Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 2*x^18 - 3*x^17 + 5*x^16 - 8*x^15 + 13*x^14 - 21*x^13 + 34*x^12 - 55*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)
gp: K = bnfinit(y^20 - y^19 + 2*y^18 - 3*y^17 + 5*y^16 - 8*y^15 + 13*y^14 - 21*y^13 + 34*y^12 - 55*y^11 + 89*y^10 + 55*y^9 + 34*y^8 + 21*y^7 + 13*y^6 + 8*y^5 + 5*y^4 + 3*y^3 + 2*y^2 + y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 + 2*x^18 - 3*x^17 + 5*x^16 - 8*x^15 + 13*x^14 - 21*x^13 + 34*x^12 - 55*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 + 2*x^18 - 3*x^17 + 5*x^16 - 8*x^15 + 13*x^14 - 21*x^13 + 34*x^12 - 55*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)
\( x^{20} - x^{19} + 2 x^{18} - 3 x^{17} + 5 x^{16} - 8 x^{15} + 13 x^{14} - 21 x^{13} + 34 x^{12} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(54296067514572573056640625\)
\(\medspace = 5^{10}\cdot 11^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.35\) | 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: | $5^{1/2}11^{9/10}\approx 19.352559831033375$ | ||
| Ramified primes: |
\(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(55=5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{55}(1,·)$, $\chi_{55}(4,·)$, $\chi_{55}(6,·)$, $\chi_{55}(9,·)$, $\chi_{55}(14,·)$, $\chi_{55}(16,·)$, $\chi_{55}(19,·)$, $\chi_{55}(21,·)$, $\chi_{55}(24,·)$, $\chi_{55}(26,·)$, $\chi_{55}(29,·)$, $\chi_{55}(31,·)$, $\chi_{55}(34,·)$, $\chi_{55}(36,·)$, $\chi_{55}(39,·)$, $\chi_{55}(41,·)$, $\chi_{55}(46,·)$, $\chi_{55}(49,·)$, $\chi_{55}(51,·)$, $\chi_{55}(54,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{89}a^{11}-\frac{34}{89}$, $\frac{1}{89}a^{12}-\frac{34}{89}a$, $\frac{1}{89}a^{13}-\frac{34}{89}a^{2}$, $\frac{1}{89}a^{14}-\frac{34}{89}a^{3}$, $\frac{1}{89}a^{15}-\frac{34}{89}a^{4}$, $\frac{1}{89}a^{16}-\frac{34}{89}a^{5}$, $\frac{1}{89}a^{17}-\frac{34}{89}a^{6}$, $\frac{1}{89}a^{18}-\frac{34}{89}a^{7}$, $\frac{1}{89}a^{19}-\frac{34}{89}a^{8}$
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
$C_{2}$, which has order $2$
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: |
\( -\frac{1}{89} a^{13} - \frac{233}{89} a^{2} \)
(order $22$)
| 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: |
$\frac{1}{89}a^{14}+\frac{233}{89}a^{3}$, $\frac{13}{89}a^{18}+\frac{8}{89}a^{17}+\frac{5}{89}a^{16}+\frac{1}{89}a^{12}+\frac{2584}{89}a^{7}+\frac{1597}{89}a^{6}+\frac{987}{89}a^{5}+\frac{144}{89}a+1$, $\frac{13}{89}a^{18}+\frac{1}{89}a^{13}+\frac{2584}{89}a^{7}+\frac{233}{89}a^{2}$, $\frac{34}{89}a^{19}-\frac{68}{89}a^{18}+\frac{102}{89}a^{17}-\frac{170}{89}a^{16}+\frac{275}{89}a^{15}-\frac{442}{89}a^{14}+\frac{715}{89}a^{13}-\frac{1156}{89}a^{12}+\frac{1870}{89}a^{11}-34a^{10}+55a^{9}-\frac{1156}{89}a^{8}-\frac{714}{89}a^{7}-\frac{442}{89}a^{6}-\frac{272}{89}a^{5}+\frac{440}{89}a^{4}-\frac{102}{89}a^{3}+\frac{165}{89}a^{2}-\frac{34}{89}a+\frac{55}{89}$, $\frac{3}{89}a^{15}+\frac{610}{89}a^{4}+1$, $\frac{8}{89}a^{17}+\frac{5}{89}a^{15}+\frac{1}{89}a^{13}+\frac{1597}{89}a^{6}+\frac{987}{89}a^{4}+\frac{233}{89}a^{2}$, $\frac{13}{89}a^{18}-\frac{5}{89}a^{17}+\frac{5}{89}a^{16}+\frac{2584}{89}a^{7}-\frac{987}{89}a^{6}+\frac{987}{89}a^{5}$, $\frac{13}{89}a^{17}+\frac{1}{89}a^{12}+\frac{2584}{89}a^{6}+\frac{144}{89}a+1$, $\frac{13}{89}a^{19}-\frac{3}{89}a^{15}-\frac{1}{89}a^{12}+\frac{2584}{89}a^{8}-\frac{610}{89}a^{4}-\frac{144}{89}a$
| 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: | \( 140644.599182 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 140644.599182 \cdot 2}{22\cdot\sqrt{54296067514572573056640625}}\cr\approx \mathstrut & 0.166396761675 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 2*x^18 - 3*x^17 + 5*x^16 - 8*x^15 + 13*x^14 - 21*x^13 + 34*x^12 - 55*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + 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^20 - x^19 + 2*x^18 - 3*x^17 + 5*x^16 - 8*x^15 + 13*x^14 - 21*x^13 + 34*x^12 - 55*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + 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^20 - x^19 + 2*x^18 - 3*x^17 + 5*x^16 - 8*x^15 + 13*x^14 - 21*x^13 + 34*x^12 - 55*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + 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^20 - x^19 + 2*x^18 - 3*x^17 + 5*x^16 - 8*x^15 + 13*x^14 - 21*x^13 + 34*x^12 - 55*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + 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
$C_2\times C_{10}$ (as 20T3):
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)
| An abelian group of order 20 |
| The 20 conjugacy class representatives for $C_2\times C_{10}$ |
| Character table for $C_2\times C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, \(\Q(\zeta_{11})\), 10.0.7368586534375.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{10}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(11\)
| 11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |