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Properties

Label 19.19.7559474410...9649.1
Degree $19$
Signature $[19, 0]$
Discriminant $457^{18}$
Root discriminant $331.07$
Ramified prime $457$
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![554402371, -1701753321, -1551959900, 3278226422, 1693728828, -2305645775, -862308300, 773799359, 209490406, -137385192, -24036830, 13251603, 1275207, -670894, -31517, 17417, 329, -216, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - x^18 - 216*x^17 + 329*x^16 + 17417*x^15 - 31517*x^14 - 670894*x^13 + 1275207*x^12 + 13251603*x^11 - 24036830*x^10 - 137385192*x^9 + 209490406*x^8 + 773799359*x^7 - 862308300*x^6 - 2305645775*x^5 + 1693728828*x^4 + 3278226422*x^3 - 1551959900*x^2 - 1701753321*x + 554402371)
 
gp: K = bnfinit(x^19 - x^18 - 216*x^17 + 329*x^16 + 17417*x^15 - 31517*x^14 - 670894*x^13 + 1275207*x^12 + 13251603*x^11 - 24036830*x^10 - 137385192*x^9 + 209490406*x^8 + 773799359*x^7 - 862308300*x^6 - 2305645775*x^5 + 1693728828*x^4 + 3278226422*x^3 - 1551959900*x^2 - 1701753321*x + 554402371, 1)
 

Normalized defining polynomial

\( x^{19} - x^{18} - 216 x^{17} + 329 x^{16} + 17417 x^{15} - 31517 x^{14} - 670894 x^{13} + 1275207 x^{12} + 13251603 x^{11} - 24036830 x^{10} - 137385192 x^{9} + 209490406 x^{8} + 773799359 x^{7} - 862308300 x^{6} - 2305645775 x^{5} + 1693728828 x^{4} + 3278226422 x^{3} - 1551959900 x^{2} - 1701753321 x + 554402371 \)

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:  \(755947441066272696677489606263668936388276269649=457^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $331.07$
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:  $457$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(457\)
Dirichlet character group:    $\lbrace$$\chi_{457}(256,·)$, $\chi_{457}(1,·)$, $\chi_{457}(68,·)$, $\chi_{457}(453,·)$, $\chi_{457}(200,·)$, $\chi_{457}(393,·)$, $\chi_{457}(215,·)$, $\chi_{457}(16,·)$, $\chi_{457}(407,·)$, $\chi_{457}(218,·)$, $\chi_{457}(347,·)$, $\chi_{457}(289,·)$, $\chi_{457}(42,·)$, $\chi_{457}(174,·)$, $\chi_{457}(241,·)$, $\chi_{457}(114,·)$, $\chi_{457}(54,·)$, $\chi_{457}(440,·)$, $\chi_{457}(185,·)$$\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}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{18} - \frac{48565805214858160384900688062775428390191621527039217474320842916670137446995456}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{17} - \frac{10426924704537712345085771343626282863172703309930471486730785172374806270655579}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{16} - \frac{40113070893093264896740747402472682024916229055168278349519993303319684200378438}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{15} - \frac{28900447009170191607394564626090272913177975644483246931228748216191838096399325}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{14} - \frac{14417729146565843611578906582113664352664795408156044363245803986801935529341471}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{13} + \frac{45821310524168705263967732973677843213527080567734994599166564506201010766635246}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{12} - \frac{41132790127421684035452761811149475728558766379912669114266388338456713608772412}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{11} - \frac{3743389818247557960195711584292590380140640953689146769669476661445095164084812}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{10} + \frac{48157269399528821213192459892300576097228239866707818807268693283876861519559340}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{9} + \frac{609893613292674474534767941251557221014146324001969927386521941084345291904495}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{8} + \frac{48527564786258675785457180533281293083230835332517312081246923983202626756271271}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{7} - \frac{566410944241927813135362983688219709852340370095571396430164413361223861690906}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{6} + \frac{43520070452054704833530778222666476907279805090896329892148657009551230920426243}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{5} - \frac{11873987390374351489138417525940782198773988864637702167085724199254307323007221}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{4} - \frac{5909179369126499240778747052804816308441875465565710444792476267317920758452412}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{3} + \frac{47314708131180935309413704055760864764586538807791989681804597175106606343877264}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a^{2} + \frac{4449973944869871689327840478813969029056618750646558642321453113956192532598426}{97508773145721215297215152412273541567940471081469801738609008394727562742806613} a + \frac{328929397604971771830118179972752413208029126312965899895514644999030977405219}{767785615320639490529253168600579067464098197491888202666212664525413879864619}$

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:  \( 438964507237879700 \) (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
457Data not computed