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Properties

Label 19.19.2999429662...7561.1
Degree $19$
Signature $[19, 0]$
Discriminant $229^{18}$
Root discriminant $172.04$
Ramified prime $229$
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![-251347, -1865379, -3318455, 4255245, 13322064, -3737798, -17492482, 3299920, 9223494, -1928935, -2225275, 559431, 248057, -74320, -11618, 4270, 213, -108, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - x^18 - 108*x^17 + 213*x^16 + 4270*x^15 - 11618*x^14 - 74320*x^13 + 248057*x^12 + 559431*x^11 - 2225275*x^10 - 1928935*x^9 + 9223494*x^8 + 3299920*x^7 - 17492482*x^6 - 3737798*x^5 + 13322064*x^4 + 4255245*x^3 - 3318455*x^2 - 1865379*x - 251347)
 
gp: K = bnfinit(x^19 - x^18 - 108*x^17 + 213*x^16 + 4270*x^15 - 11618*x^14 - 74320*x^13 + 248057*x^12 + 559431*x^11 - 2225275*x^10 - 1928935*x^9 + 9223494*x^8 + 3299920*x^7 - 17492482*x^6 - 3737798*x^5 + 13322064*x^4 + 4255245*x^3 - 3318455*x^2 - 1865379*x - 251347, 1)
 

Normalized defining polynomial

\( x^{19} - x^{18} - 108 x^{17} + 213 x^{16} + 4270 x^{15} - 11618 x^{14} - 74320 x^{13} + 248057 x^{12} + 559431 x^{11} - 2225275 x^{10} - 1928935 x^{9} + 9223494 x^{8} + 3299920 x^{7} - 17492482 x^{6} - 3737798 x^{5} + 13322064 x^{4} + 4255245 x^{3} - 3318455 x^{2} - 1865379 x - 251347 \)

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:  \(2999429662895796650415561622892044448017561=229^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $172.04$
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:  $229$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(229\)
Dirichlet character group:    $\lbrace$$\chi_{229}(1,·)$, $\chi_{229}(161,·)$, $\chi_{229}(203,·)$, $\chi_{229}(16,·)$, $\chi_{229}(17,·)$, $\chi_{229}(214,·)$, $\chi_{229}(57,·)$, $\chi_{229}(218,·)$, $\chi_{229}(27,·)$, $\chi_{229}(225,·)$, $\chi_{229}(165,·)$, $\chi_{229}(104,·)$, $\chi_{229}(42,·)$, $\chi_{229}(43,·)$, $\chi_{229}(44,·)$, $\chi_{229}(53,·)$, $\chi_{229}(121,·)$, $\chi_{229}(60,·)$, $\chi_{229}(61,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{89} a^{17} - \frac{25}{89} a^{16} + \frac{12}{89} a^{15} - \frac{1}{89} a^{14} - \frac{42}{89} a^{13} + \frac{16}{89} a^{12} + \frac{13}{89} a^{11} + \frac{32}{89} a^{10} + \frac{29}{89} a^{8} - \frac{22}{89} a^{7} + \frac{26}{89} a^{6} + \frac{35}{89} a^{5} - \frac{36}{89} a^{4} + \frac{19}{89} a^{3} + \frac{13}{89} a^{2} - \frac{21}{89} a - \frac{41}{89}$, $\frac{1}{890586960280420453431275712528280672450515757006003} a^{18} + \frac{3158251944488249744471091015124673255424549899629}{890586960280420453431275712528280672450515757006003} a^{17} + \frac{240111493841675190951325842940869652459576214837038}{890586960280420453431275712528280672450515757006003} a^{16} + \frac{82891693721274572165544450260198205099418245692767}{890586960280420453431275712528280672450515757006003} a^{15} + \frac{154448248615280109868036493118948889088586906547361}{890586960280420453431275712528280672450515757006003} a^{14} + \frac{402470996514117682070632249077427052338514902617461}{890586960280420453431275712528280672450515757006003} a^{13} + \frac{360324228715855209482352512526943815187770330811492}{890586960280420453431275712528280672450515757006003} a^{12} - \frac{28963255772145319006477119100114876643409346163087}{890586960280420453431275712528280672450515757006003} a^{11} + \frac{91013038000767575411237257663076051933583872513976}{890586960280420453431275712528280672450515757006003} a^{10} - \frac{157069002081559453599676362301069614493982256612206}{890586960280420453431275712528280672450515757006003} a^{9} - \frac{107927899873438593350189430971023698002343552671043}{890586960280420453431275712528280672450515757006003} a^{8} + \frac{153013227850143101724964826937353775235945408590684}{890586960280420453431275712528280672450515757006003} a^{7} - \frac{124202511331908740040441685048463388332777904672437}{890586960280420453431275712528280672450515757006003} a^{6} - \frac{430149601646763526787713114520136663593379590826473}{890586960280420453431275712528280672450515757006003} a^{5} - \frac{88859091827668967645964185573331988301046596056447}{890586960280420453431275712528280672450515757006003} a^{4} - \frac{74328542359361374978876415048064422409122666571064}{890586960280420453431275712528280672450515757006003} a^{3} + \frac{267562471565065508521694146795518350469503346260966}{890586960280420453431275712528280672450515757006003} a^{2} + \frac{427841418484345012156785343600955898547504089252826}{890586960280420453431275712528280672450515757006003} a - \frac{111289245552412705258148912329203335089623628729263}{890586960280420453431275712528280672450515757006003}$

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:  \( 1038259438940000 \) (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
229Data not computed