Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^19 + 19*x^17 + 152*x^15 + 665*x^13 + 1729*x^11 + 2717*x^9 + 2508*x^7 + 1254*x^5 + 285*x^3 + 19*x - 1)
gp: K = bnfinit(y^19 + 19*y^17 + 152*y^15 + 665*y^13 + 1729*y^11 + 2717*y^9 + 2508*y^7 + 1254*y^5 + 285*y^3 + 19*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 + 19*x^17 + 152*x^15 + 665*x^13 + 1729*x^11 + 2717*x^9 + 2508*x^7 + 1254*x^5 + 285*x^3 + 19*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 + 19*x^17 + 152*x^15 + 665*x^13 + 1729*x^11 + 2717*x^9 + 2508*x^7 + 1254*x^5 + 285*x^3 + 19*x - 1)
\( x^{19} + 19 x^{17} + 152 x^{15} + 665 x^{13} + 1729 x^{11} + 2717 x^{9} + 2508 x^{7} + 1254 x^{5} + \cdots - 1 \)
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: | $[1, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-3864100889961549978757771484375\)
\(\medspace = -\,5^{9}\cdot 19^{19}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(40.72\) | 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: | $5^{1/2}19^{359/342}\approx 49.18156889081904$ | ||
| Ramified primes: |
\(5\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-95}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a$, $a^{17}+17a^{15}+119a^{13}-a^{12}+441a^{11}-12a^{10}+925a^{9}-54a^{8}+1086a^{7}-111a^{6}+658a^{5}-100a^{4}+169a^{3}-31a^{2}+11a-3$, $a^{15}+a^{14}+14a^{13}+14a^{12}+76a^{11}+76a^{10}+201a^{9}+201a^{8}+268a^{7}+268a^{6}+171a^{5}+171a^{4}+45a^{3}+45a^{2}+4a+3$, $a^{17}+16a^{15}+103a^{13}+339a^{11}-a^{10}+596a^{9}-10a^{8}+525a^{7}-35a^{6}+180a^{5}-50a^{4}-a^{3}-25a^{2}-3a-2$, $a^{17}+18a^{15}-a^{14}+133a^{13}-14a^{12}+520a^{11}-77a^{10}+1156a^{9}-210a^{8}+1461a^{7}-294a^{6}+995a^{5}-197a^{4}+326a^{3}-53a^{2}+39a-4$, $a^{17}+19a^{15}+151a^{13}+a^{12}+649a^{11}+12a^{10}+1626a^{9}+54a^{8}+2376a^{7}+111a^{6}+1895a^{5}+99a^{4}+681a^{3}+26a^{2}+54a-2$, $a^{17}+17a^{15}-a^{14}+118a^{13}-14a^{12}+430a^{11}-76a^{10}+882a^{9}-201a^{8}+1019a^{7}-268a^{6}+635a^{5}-170a^{4}+194a^{3}-41a^{2}+22a-2$, $a^{15}+16a^{13}+105a^{11}+363a^{9}-a^{8}+704a^{7}-8a^{6}+749a^{5}-20a^{4}+391a^{3}-16a^{2}+75a-2$, $a^{17}+a^{16}+15a^{15}+15a^{14}+91a^{13}+91a^{12}+286a^{11}+286a^{10}+494a^{9}+494a^{8}+455a^{7}+455a^{6}+194a^{5}+194a^{4}+23a^{3}+23a^{2}-a-2$
| 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: | \( 123041861.807 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 123041861.807 \cdot 1}{2\cdot\sqrt{3864100889961549978757771484375}}\cr\approx \mathstrut & 0.955316944005 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^19 + 19*x^17 + 152*x^15 + 665*x^13 + 1729*x^11 + 2717*x^9 + 2508*x^7 + 1254*x^5 + 285*x^3 + 19*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 + 19*x^17 + 152*x^15 + 665*x^13 + 1729*x^11 + 2717*x^9 + 2508*x^7 + 1254*x^5 + 285*x^3 + 19*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 + 19*x^17 + 152*x^15 + 665*x^13 + 1729*x^11 + 2717*x^9 + 2508*x^7 + 1254*x^5 + 285*x^3 + 19*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 + 19*x^17 + 152*x^15 + 665*x^13 + 1729*x^11 + 2717*x^9 + 2508*x^7 + 1254*x^5 + 285*x^3 + 19*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 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 $\Q$. |
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 | ${\href{/padicField/2.9.0.1}{9} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.9.0.1}{9} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.3.0.1}{3} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.9.0.1}{9} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/17.1.0.1}{1} }$ | R | $18{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $19$ | $18{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.9.0.1}{9} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# 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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.18.9.1 | $x^{18} + 180 x^{17} + 14445 x^{16} + 679200 x^{15} + 20664900 x^{14} + 423486000 x^{13} + 5887570504 x^{12} + 54397260480 x^{11} + 316143109712 x^{10} + 1034969211206 x^{9} + 1580754753720 x^{8} + 1360718216520 x^{7} + 746510415004 x^{6} + 357128191140 x^{5} + 552895560364 x^{4} + 1314509471572 x^{3} + 1121303668936 x^{2} + 1315877100296 x + 1500010785049$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ | |
|
\(19\)
| 19.19.19.10 | $x^{19} + 19 x + 19$ | $19$ | $1$ | $19$ | $F_{19}$ | $[19/18]_{18}$ |