Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 2*x^18 + 7*x^16 - 10*x^15 - x^14 + 19*x^13 - 19*x^12 - 9*x^11 + 33*x^10 - 19*x^9 - 20*x^8 + 33*x^7 - 9*x^6 - 18*x^5 + 17*x^4 - 3*x^3 - 5*x^2 + 4*x - 1)
gp: K = bnfinit(y^19 - 2*y^18 + 7*y^16 - 10*y^15 - y^14 + 19*y^13 - 19*y^12 - 9*y^11 + 33*y^10 - 19*y^9 - 20*y^8 + 33*y^7 - 9*y^6 - 18*y^5 + 17*y^4 - 3*y^3 - 5*y^2 + 4*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 2*x^18 + 7*x^16 - 10*x^15 - x^14 + 19*x^13 - 19*x^12 - 9*x^11 + 33*x^10 - 19*x^9 - 20*x^8 + 33*x^7 - 9*x^6 - 18*x^5 + 17*x^4 - 3*x^3 - 5*x^2 + 4*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 - 2*x^18 + 7*x^16 - 10*x^15 - x^14 + 19*x^13 - 19*x^12 - 9*x^11 + 33*x^10 - 19*x^9 - 20*x^8 + 33*x^7 - 9*x^6 - 18*x^5 + 17*x^4 - 3*x^3 - 5*x^2 + 4*x - 1)
\( x^{19} - 2 x^{18} + 7 x^{16} - 10 x^{15} - x^{14} + 19 x^{13} - 19 x^{12} - 9 x^{11} + 33 x^{10} + \cdots - 1 \)
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: | $[3, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(7878347336355332175700013824\)
\(\medspace = 2^{8}\cdot 317\cdot 25373\cdot 137363\cdot 27854432176313\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.39\) | 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: | $2^{4/3}317^{1/2}25373^{1/2}137363^{1/2}27854432176313^{1/2}\approx 13978832686367.559$ | ||
| Ramified primes: |
\(2\), \(317\), \(25373\), \(137363\), \(27854432176313\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{30774\!\cdots\!28179}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5}a^{18}+\frac{1}{5}a^{17}-\frac{2}{5}a^{16}+\frac{1}{5}a^{15}-\frac{2}{5}a^{14}-\frac{2}{5}a^{13}-\frac{2}{5}a^{12}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{2}{5}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| 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: | $10$ | 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: |
$a$, $a^{17}-a^{16}-2a^{15}+6a^{14}-2a^{13}-9a^{12}+12a^{11}+2a^{10}-19a^{9}+12a^{8}+12a^{7}-20a^{6}+a^{5}+12a^{4}-7a^{3}-2a^{2}+2a-1$, $a^{17}-2a^{16}+6a^{14}-8a^{13}-a^{12}+13a^{11}-11a^{10}-8a^{9}+20a^{8}-8a^{7}-12a^{6}+13a^{5}-a^{4}-6a^{3}+4a^{2}-2a+1$, $\frac{1}{5}a^{18}+\frac{1}{5}a^{17}-\frac{7}{5}a^{16}+\frac{11}{5}a^{15}+\frac{3}{5}a^{14}-\frac{27}{5}a^{13}+\frac{33}{5}a^{12}-\frac{39}{5}a^{10}+\frac{31}{5}a^{9}+\frac{19}{5}a^{8}-\frac{53}{5}a^{7}+\frac{29}{5}a^{6}+\frac{8}{5}a^{5}-\frac{29}{5}a^{4}+4a^{3}-\frac{8}{5}a^{2}-\frac{4}{5}a+\frac{2}{5}$, $\frac{22}{5}a^{18}-\frac{33}{5}a^{17}-\frac{9}{5}a^{16}+\frac{132}{5}a^{15}-\frac{144}{5}a^{14}-\frac{54}{5}a^{13}+\frac{306}{5}a^{12}-44a^{11}-\frac{223}{5}a^{10}+\frac{457}{5}a^{9}-\frac{152}{5}a^{8}-\frac{356}{5}a^{7}+\frac{353}{5}a^{6}-\frac{14}{5}a^{5}-\frac{243}{5}a^{4}+23a^{3}-\frac{16}{5}a^{2}-\frac{43}{5}a+\frac{9}{5}$, $\frac{9}{5}a^{18}-\frac{11}{5}a^{17}-\frac{8}{5}a^{16}+\frac{44}{5}a^{15}-\frac{28}{5}a^{14}-\frac{38}{5}a^{13}+\frac{62}{5}a^{12}+3a^{11}-\frac{101}{5}a^{10}+\frac{29}{5}a^{9}+\frac{96}{5}a^{8}-\frac{77}{5}a^{7}-\frac{89}{5}a^{6}+\frac{97}{5}a^{5}+\frac{49}{5}a^{4}-25a^{3}+\frac{8}{5}a^{2}+\frac{59}{5}a-\frac{22}{5}$, $\frac{4}{5}a^{18}-\frac{6}{5}a^{17}-\frac{3}{5}a^{16}+\frac{24}{5}a^{15}-\frac{23}{5}a^{14}-\frac{13}{5}a^{13}+\frac{47}{5}a^{12}-5a^{11}-\frac{36}{5}a^{10}+\frac{44}{5}a^{9}+\frac{11}{5}a^{8}-\frac{47}{5}a^{7}-\frac{4}{5}a^{6}+\frac{37}{5}a^{5}-\frac{16}{5}a^{4}-6a^{3}+\frac{13}{5}a^{2}+\frac{9}{5}a-\frac{7}{5}$, $\frac{11}{5}a^{18}+\frac{1}{5}a^{17}-\frac{17}{5}a^{16}+\frac{41}{5}a^{15}+\frac{13}{5}a^{14}-\frac{67}{5}a^{13}+\frac{48}{5}a^{12}+9a^{11}-\frac{134}{5}a^{10}-\frac{4}{5}a^{9}+\frac{84}{5}a^{8}-\frac{98}{5}a^{7}-\frac{46}{5}a^{6}+\frac{73}{5}a^{5}-\frac{4}{5}a^{4}-2a^{3}+\frac{17}{5}a^{2}+\frac{1}{5}a-\frac{3}{5}$, $2a^{18}-4a^{17}+14a^{15}-21a^{14}-a^{13}+38a^{12}-41a^{11}-15a^{10}+65a^{9}-40a^{8}-38a^{7}+68a^{6}-20a^{5}-33a^{4}+32a^{3}-4a^{2}-7a+6$, $\frac{16}{5}a^{18}-\frac{39}{5}a^{17}+\frac{23}{5}a^{16}+\frac{91}{5}a^{15}-\frac{192}{5}a^{14}+\frac{93}{5}a^{13}+\frac{208}{5}a^{12}-71a^{11}+\frac{56}{5}a^{10}+\frac{396}{5}a^{9}-\frac{431}{5}a^{8}-\frac{43}{5}a^{7}+\frac{394}{5}a^{6}-\frac{277}{5}a^{5}-\frac{64}{5}a^{4}+35a^{3}-\frac{118}{5}a^{2}+\frac{26}{5}a+\frac{2}{5}$
| 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: | \( 6223827.226 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{8}\cdot 6223827.226 \cdot 1}{2\cdot\sqrt{7878347336355332175700013824}}\cr\approx \mathstrut & 0.6813008086 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^19 - 2*x^18 + 7*x^16 - 10*x^15 - x^14 + 19*x^13 - 19*x^12 - 9*x^11 + 33*x^10 - 19*x^9 - 20*x^8 + 33*x^7 - 9*x^6 - 18*x^5 + 17*x^4 - 3*x^3 - 5*x^2 + 4*x - 1)
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^18 + 7*x^16 - 10*x^15 - x^14 + 19*x^13 - 19*x^12 - 9*x^11 + 33*x^10 - 19*x^9 - 20*x^8 + 33*x^7 - 9*x^6 - 18*x^5 + 17*x^4 - 3*x^3 - 5*x^2 + 4*x - 1, 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^18 + 7*x^16 - 10*x^15 - x^14 + 19*x^13 - 19*x^12 - 9*x^11 + 33*x^10 - 19*x^9 - 20*x^8 + 33*x^7 - 9*x^6 - 18*x^5 + 17*x^4 - 3*x^3 - 5*x^2 + 4*x - 1);
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^18 + 7*x^16 - 10*x^15 - x^14 + 19*x^13 - 19*x^12 - 9*x^11 + 33*x^10 - 19*x^9 - 20*x^8 + 33*x^7 - 9*x^6 - 18*x^5 + 17*x^4 - 3*x^3 - 5*x^2 + 4*x - 1);
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 | ${\href{/padicField/3.7.0.1}{7} }^{2}{,}\,{\href{/padicField/3.5.0.1}{5} }$ | $17{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.7.0.1}{7} }$ | $18{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/47.4.0.1}{4} }$ | $17{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.6.8.1 | $x^{6} + 2 x^{3} + 2$ | $6$ | $1$ | $8$ | $D_{6}$ | $[2]_{3}^{2}$ | |
| 2.9.0.1 | $x^{9} + x^{4} + 1$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
|
\(317\)
| $\Q_{317}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(25373\)
| $\Q_{25373}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(137363\)
| $\Q_{137363}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{137363}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
|
\(27854432176313\)
| $\Q_{27854432176313}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |