Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^19 + 2*x - 2)
gp: K = bnfinit(y^19 + 2*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 + 2*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 + 2*x - 2)
\( x^{19} + 2x - 2 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $19$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-539259691810398316599660249088\)
\(\medspace = -\,2^{18}\cdot 7\cdot 47\cdot 1031\cdot 108139\cdot 56081694178207\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(36.71\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2^{18/19}7^{1/2}47^{1/2}1031^{1/2}108139^{1/2}56081694178207^{1/2}\approx 2765765342919.86$ | ||
| Ramified primes: |
\(2\), \(7\), \(47\), \(1031\), \(108139\), \(56081694178207\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-20571\!\cdots\!74827}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$a-1$, $a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{2}+a+3$, $a^{18}+a^{17}+a^{16}+a^{15}-a^{10}-a^{9}-a^{7}-a^{6}-a^{5}-a^{2}+3$, $a^{18}+a^{16}+a^{14}+a^{12}+a^{10}+a^{8}+a^{6}+a^{4}+a+1$, $a^{13}-a^{8}+a^{7}-a^{2}+1$, $2a^{18}+2a^{17}+2a^{16}+2a^{15}+a^{14}+a^{13}+a^{8}+a^{7}+2a^{6}+a^{5}+a^{4}+a^{3}-a^{2}+3$, $a^{18}+a^{17}-a^{15}-a^{14}-a^{13}+a^{11}+a^{10}+a^{9}+a^{8}-a^{4}-a^{2}-a+3$, $a^{15}-a^{14}+a^{13}-a^{9}+2a^{8}-a^{7}+a^{6}-a^{4}+a^{3}-2a^{2}+3a-1$, $a^{18}+a^{17}+a^{16}-a^{14}-a^{13}-a^{12}-a^{8}-2a^{7}-a^{6}-a^{5}+a^{4}+a^{3}+a^{2}+1$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 30572994.7677 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 30572994.7677 \cdot 1}{2\cdot\sqrt{539259691810398316599660249088}}\cr\approx \mathstrut & 0.635415519082 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^19 + 2*x - 2)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^19 + 2*x - 2, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^19 + 2*x - 2);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 + 2*x - 2);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A non-solvable group of order 121645100408832000 |
| The 490 conjugacy class representatives for $S_{19}$ are not computed |
| Character table for $S_{19}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $16{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.13.0.1}{13} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ | $18{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.9.0.1}{9} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | R | $15{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.8.0.1}{8} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.19.18.1 | $x^{19} + 2$ | $19$ | $1$ | $18$ | $F_{19}$ | $[\ ]_{19}^{18}$ |
|
\(7\)
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
|
\(47\)
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 47.4.0.1 | $x^{4} + 8 x^{2} + 40 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 47.4.0.1 | $x^{4} + 8 x^{2} + 40 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 47.9.0.1 | $x^{9} + x^{3} + 19 x^{2} + x + 42$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
|
\(1031\)
| $\Q_{1031}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1031}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(108139\)
| $\Q_{108139}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
|
\(56081694178207\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |