Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 110*x^18 + 5060*x^16 + 127600*x^14 + 1944800*x^12 + 18612000*x^10 + 112288000*x^8 + 416240000*x^6 + 890560000*x^4 + 968000000*x^2 + 387200000)
gp: K = bnfinit(y^20 + 110*y^18 + 5060*y^16 + 127600*y^14 + 1944800*y^12 + 18612000*y^10 + 112288000*y^8 + 416240000*y^6 + 890560000*y^4 + 968000000*y^2 + 387200000, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 110*x^18 + 5060*x^16 + 127600*x^14 + 1944800*x^12 + 18612000*x^10 + 112288000*x^8 + 416240000*x^6 + 890560000*x^4 + 968000000*x^2 + 387200000);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 110*x^18 + 5060*x^16 + 127600*x^14 + 1944800*x^12 + 18612000*x^10 + 112288000*x^8 + 416240000*x^6 + 890560000*x^4 + 968000000*x^2 + 387200000)
\( x^{20} + 110 x^{18} + 5060 x^{16} + 127600 x^{14} + 1944800 x^{12} + 18612000 x^{10} + 112288000 x^{8} + \cdots + 387200000 \)
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: |
\(182187370528513441169408000000000000000\)
\(\medspace = 2^{30}\cdot 5^{15}\cdot 11^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(81.85\) | 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: | $2^{3/2}5^{3/4}11^{9/10}\approx 81.85136255739549$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(440=2^{3}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(43,·)$, $\chi_{440}(9,·)$, $\chi_{440}(81,·)$, $\chi_{440}(83,·)$, $\chi_{440}(89,·)$, $\chi_{440}(347,·)$, $\chi_{440}(361,·)$, $\chi_{440}(289,·)$, $\chi_{440}(227,·)$, $\chi_{440}(401,·)$, $\chi_{440}(169,·)$, $\chi_{440}(107,·)$, $\chi_{440}(387,·)$, $\chi_{440}(307,·)$, $\chi_{440}(49,·)$, $\chi_{440}(403,·)$, $\chi_{440}(201,·)$, $\chi_{440}(283,·)$, $\chi_{440}(123,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{20}a^{4}$, $\frac{1}{20}a^{5}$, $\frac{1}{40}a^{6}$, $\frac{1}{40}a^{7}$, $\frac{1}{400}a^{8}$, $\frac{1}{400}a^{9}$, $\frac{1}{8800}a^{10}$, $\frac{1}{8800}a^{11}$, $\frac{1}{88000}a^{12}$, $\frac{1}{88000}a^{13}$, $\frac{1}{176000}a^{14}$, $\frac{1}{176000}a^{15}$, $\frac{1}{1760000}a^{16}$, $\frac{1}{1760000}a^{17}$, $\frac{1}{38720000}a^{18}-\frac{1}{1100}a^{8}$, $\frac{1}{38720000}a^{19}-\frac{1}{1100}a^{9}$
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
$C_{2}\times C_{2}\times C_{62564}$, which has order $250256$ (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: |
$\frac{7}{7744000}a^{18}+\frac{1}{11000}a^{16}+\frac{653}{176000}a^{14}+\frac{1403}{17600}a^{12}+\frac{863}{880}a^{10}+\frac{3099}{440}a^{8}+\frac{1153}{40}a^{6}+\frac{251}{4}a^{4}+64a^{2}+22$, $\frac{3}{3520000}a^{18}+\frac{19}{220000}a^{16}+\frac{57}{16000}a^{14}+\frac{171}{2200}a^{12}+\frac{8607}{8800}a^{10}+\frac{2907}{400}a^{8}+\frac{627}{20}a^{6}+\frac{741}{10}a^{4}+\frac{171}{2}a^{2}+37$, $\frac{1}{3520000}a^{18}+\frac{3}{110000}a^{16}+\frac{23}{22000}a^{14}+\frac{227}{11000}a^{12}+\frac{249}{1100}a^{10}+\frac{139}{100}a^{8}+\frac{91}{20}a^{6}+7a^{4}+4a^{2}+1$, $\frac{1}{3520000}a^{18}+\frac{13}{440000}a^{16}+\frac{111}{88000}a^{14}+\frac{507}{17600}a^{12}+\frac{339}{880}a^{10}+\frac{621}{200}a^{8}+\frac{597}{40}a^{6}+\frac{202}{5}a^{4}+53a^{2}+22$, $\frac{1}{3520000}a^{18}+\frac{13}{440000}a^{16}+\frac{111}{88000}a^{14}+\frac{507}{17600}a^{12}+\frac{339}{880}a^{10}+\frac{621}{200}a^{8}+\frac{597}{40}a^{6}+\frac{202}{5}a^{4}+53a^{2}+23$, $\frac{3}{7744000}a^{18}+\frac{9}{220000}a^{16}+\frac{313}{176000}a^{14}+\frac{3647}{88000}a^{12}+\frac{199}{352}a^{10}+\frac{4069}{880}a^{8}+\frac{89}{4}a^{6}+\frac{1177}{20}a^{4}+\frac{147}{2}a^{2}+31$, $\frac{31}{19360000}a^{18}+\frac{7}{44000}a^{16}+\frac{9}{1408}a^{14}+\frac{1481}{11000}a^{12}+\frac{1293}{800}a^{10}+\frac{12401}{1100}a^{8}+\frac{223}{5}a^{6}+\frac{467}{5}a^{4}+\frac{183}{2}a^{2}+33$, $\frac{17}{9680000}a^{18}+\frac{39}{220000}a^{16}+\frac{2}{275}a^{14}+\frac{2771}{17600}a^{12}+\frac{1567}{800}a^{10}+\frac{62967}{4400}a^{8}+\frac{2407}{40}a^{6}+\frac{2737}{20}a^{4}+\frac{299}{2}a^{2}+60$, $\frac{3}{7744000}a^{18}+\frac{1}{27500}a^{16}+\frac{59}{44000}a^{14}+\frac{1083}{44000}a^{12}+\frac{1019}{4400}a^{10}+\frac{2071}{2200}a^{8}-\frac{1}{2}a^{6}-\frac{157}{10}a^{4}-\frac{77}{2}a^{2}-23$
| 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: | \( 140644.599182 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 140644.599182 \cdot 250256}{2\cdot\sqrt{182187370528513441169408000000000000000}}\cr\approx \mathstrut & 0.125030829486 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 + 110*x^18 + 5060*x^16 + 127600*x^14 + 1944800*x^12 + 18612000*x^10 + 112288000*x^8 + 416240000*x^6 + 890560000*x^4 + 968000000*x^2 + 387200000)
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 + 110*x^18 + 5060*x^16 + 127600*x^14 + 1944800*x^12 + 18612000*x^10 + 112288000*x^8 + 416240000*x^6 + 890560000*x^4 + 968000000*x^2 + 387200000, 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 + 110*x^18 + 5060*x^16 + 127600*x^14 + 1944800*x^12 + 18612000*x^10 + 112288000*x^8 + 416240000*x^6 + 890560000*x^4 + 968000000*x^2 + 387200000);
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 + 110*x^18 + 5060*x^16 + 127600*x^14 + 1944800*x^12 + 18612000*x^10 + 112288000*x^8 + 416240000*x^6 + 890560000*x^4 + 968000000*x^2 + 387200000);
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 cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.0.968000.5, \(\Q(\zeta_{11})^+\), 10.10.669871503125.1 |
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 | R | $20$ | R | $20$ | R | $20$ | $20$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $20$ | $2$ | $10$ | $30$ | |||
|
\(5\)
| Deg $20$ | $4$ | $5$ | $15$ | |||
|
\(11\)
| 11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |