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Properties

Label 19.19.1084250508...1281.1
Degree $19$
Signature $[19, 0]$
Discriminant $19^{36}$
Root discriminant $264.79$
Ramified prime $19$
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![-221874931, -137550709, 1138104275, 868638105, -1264657480, -861915924, 582575340, 339213156, -126730380, -66283229, 13391960, 6916190, -673436, -385833, 15580, 11476, -133, -171, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 171*x^17 - 133*x^16 + 11476*x^15 + 15580*x^14 - 385833*x^13 - 673436*x^12 + 6916190*x^11 + 13391960*x^10 - 66283229*x^9 - 126730380*x^8 + 339213156*x^7 + 582575340*x^6 - 861915924*x^5 - 1264657480*x^4 + 868638105*x^3 + 1138104275*x^2 - 137550709*x - 221874931)
 
gp: K = bnfinit(x^19 - 171*x^17 - 133*x^16 + 11476*x^15 + 15580*x^14 - 385833*x^13 - 673436*x^12 + 6916190*x^11 + 13391960*x^10 - 66283229*x^9 - 126730380*x^8 + 339213156*x^7 + 582575340*x^6 - 861915924*x^5 - 1264657480*x^4 + 868638105*x^3 + 1138104275*x^2 - 137550709*x - 221874931, 1)
 

Normalized defining polynomial

\( x^{19} - 171 x^{17} - 133 x^{16} + 11476 x^{15} + 15580 x^{14} - 385833 x^{13} - 673436 x^{12} + 6916190 x^{11} + 13391960 x^{10} - 66283229 x^{9} - 126730380 x^{8} + 339213156 x^{7} + 582575340 x^{6} - 861915924 x^{5} - 1264657480 x^{4} + 868638105 x^{3} + 1138104275 x^{2} - 137550709 x - 221874931 \)

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:  \(10842505080063916320800450434338728415281531281=19^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $264.79$
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:  $19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(361=19^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{361}(1,·)$, $\chi_{361}(324,·)$, $\chi_{361}(134,·)$, $\chi_{361}(267,·)$, $\chi_{361}(77,·)$, $\chi_{361}(210,·)$, $\chi_{361}(20,·)$, $\chi_{361}(343,·)$, $\chi_{361}(153,·)$, $\chi_{361}(286,·)$, $\chi_{361}(96,·)$, $\chi_{361}(229,·)$, $\chi_{361}(39,·)$, $\chi_{361}(172,·)$, $\chi_{361}(305,·)$, $\chi_{361}(115,·)$, $\chi_{361}(248,·)$, $\chi_{361}(58,·)$, $\chi_{361}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{18} + \frac{385773302914194215958708474679153265996618803786785791891677387642124790975}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{17} - \frac{137758447500011035831700065916657168294693413641717360633740737929262364770}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{16} - \frac{275922122987580811277954141668589623498080180458499245749456794283976807730}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{15} - \frac{204932227288395113015363329992952994151287479770007783831620436828977132739}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{14} - \frac{403926988808031051792988039201735737440254238912850999914461843794759178817}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{13} + \frac{209254334101372477745862724984160008750727627142603007948133143046831245689}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{12} - \frac{382994006198842969114166023920839637893244660930891600894002623042470925244}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{11} - \frac{147958123676817920593698409000944500763618440431252886918498987251904466694}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{10} + \frac{71907278105373126737665145449571000495313397272033945456101313523015933645}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{9} + \frac{26020501230728326903654555878424656307616254382367417310027893035756790015}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{8} + \frac{234789931242068185541713682786818270535616275400365628499821258418223992753}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{7} + \frac{170156478862607699222752319663304229901253170858559621436897352853431391570}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{6} + \frac{51434669223691623854469917932221728632389097315306253574043964649863529533}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{5} + \frac{180118104386827346617145767626948229546711162091188905441111616977919351661}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{4} - \frac{126732392845067272955566618652720270512482015664048200485297892429510262733}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{3} - \frac{80302551402194570398713456053285295356331915373054352635337553228869931531}{887050307246332877087962564975981866958830166303191353650070211476081622289} a^{2} - \frac{333674235764421454047528683730568883486591779237089219452485340353240679082}{887050307246332877087962564975981866958830166303191353650070211476081622289} a + \frac{255024674187499033139330176828287511129716356017932406309033667079483395219}{887050307246332877087962564975981866958830166303191353650070211476081622289}$

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:  \( 50751893334000000 \) (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 $19$ $19$ $19$ $19$ $19$ $19$ $19$ R $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$

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
19Data not computed