032968148360703355647974765822798880262361977322034857} a + \frac{426375004940399879955697302544734356256780459797670746579985718177140055}{2601127624625722985032968148360703355647974765822798880262361977322034857}$

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:  \( 5743380709748059000 \) (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 $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$

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
571Data not computed