Properties

Label 19.1.75613185918...8064.1
Degree $19$
Signature $[1, 9]$
Discriminant $-\,2^{18}\cdot 19^{16}$
Root discriminant $23.02$
Ramified primes $2, 19$
Class number $1$
Class group Trivial
Galois Group $F_{19}$ (as 19T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -31, -22, 43, 98, 120, -128, -264, -70, 338, 0, -25, -18, 108, -74, 43, -12, 8, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 2*x^18 + 8*x^17 - 12*x^16 + 43*x^15 - 74*x^14 + 108*x^13 - 18*x^12 - 25*x^11 + 338*x^9 - 70*x^8 - 264*x^7 - 128*x^6 + 120*x^5 + 98*x^4 + 43*x^3 - 22*x^2 - 31*x - 8)
gp: K = bnfinit(x^19 - 2*x^18 + 8*x^17 - 12*x^16 + 43*x^15 - 74*x^14 + 108*x^13 - 18*x^12 - 25*x^11 + 338*x^9 - 70*x^8 - 264*x^7 - 128*x^6 + 120*x^5 + 98*x^4 + 43*x^3 - 22*x^2 - 31*x - 8, 1)

Normalized defining polynomial

\(x^{19} \) \(\mathstrut -\mathstrut 2 x^{18} \) \(\mathstrut +\mathstrut 8 x^{17} \) \(\mathstrut -\mathstrut 12 x^{16} \) \(\mathstrut +\mathstrut 43 x^{15} \) \(\mathstrut -\mathstrut 74 x^{14} \) \(\mathstrut +\mathstrut 108 x^{13} \) \(\mathstrut -\mathstrut 18 x^{12} \) \(\mathstrut -\mathstrut 25 x^{11} \) \(\mathstrut +\mathstrut 338 x^{9} \) \(\mathstrut -\mathstrut 70 x^{8} \) \(\mathstrut -\mathstrut 264 x^{7} \) \(\mathstrut -\mathstrut 128 x^{6} \) \(\mathstrut +\mathstrut 120 x^{5} \) \(\mathstrut +\mathstrut 98 x^{4} \) \(\mathstrut +\mathstrut 43 x^{3} \) \(\mathstrut -\mathstrut 22 x^{2} \) \(\mathstrut -\mathstrut 31 x \) \(\mathstrut -\mathstrut 8 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $19$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[1, 9]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-75613185918270483380568064=-\,2^{18}\cdot 19^{16}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $23.02$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 19$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{39400150873630551157999} a^{18} - \frac{12080147394465471598402}{39400150873630551157999} a^{17} - \frac{9753733715124784245249}{39400150873630551157999} a^{16} - \frac{9235379841661719531612}{39400150873630551157999} a^{15} - \frac{8557727836047276750604}{39400150873630551157999} a^{14} - \frac{3769599737957497726074}{39400150873630551157999} a^{13} - \frac{19063720987888872194204}{39400150873630551157999} a^{12} - \frac{5806182467773850213415}{39400150873630551157999} a^{11} - \frac{8612257376902739598163}{39400150873630551157999} a^{10} + \frac{17862669450433223958046}{39400150873630551157999} a^{9} - \frac{5688014749199544505380}{39400150873630551157999} a^{8} - \frac{13839704170299550729659}{39400150873630551157999} a^{7} - \frac{10932224163403606145946}{39400150873630551157999} a^{6} - \frac{2172471404164941776342}{39400150873630551157999} a^{5} + \frac{17614982574680198035959}{39400150873630551157999} a^{4} - \frac{6161996356261955966797}{39400150873630551157999} a^{3} + \frac{17496426713903776840932}{39400150873630551157999} a^{2} + \frac{2488339540236741096796}{39400150873630551157999} a - \frac{16292170091273999535750}{39400150873630551157999}$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 540383.098056 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$F_{19}$ (as 19T6):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 342
The 19 conjugacy class representatives for $F_{19}$
Character table for $F_{19}$

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 R $18{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R $18{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2Data not computed
19Data not computed

Additional information

The polynomial was contributed by Noam Elkies.