Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 19*x^18 + 18*x^17 + 153*x^16 - 136*x^15 - 680*x^14 + 560*x^13 + 1820*x^12 - 1365*x^11 - 3003*x^10 + 2002*x^9 + 3003*x^8 - 1716*x^7 - 1716*x^6 + 792*x^5 + 495*x^4 - 165*x^3 - 55*x^2 + 10*x + 1)
gp: K = bnfinit(y^20 - y^19 - 19*y^18 + 18*y^17 + 153*y^16 - 136*y^15 - 680*y^14 + 560*y^13 + 1820*y^12 - 1365*y^11 - 3003*y^10 + 2002*y^9 + 3003*y^8 - 1716*y^7 - 1716*y^6 + 792*y^5 + 495*y^4 - 165*y^3 - 55*y^2 + 10*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 - 19*x^18 + 18*x^17 + 153*x^16 - 136*x^15 - 680*x^14 + 560*x^13 + 1820*x^12 - 1365*x^11 - 3003*x^10 + 2002*x^9 + 3003*x^8 - 1716*x^7 - 1716*x^6 + 792*x^5 + 495*x^4 - 165*x^3 - 55*x^2 + 10*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 - 19*x^18 + 18*x^17 + 153*x^16 - 136*x^15 - 680*x^14 + 560*x^13 + 1820*x^12 - 1365*x^11 - 3003*x^10 + 2002*x^9 + 3003*x^8 - 1716*x^7 - 1716*x^6 + 792*x^5 + 495*x^4 - 165*x^3 - 55*x^2 + 10*x + 1)
\( x^{20} - x^{19} - 19 x^{18} + 18 x^{17} + 153 x^{16} - 136 x^{15} - 680 x^{14} + 560 x^{13} + 1820 x^{12} + \cdots + 1 \)
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: | $[20, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4394336169668803158610484050361\)
\(\medspace = 41^{19}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.05\) | 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: | $41^{19/20}\approx 34.052159886341$ | ||
| Ramified primes: |
\(41\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{41}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{41}(1,·)$, $\chi_{41}(2,·)$, $\chi_{41}(4,·)$, $\chi_{41}(5,·)$, $\chi_{41}(8,·)$, $\chi_{41}(9,·)$, $\chi_{41}(10,·)$, $\chi_{41}(16,·)$, $\chi_{41}(18,·)$, $\chi_{41}(20,·)$, $\chi_{41}(21,·)$, $\chi_{41}(23,·)$, $\chi_{41}(25,·)$, $\chi_{41}(31,·)$, $\chi_{41}(32,·)$, $\chi_{41}(33,·)$, $\chi_{41}(36,·)$, $\chi_{41}(37,·)$, $\chi_{41}(39,·)$, $\chi_{41}(40,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$
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: | $19$ | 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: |
$a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+2$, $a^{7}-7a^{5}+14a^{3}-7a$, $a^{19}-18a^{17}+136a^{15}-560a^{13}+1365a^{11}-2002a^{9}+1716a^{7}-792a^{5}+165a^{3}-10a$, $a^{13}-12a^{11}+55a^{9}-120a^{7}+126a^{5}-56a^{3}+7a$, $a^{19}-a^{18}-18a^{17}+18a^{16}+135a^{15}-135a^{14}-545a^{13}+545a^{12}+1275a^{11}-1275a^{10}-1728a^{9}+1728a^{8}+1275a^{7}-1275a^{6}-441a^{5}+441a^{4}+54a^{3}-54a^{2}-a+1$, $a-1$, $a^{18}-17a^{16}+120a^{14}-455a^{12}+1001a^{10}-1288a^{8}+931a^{6}-345a^{4}-a^{3}+55a^{2}+2a-2$, $a^{10}-a^{9}-9a^{8}+8a^{7}+28a^{6}-21a^{5}-35a^{4}+20a^{3}+15a^{2}-5a-1$, $a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a$, $a^{15}-a^{14}-14a^{13}+13a^{12}+78a^{11}-66a^{10}-220a^{9}+165a^{8}+330a^{7}-210a^{6}-252a^{5}+126a^{4}+84a^{3}-28a^{2}-8a+1$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-a-2$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-2$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{3}-3a$, $a$, $a^{12}-12a^{10}+54a^{8}-112a^{6}-a^{5}+105a^{4}+5a^{3}-36a^{2}-5a+2$, $a^{3}-2a$, $a^{8}-8a^{6}+20a^{4}-16a^{2}+2$, $a^{17}-a^{16}-16a^{15}+15a^{14}+105a^{13}-91a^{12}-364a^{11}+287a^{10}+714a^{9}-504a^{8}-784a^{7}+490a^{6}+441a^{5}-245a^{4}-99a^{3}+50a^{2}+2a-1$
| 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: | \( 592074817.62 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 592074817.62 \cdot 1}{2\cdot\sqrt{4394336169668803158610484050361}}\cr\approx \mathstrut & 0.14808118074 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 19*x^18 + 18*x^17 + 153*x^16 - 136*x^15 - 680*x^14 + 560*x^13 + 1820*x^12 - 1365*x^11 - 3003*x^10 + 2002*x^9 + 3003*x^8 - 1716*x^7 - 1716*x^6 + 792*x^5 + 495*x^4 - 165*x^3 - 55*x^2 + 10*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 - 19*x^18 + 18*x^17 + 153*x^16 - 136*x^15 - 680*x^14 + 560*x^13 + 1820*x^12 - 1365*x^11 - 3003*x^10 + 2002*x^9 + 3003*x^8 - 1716*x^7 - 1716*x^6 + 792*x^5 + 495*x^4 - 165*x^3 - 55*x^2 + 10*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 - 19*x^18 + 18*x^17 + 153*x^16 - 136*x^15 - 680*x^14 + 560*x^13 + 1820*x^12 - 1365*x^11 - 3003*x^10 + 2002*x^9 + 3003*x^8 - 1716*x^7 - 1716*x^6 + 792*x^5 + 495*x^4 - 165*x^3 - 55*x^2 + 10*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 - 19*x^18 + 18*x^17 + 153*x^16 - 136*x^15 - 680*x^14 + 560*x^13 + 1820*x^12 - 1365*x^11 - 3003*x^10 + 2002*x^9 + 3003*x^8 - 1716*x^7 - 1716*x^6 + 792*x^5 + 495*x^4 - 165*x^3 - 55*x^2 + 10*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 cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{41}) \), 4.4.68921.1, 5.5.2825761.1, 10.10.327381934393961.1 |
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 | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{5}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | $20$ | $20$ | $20$ | $20$ | $20$ | ${\href{/padicField/23.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/43.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(41\)
| 41.20.19.1 | $x^{20} + 41$ | $20$ | $1$ | $19$ | 20T1 | $[\ ]_{20}$ |