Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 3*x^18 + 12*x^17 + 7*x^16 - 52*x^15 + 19*x^14 + 134*x^13 - 150*x^12 - 154*x^11 + 362*x^10 - 36*x^9 - 384*x^8 + 260*x^7 + 134*x^6 - 220*x^5 + 53*x^4 + 48*x^3 - 37*x^2 + 10*x - 1)
gp: K = bnfinit(y^20 - 2*y^19 - 3*y^18 + 12*y^17 + 7*y^16 - 52*y^15 + 19*y^14 + 134*y^13 - 150*y^12 - 154*y^11 + 362*y^10 - 36*y^9 - 384*y^8 + 260*y^7 + 134*y^6 - 220*y^5 + 53*y^4 + 48*y^3 - 37*y^2 + 10*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 - 3*x^18 + 12*x^17 + 7*x^16 - 52*x^15 + 19*x^14 + 134*x^13 - 150*x^12 - 154*x^11 + 362*x^10 - 36*x^9 - 384*x^8 + 260*x^7 + 134*x^6 - 220*x^5 + 53*x^4 + 48*x^3 - 37*x^2 + 10*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 - 3*x^18 + 12*x^17 + 7*x^16 - 52*x^15 + 19*x^14 + 134*x^13 - 150*x^12 - 154*x^11 + 362*x^10 - 36*x^9 - 384*x^8 + 260*x^7 + 134*x^6 - 220*x^5 + 53*x^4 + 48*x^3 - 37*x^2 + 10*x - 1)
\( x^{20} - 2 x^{19} - 3 x^{18} + 12 x^{17} + 7 x^{16} - 52 x^{15} + 19 x^{14} + 134 x^{13} - 150 x^{12} + \cdots - 1 \)
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: | $[2, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-229772481637155475554304\)
\(\medspace = -\,2^{20}\cdot 503\cdot 617\cdot 840277^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.73\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(503\), \(617\), \(840277\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-310351}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$
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: | $10$ | 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: |
$2584a^{19}-3571a^{18}-9959a^{17}+24853a^{16}+33448a^{15}-113696a^{14}-21172a^{13}+333171a^{12}-181689a^{11}-510226a^{10}+620071a^{9}+290201a^{8}-812902a^{7}+169439a^{6}+450975a^{5}-289762a^{4}-42131a^{3}+97994a^{2}-35045a+4181$, $8239a^{19}-12597a^{18}-30693a^{17}+84451a^{16}+97634a^{15}-382769a^{14}-24449a^{13}+1094087a^{12}-719467a^{11}-1612753a^{10}+2223683a^{9}+759623a^{8}-2812424a^{7}+809785a^{6}+1495645a^{5}-1106439a^{4}-91303a^{3}+354285a^{2}-136439a+17166$, $10799a^{19}-16135a^{18}-40563a^{17}+109075a^{16}+130786a^{15}-495430a^{14}-45488a^{13}+1424257a^{12}-899350a^{11}-2118577a^{10}+2837898a^{9}+1047748a^{8}-3617974a^{7}+976982a^{6}+1942872a^{5}-1393189a^{4}-133392a^{3}+451353a^{2}-171056a+21280$, $175a^{19}-307a^{18}-602a^{17}+1955a^{16}+1711a^{15}-8698a^{14}+1172a^{13}+23821a^{12}-20406a^{11}-32193a^{10}+55628a^{9}+7696a^{8}-65832a^{7}+29200a^{6}+31276a^{5}-31028a^{4}+1295a^{3}+8981a^{2}-4227a+624$, $12013a^{19}-18157a^{18}-44894a^{17}+122211a^{16}+143729a^{15}-554348a^{14}-42308a^{13}+1588548a^{12}-1026327a^{11}-2349663a^{10}+3200779a^{9}+1128115a^{8}-4060050a^{7}+1142834a^{6}+2164930a^{5}-1586201a^{4}-136046a^{3}+509793a^{2}-195956a+24665$, $a^{19}-2a^{18}-3a^{17}+12a^{16}+7a^{15}-52a^{14}+19a^{13}+134a^{12}-150a^{11}-154a^{10}+362a^{9}-36a^{8}-384a^{7}+260a^{6}+134a^{5}-220a^{4}+53a^{3}+48a^{2}-37a+9$, $9150a^{19}-13530a^{18}-34516a^{17}+91817a^{16}+111970a^{15}-417521a^{14}-44023a^{13}+1203592a^{12}-744708a^{11}-1798776a^{10}+2374680a^{9}+911120a^{8}-3040279a^{7}+791730a^{6}+1641586a^{5}-1156567a^{4}-119861a^{3}+377133a^{2}-141510a+17479$, $18656a^{19}-27885a^{18}-70033a^{17}+188462a^{16}+225713a^{15}-855870a^{14}-77583a^{13}+2459802a^{12}-1556132a^{11}-3656409a^{10}+4905621a^{9}+1802027a^{8}-6249931a^{7}+1697131a^{6}+3351877a^{5}-2411898a^{4}-226163a^{3}+780352a^{2}-296838a+37074$, $21151a^{19}-32220a^{18}-78803a^{17}+216244a^{16}+251097a^{15}-980110a^{14}-65168a^{13}+2802916a^{12}-1836866a^{11}-4131949a^{10}+5687209a^{9}+1947844a^{8}-7192776a^{7}+2072333a^{6}+3820627a^{5}-2832738a^{4}-228233a^{3}+906387a^{2}-350811a+44395$, $14439a^{19}-21698a^{18}-54118a^{17}+146366a^{16}+173904a^{15}-664431a^{14}-56249a^{13}+1907234a^{12}-1217138a^{11}-2830094a^{10}+3819662a^{9}+1381774a^{8}-4858814a^{7}+1336030a^{6}+2601404a^{5}-1882402a^{4}-172269a^{3}+607703a^{2}-231716a+28983$
| 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: | \( 2859.44274827 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 2859.44274827 \cdot 1}{2\cdot\sqrt{229772481637155475554304}}\cr\approx \mathstrut & 0.182087951041 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 3*x^18 + 12*x^17 + 7*x^16 - 52*x^15 + 19*x^14 + 134*x^13 - 150*x^12 - 154*x^11 + 362*x^10 - 36*x^9 - 384*x^8 + 260*x^7 + 134*x^6 - 220*x^5 + 53*x^4 + 48*x^3 - 37*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 - 2*x^19 - 3*x^18 + 12*x^17 + 7*x^16 - 52*x^15 + 19*x^14 + 134*x^13 - 150*x^12 - 154*x^11 + 362*x^10 - 36*x^9 - 384*x^8 + 260*x^7 + 134*x^6 - 220*x^5 + 53*x^4 + 48*x^3 - 37*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 - 2*x^19 - 3*x^18 + 12*x^17 + 7*x^16 - 52*x^15 + 19*x^14 + 134*x^13 - 150*x^12 - 154*x^11 + 362*x^10 - 36*x^9 - 384*x^8 + 260*x^7 + 134*x^6 - 220*x^5 + 53*x^4 + 48*x^3 - 37*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 - 2*x^19 - 3*x^18 + 12*x^17 + 7*x^16 - 52*x^15 + 19*x^14 + 134*x^13 - 150*x^12 - 154*x^11 + 362*x^10 - 36*x^9 - 384*x^8 + 260*x^7 + 134*x^6 - 220*x^5 + 53*x^4 + 48*x^3 - 37*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
$C_2^{10}.S_{10}$ (as 20T1110):
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 3715891200 |
| The 481 conjugacy class representatives for $C_2^{10}.S_{10}$ |
| Character table for $C_2^{10}.S_{10}$ |
Intermediate fields
| 10.2.860443648.1 |
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]
Sibling fields
| Degree 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | $16{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | $20$ | ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.7.0.1}{7} }^{2}{,}\,{\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.9.0.1}{9} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 $20$ | $2$ | $10$ | $20$ | |||
|
\(503\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
|
\(617\)
| $\Q_{617}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{617}$ | $x$ | $1$ | $1$ | $0$ | 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
|
\(840277\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |