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Properties

Label 19.19.1603710409...5561.1
Degree $19$
Signature $[19, 0]$
Discriminant $11^{18}\cdot 19^{16}$
Root discriminant $115.73$
Ramified primes $11, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{19}:C_{9}$ (as 19T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1726, -118143, -707173, -646414, 2646791, 2280774, -5156662, -573488, 3261556, -382602, -899712, 188431, 122460, -31377, -8370, 2391, 265, -82, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 3*x^18 - 82*x^17 + 265*x^16 + 2391*x^15 - 8370*x^14 - 31377*x^13 + 122460*x^12 + 188431*x^11 - 899712*x^10 - 382602*x^9 + 3261556*x^8 - 573488*x^7 - 5156662*x^6 + 2280774*x^5 + 2646791*x^4 - 646414*x^3 - 707173*x^2 - 118143*x - 1726)
 
gp: K = bnfinit(x^19 - 3*x^18 - 82*x^17 + 265*x^16 + 2391*x^15 - 8370*x^14 - 31377*x^13 + 122460*x^12 + 188431*x^11 - 899712*x^10 - 382602*x^9 + 3261556*x^8 - 573488*x^7 - 5156662*x^6 + 2280774*x^5 + 2646791*x^4 - 646414*x^3 - 707173*x^2 - 118143*x - 1726, 1)
 

Normalized defining polynomial

\( x^{19} - 3 x^{18} - 82 x^{17} + 265 x^{16} + 2391 x^{15} - 8370 x^{14} - 31377 x^{13} + 122460 x^{12} + 188431 x^{11} - 899712 x^{10} - 382602 x^{9} + 3261556 x^{8} - 573488 x^{7} - 5156662 x^{6} + 2280774 x^{5} + 2646791 x^{4} - 646414 x^{3} - 707173 x^{2} - 118143 x - 1726 \)

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:  \(1603710409222789970596991622167567965561=11^{18}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.73$
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:  $11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
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}{11378201340033166426721709451504679262630031767809} a^{18} - \frac{3675163336235798352479084791453650832471473508782}{11378201340033166426721709451504679262630031767809} a^{17} + \frac{5319865113822608887274159675639258766822831699057}{11378201340033166426721709451504679262630031767809} a^{16} + \frac{1355457357139475342203529243862186610274801948490}{11378201340033166426721709451504679262630031767809} a^{15} + \frac{4984633128148291382650654732931977329314953044383}{11378201340033166426721709451504679262630031767809} a^{14} + \frac{3691871479457767754159736644683876637624512673829}{11378201340033166426721709451504679262630031767809} a^{13} + \frac{4108258552450729520323912769394863725610409377380}{11378201340033166426721709451504679262630031767809} a^{12} + \frac{1946354145870029644297270627927574904906278903075}{11378201340033166426721709451504679262630031767809} a^{11} - \frac{2779751378735476760419135915144691755991116212571}{11378201340033166426721709451504679262630031767809} a^{10} + \frac{4157772685435773082397470147649767196501698120224}{11378201340033166426721709451504679262630031767809} a^{9} + \frac{1798143783657625311257098823674799011802287514238}{11378201340033166426721709451504679262630031767809} a^{8} + \frac{2111648015647432353674711422382664102776036214694}{11378201340033166426721709451504679262630031767809} a^{7} - \frac{194472132101695988835219938302418752549321510951}{11378201340033166426721709451504679262630031767809} a^{6} + \frac{3256266079514865396647492092761847760171290295233}{11378201340033166426721709451504679262630031767809} a^{5} - \frac{3979293613803319607056230104335152782290189971449}{11378201340033166426721709451504679262630031767809} a^{4} - \frac{1080469278308936043700156502620256614435008538565}{11378201340033166426721709451504679262630031767809} a^{3} - \frac{4374123694143762345582320110574832578772971949160}{11378201340033166426721709451504679262630031767809} a^{2} + \frac{5556307363144216024802928873056026143972746165659}{11378201340033166426721709451504679262630031767809} a + \frac{4234182000917607614954138609222161292736131330116}{11378201340033166426721709451504679262630031767809}$

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:  \( 163320198457000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{19}:C_3.C_3$ (as 19T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 171
The 11 conjugacy class representatives for $C_{19}:C_{9}$
Character table for $C_{19}:C_{9}$

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
11Data not computed
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$