Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^18 + 6*x^17 - 5*x^16 - 5*x^15 + 17*x^14 - 9*x^13 - 10*x^12 + 24*x^11 - 13*x^10 - 9*x^9 + 30*x^8 - 26*x^7 + x^6 + 25*x^5 - 16*x^4 - 4*x^3 + 11*x^2 - 4*x + 1)
gp: K = bnfinit(y^20 - y^19 - y^18 + 6*y^17 - 5*y^16 - 5*y^15 + 17*y^14 - 9*y^13 - 10*y^12 + 24*y^11 - 13*y^10 - 9*y^9 + 30*y^8 - 26*y^7 + y^6 + 25*y^5 - 16*y^4 - 4*y^3 + 11*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 - x^18 + 6*x^17 - 5*x^16 - 5*x^15 + 17*x^14 - 9*x^13 - 10*x^12 + 24*x^11 - 13*x^10 - 9*x^9 + 30*x^8 - 26*x^7 + x^6 + 25*x^5 - 16*x^4 - 4*x^3 + 11*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 - x^18 + 6*x^17 - 5*x^16 - 5*x^15 + 17*x^14 - 9*x^13 - 10*x^12 + 24*x^11 - 13*x^10 - 9*x^9 + 30*x^8 - 26*x^7 + x^6 + 25*x^5 - 16*x^4 - 4*x^3 + 11*x^2 - 4*x + 1)
\( x^{20} - x^{19} - x^{18} + 6 x^{17} - 5 x^{16} - 5 x^{15} + 17 x^{14} - 9 x^{13} - 10 x^{12} + 24 x^{11} + \cdots + 1 \)
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: |
\(686255883923847255777801\)
\(\medspace = 3^{10}\cdot 7^{8}\cdot 17^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.55\) | 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: | $3^{1/2}7^{1/2}17^{1/2}\approx 18.894443627691185$ | ||
| Ramified primes: |
\(3\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{4199}a^{18}-\frac{1226}{4199}a^{17}+\frac{1110}{4199}a^{16}+\frac{1523}{4199}a^{15}+\frac{1467}{4199}a^{14}-\frac{531}{4199}a^{13}+\frac{1622}{4199}a^{12}+\frac{1158}{4199}a^{11}+\frac{135}{4199}a^{10}-\frac{426}{4199}a^{9}-\frac{81}{323}a^{8}+\frac{1091}{4199}a^{7}+\frac{150}{4199}a^{6}-\frac{1796}{4199}a^{5}+\frac{92}{247}a^{4}+\frac{610}{4199}a^{3}+\frac{1380}{4199}a^{2}+\frac{87}{4199}a+\frac{973}{4199}$, $\frac{1}{348517}a^{19}-\frac{6}{348517}a^{18}+\frac{9062}{26809}a^{17}-\frac{554}{348517}a^{16}+\frac{41360}{348517}a^{15}+\frac{29828}{348517}a^{14}+\frac{113821}{348517}a^{13}-\frac{27124}{348517}a^{12}-\frac{165929}{348517}a^{11}-\frac{5277}{18343}a^{10}+\frac{20898}{348517}a^{9}-\frac{91053}{348517}a^{8}+\frac{159649}{348517}a^{7}+\frac{139214}{348517}a^{6}-\frac{106852}{348517}a^{5}-\frac{140422}{348517}a^{4}-\frac{157205}{348517}a^{3}+\frac{138455}{348517}a^{2}+\frac{128108}{348517}a-\frac{152421}{348517}$
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
Trivial group, which has order $1$
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: |
\( -\frac{5818}{18343} a^{19} + \frac{9759}{18343} a^{18} + \frac{7321}{18343} a^{17} - \frac{39226}{18343} a^{16} + \frac{44280}{18343} a^{15} + \frac{34695}{18343} a^{14} - \frac{7034}{1079} a^{13} + \frac{79129}{18343} a^{12} + \frac{80689}{18343} a^{11} - \frac{162356}{18343} a^{10} + \frac{104060}{18343} a^{9} + \frac{68491}{18343} a^{8} - \frac{201595}{18343} a^{7} + \frac{196194}{18343} a^{6} - \frac{8882}{18343} a^{5} - \frac{174412}{18343} a^{4} + \frac{122283}{18343} a^{3} + \frac{56020}{18343} a^{2} - \frac{79743}{18343} a + \frac{29314}{18343} \)
(order $6$)
| 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: |
$a$, $\frac{5166}{18343}a^{19}-\frac{2031}{348517}a^{18}-\frac{158632}{348517}a^{17}+\frac{412810}{348517}a^{16}+\frac{15358}{348517}a^{15}-\frac{715798}{348517}a^{14}+\frac{905048}{348517}a^{13}+\frac{481303}{348517}a^{12}-\frac{1078746}{348517}a^{11}+\frac{1022774}{348517}a^{10}+\frac{32195}{26809}a^{9}-\frac{1317970}{348517}a^{8}+\frac{1635194}{348517}a^{7}-\frac{35914}{348517}a^{6}-\frac{1219098}{348517}a^{5}+\frac{1470270}{348517}a^{4}+\frac{716893}{348517}a^{3}-\frac{898030}{348517}a^{2}+\frac{343998}{348517}a+\frac{18036}{26809}$, $\frac{114652}{348517}a^{19}-\frac{138950}{348517}a^{18}-\frac{123908}{348517}a^{17}+\frac{750024}{348517}a^{16}-\frac{635773}{348517}a^{15}-\frac{601515}{348517}a^{14}+\frac{171981}{26809}a^{13}-\frac{1093779}{348517}a^{12}-\frac{102602}{26809}a^{11}+\frac{3143654}{348517}a^{10}-\frac{1453852}{348517}a^{9}-\frac{1191689}{348517}a^{8}+\frac{3963331}{348517}a^{7}-\frac{2911886}{348517}a^{6}-\frac{55596}{348517}a^{5}+\frac{3395521}{348517}a^{4}-\frac{1763090}{348517}a^{3}-\frac{915088}{348517}a^{2}+\frac{1344901}{348517}a-\frac{8927}{18343}$, $\frac{38549}{348517}a^{19}+\frac{15050}{348517}a^{18}-\frac{4414}{20501}a^{17}+\frac{107903}{348517}a^{16}+\frac{98985}{348517}a^{15}-\frac{17177}{20501}a^{14}+\frac{90427}{348517}a^{13}+\frac{465927}{348517}a^{12}-\frac{233591}{348517}a^{11}-\frac{186049}{348517}a^{10}+\frac{483805}{348517}a^{9}-\frac{190974}{348517}a^{8}-\frac{97705}{348517}a^{7}+\frac{453035}{348517}a^{6}-\frac{429593}{348517}a^{5}-\frac{139420}{348517}a^{4}+\frac{1022615}{348517}a^{3}-\frac{433932}{348517}a^{2}-\frac{575241}{348517}a+\frac{232507}{348517}$, $\frac{50599}{348517}a^{19}-\frac{8822}{26809}a^{18}-\frac{6377}{348517}a^{17}+\frac{427194}{348517}a^{16}-\frac{586620}{348517}a^{15}-\frac{103134}{348517}a^{14}+\frac{1442170}{348517}a^{13}-\frac{1319065}{348517}a^{12}-\frac{528670}{348517}a^{11}+\frac{135669}{20501}a^{10}-\frac{1692642}{348517}a^{9}-\frac{409958}{348517}a^{8}+\frac{2717425}{348517}a^{7}-\frac{216787}{26809}a^{6}+\frac{813697}{348517}a^{5}+\frac{2352616}{348517}a^{4}-\frac{2428653}{348517}a^{3}-\frac{201865}{348517}a^{2}+\frac{1517774}{348517}a-\frac{575495}{348517}$, $\frac{42261}{348517}a^{19}+\frac{4521}{20501}a^{18}-\frac{87823}{348517}a^{17}+\frac{5207}{26809}a^{16}+\frac{426403}{348517}a^{15}-\frac{430004}{348517}a^{14}-\frac{192915}{348517}a^{13}+\frac{1306077}{348517}a^{12}-\frac{553778}{348517}a^{11}-\frac{645062}{348517}a^{10}+\frac{95345}{18343}a^{9}-\frac{827123}{348517}a^{8}-\frac{586333}{348517}a^{7}+\frac{127795}{20501}a^{6}-\frac{1595445}{348517}a^{5}-\frac{45603}{26809}a^{4}+\frac{2693889}{348517}a^{3}-\frac{927205}{348517}a^{2}-\frac{768034}{348517}a+\frac{693252}{348517}$, $\frac{489}{4199}a^{19}-\frac{1532}{4199}a^{18}-\frac{24}{247}a^{17}+\frac{4619}{4199}a^{16}-\frac{599}{323}a^{15}-\frac{10}{19}a^{14}+\frac{16281}{4199}a^{13}-\frac{17605}{4199}a^{12}-\frac{7700}{4199}a^{11}+\frac{24426}{4199}a^{10}-\frac{23233}{4199}a^{9}-\frac{283}{221}a^{8}+\frac{26893}{4199}a^{7}-\frac{35983}{4199}a^{6}+\frac{7335}{4199}a^{5}+\frac{25933}{4199}a^{4}-\frac{32921}{4199}a^{3}-\frac{1160}{4199}a^{2}+\frac{12731}{4199}a-\frac{6347}{4199}$, $\frac{112441}{348517}a^{19}-\frac{91405}{348517}a^{18}-\frac{106272}{348517}a^{17}+\frac{49399}{26809}a^{16}-\frac{493395}{348517}a^{15}-\frac{580665}{348517}a^{14}+\frac{1779014}{348517}a^{13}-\frac{881204}{348517}a^{12}-\frac{67091}{20501}a^{11}+\frac{2568245}{348517}a^{10}-\frac{1271709}{348517}a^{9}-\frac{1166351}{348517}a^{8}+\frac{3088632}{348517}a^{7}-\frac{2710917}{348517}a^{6}+\frac{22075}{348517}a^{5}+\frac{10460}{1411}a^{4}-\frac{1655114}{348517}a^{3}-\frac{437425}{348517}a^{2}+\frac{1288054}{348517}a-\frac{286886}{348517}$, $\frac{102642}{348517}a^{19}-\frac{18418}{348517}a^{18}-\frac{173730}{348517}a^{17}+\frac{43502}{26809}a^{16}-\frac{5786}{20501}a^{15}-\frac{897880}{348517}a^{14}+\frac{1483809}{348517}a^{13}+\frac{52876}{348517}a^{12}-\frac{1675920}{348517}a^{11}+\frac{2052395}{348517}a^{10}-\frac{199093}{348517}a^{9}-\frac{1797556}{348517}a^{8}+\frac{2633675}{348517}a^{7}-\frac{1341449}{348517}a^{6}-\frac{1671717}{348517}a^{5}+\frac{221151}{26809}a^{4}-\frac{52980}{18343}a^{3}-\frac{660418}{348517}a^{2}+\frac{459685}{348517}a+\frac{108774}{348517}$
| 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: | \( 12584.3597894 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 12584.3597894 \cdot 1}{6\cdot\sqrt{686255883923847255777801}}\cr\approx \mathstrut & 0.242792640885 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^18 + 6*x^17 - 5*x^16 - 5*x^15 + 17*x^14 - 9*x^13 - 10*x^12 + 24*x^11 - 13*x^10 - 9*x^9 + 30*x^8 - 26*x^7 + x^6 + 25*x^5 - 16*x^4 - 4*x^3 + 11*x^2 - 4*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 - x^18 + 6*x^17 - 5*x^16 - 5*x^15 + 17*x^14 - 9*x^13 - 10*x^12 + 24*x^11 - 13*x^10 - 9*x^9 + 30*x^8 - 26*x^7 + x^6 + 25*x^5 - 16*x^4 - 4*x^3 + 11*x^2 - 4*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 - x^18 + 6*x^17 - 5*x^16 - 5*x^15 + 17*x^14 - 9*x^13 - 10*x^12 + 24*x^11 - 13*x^10 - 9*x^9 + 30*x^8 - 26*x^7 + x^6 + 25*x^5 - 16*x^4 - 4*x^3 + 11*x^2 - 4*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 - x^18 + 6*x^17 - 5*x^16 - 5*x^15 + 17*x^14 - 9*x^13 - 10*x^12 + 24*x^11 - 13*x^10 - 9*x^9 + 30*x^8 - 26*x^7 + x^6 + 25*x^5 - 16*x^4 - 4*x^3 + 11*x^2 - 4*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
$C_2\times D_{10}$ (as 20T8):
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 40 |
| The 16 conjugacy class representatives for $C_2\times D_{10}$ |
| Character table for $C_2\times D_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-3}, \sqrt{17})\), 5.1.14161.1, 10.0.828405627651.1, 10.0.48729742803.1, 10.2.3409076657.1 |
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
| Galois closure: | deg 40 |
| Degree 20 siblings: | 20.0.116354803848679984543641.1, 20.4.33626538312268515533112249.1, deg 20 |
| Minimal sibling: | 20.0.116354803848679984543641.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{10}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.2.0.1}{2} }^{10}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{10}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{10}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.20.10.1 | $x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ |
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(17\)
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |