Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + x^18 - 4*x^17 + 2*x^16 - 4*x^15 + 7*x^14 - 6*x^13 + 8*x^12 - 6*x^11 + 7*x^10 - 6*x^9 + 4*x^8 - 2*x^7 - 2*x^5 - x^4 + x^2 + 1)
gp: K = bnfinit(y^20 + y^18 - 4*y^17 + 2*y^16 - 4*y^15 + 7*y^14 - 6*y^13 + 8*y^12 - 6*y^11 + 7*y^10 - 6*y^9 + 4*y^8 - 2*y^7 - 2*y^5 - y^4 + y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + x^18 - 4*x^17 + 2*x^16 - 4*x^15 + 7*x^14 - 6*x^13 + 8*x^12 - 6*x^11 + 7*x^10 - 6*x^9 + 4*x^8 - 2*x^7 - 2*x^5 - x^4 + x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + x^18 - 4*x^17 + 2*x^16 - 4*x^15 + 7*x^14 - 6*x^13 + 8*x^12 - 6*x^11 + 7*x^10 - 6*x^9 + 4*x^8 - 2*x^7 - 2*x^5 - x^4 + x^2 + 1)
\( x^{20} + x^{18} - 4 x^{17} + 2 x^{16} - 4 x^{15} + 7 x^{14} - 6 x^{13} + 8 x^{12} - 6 x^{11} + 7 x^{10} + \cdots + 1 \)
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: |
\(3020631315406201552896\)
\(\medspace = 2^{20}\cdot 3^{10}\cdot 220873^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.86\) | 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: |
\(2\), \(3\), \(220873\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5}a^{18}+\frac{2}{5}a^{17}+\frac{1}{5}a^{16}-\frac{2}{5}a^{14}+\frac{2}{5}a^{13}-\frac{1}{5}a^{12}-\frac{1}{5}a^{11}-\frac{2}{5}a^{9}-\frac{2}{5}a^{8}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{1745}a^{19}+\frac{12}{349}a^{18}-\frac{238}{1745}a^{17}+\frac{723}{1745}a^{16}-\frac{592}{1745}a^{15}-\frac{624}{1745}a^{14}-\frac{18}{349}a^{13}-\frac{869}{1745}a^{12}+\frac{567}{1745}a^{11}-\frac{537}{1745}a^{10}-\frac{803}{1745}a^{9}-\frac{373}{1745}a^{8}-\frac{738}{1745}a^{7}+\frac{41}{1745}a^{6}-\frac{332}{1745}a^{5}+\frac{64}{349}a^{4}-\frac{694}{1745}a^{3}+\frac{589}{1745}a^{2}+\frac{158}{349}a-\frac{413}{1745}$
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 $1$ (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: |
\( \frac{57}{349} a^{19} - \frac{70}{349} a^{18} + \frac{45}{349} a^{17} - \frac{320}{349} a^{16} + \frac{458}{349} a^{15} - \frac{319}{349} a^{14} + \frac{803}{349} a^{13} - \frac{1022}{349} a^{12} + \frac{909}{349} a^{11} - \frac{944}{349} a^{10} + \frac{995}{349} a^{9} - \frac{1019}{349} a^{8} + \frac{512}{349} a^{7} - \frac{106}{349} a^{6} + \frac{271}{349} a^{5} + \frac{92}{349} a^{4} - \frac{121}{349} a^{3} + \frac{418}{349} a^{2} + \frac{9}{349} a - \frac{158}{349} \)
(order $12$)
| 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: |
$\frac{158}{349}a^{19}+\frac{57}{349}a^{18}+\frac{88}{349}a^{17}-\frac{587}{349}a^{16}-\frac{4}{349}a^{15}-\frac{174}{349}a^{14}+\frac{787}{349}a^{13}-\frac{145}{349}a^{12}+\frac{242}{349}a^{11}-\frac{39}{349}a^{10}+\frac{162}{349}a^{9}+\frac{47}{349}a^{8}-\frac{387}{349}a^{7}+\frac{196}{349}a^{6}-\frac{106}{349}a^{5}-\frac{45}{349}a^{4}-\frac{66}{349}a^{3}-\frac{121}{349}a^{2}+\frac{576}{349}a+\frac{9}{349}$, $\frac{1416}{1745}a^{19}+\frac{153}{1745}a^{18}+\frac{1173}{1745}a^{17}-\frac{5084}{1745}a^{16}+\frac{2818}{1745}a^{15}-\frac{751}{349}a^{14}+\frac{8321}{1745}a^{13}-\frac{7957}{1745}a^{12}+\frac{8199}{1745}a^{11}-\frac{6552}{1745}a^{10}+\frac{8021}{1745}a^{9}-\frac{6064}{1745}a^{8}+\frac{5831}{1745}a^{7}-\frac{185}{349}a^{6}+\frac{689}{1745}a^{5}-\frac{2674}{1745}a^{4}+\frac{778}{1745}a^{3}-\frac{784}{1745}a^{2}+\frac{1491}{1745}a-\frac{931}{1745}$, $\frac{588}{1745}a^{19}+\frac{1078}{1745}a^{18}+\frac{1052}{1745}a^{17}-\frac{1703}{1745}a^{16}-\frac{2586}{1745}a^{15}-\frac{1858}{1745}a^{14}+\frac{2571}{1745}a^{13}+\frac{272}{349}a^{12}+\frac{1148}{1745}a^{11}+\frac{89}{1745}a^{10}+\frac{565}{349}a^{9}+\frac{179}{349}a^{8}-\frac{167}{349}a^{7}+\frac{27}{1745}a^{6}-\frac{25}{349}a^{5}-\frac{649}{1745}a^{4}-\frac{437}{349}a^{3}+\frac{124}{1745}a^{2}+\frac{1}{1745}a+\frac{758}{1745}$, $\frac{1114}{1745}a^{19}+\frac{879}{1745}a^{18}+\frac{806}{1745}a^{17}-\frac{3909}{1745}a^{16}-\frac{1623}{1745}a^{15}-\frac{1324}{1745}a^{14}+\frac{5138}{1745}a^{13}+\frac{12}{349}a^{12}+\frac{1344}{1745}a^{11}+\frac{317}{1745}a^{10}+\frac{338}{349}a^{9}+\frac{167}{349}a^{8}-\frac{536}{349}a^{7}+\frac{3096}{1745}a^{6}-\frac{540}{349}a^{5}-\frac{547}{1745}a^{4}-\frac{784}{349}a^{3}+\frac{1422}{1745}a^{2}+\frac{1278}{1745}a+\frac{249}{1745}$, $\frac{427}{1745}a^{19}-\frac{1253}{1745}a^{18}-\frac{67}{1745}a^{17}-\frac{2587}{1745}a^{16}+\frac{5476}{1745}a^{15}-\frac{1557}{1745}a^{14}+\frac{5544}{1745}a^{13}-\frac{1830}{349}a^{12}+\frac{5487}{1745}a^{11}-\frac{5939}{1745}a^{10}+\frac{1154}{349}a^{9}-\frac{1212}{349}a^{8}+\frac{423}{349}a^{7}-\frac{292}{1745}a^{6}-\frac{14}{349}a^{5}+\frac{879}{1745}a^{4}+\frac{202}{349}a^{3}+\frac{2666}{1745}a^{2}+\frac{894}{1745}a-\frac{2898}{1745}$, $\frac{1098}{1745}a^{19}-\frac{86}{349}a^{18}+\frac{426}{1745}a^{17}-\frac{5356}{1745}a^{16}+\frac{2614}{1745}a^{15}-\frac{2857}{1745}a^{14}+\frac{1874}{349}a^{13}-\frac{6627}{1745}a^{12}+\frac{6581}{1745}a^{11}-\frac{6796}{1745}a^{10}+\frac{6511}{1745}a^{9}-\frac{6459}{1745}a^{8}+\frac{1101}{1745}a^{7}-\frac{2097}{1745}a^{6}-\frac{1576}{1745}a^{5}-\frac{575}{349}a^{4}-\frac{1192}{1745}a^{3}+\frac{2817}{1745}a^{2}+\frac{729}{349}a+\frac{226}{1745}$, $\frac{1627}{1745}a^{19}+\frac{1296}{1745}a^{18}+\frac{1211}{1745}a^{17}-\frac{5393}{1745}a^{16}-\frac{1689}{1745}a^{15}-\frac{490}{349}a^{14}+\frac{6432}{1745}a^{13}-\frac{1809}{1745}a^{12}+\frac{3243}{1745}a^{11}+\frac{546}{1745}a^{10}+\frac{4712}{1745}a^{9}-\frac{658}{1745}a^{8}-\frac{1213}{1745}a^{7}+\frac{219}{349}a^{6}-\frac{1657}{1745}a^{5}-\frac{3558}{1745}a^{4}-\frac{3264}{1745}a^{3}-\frac{1098}{1745}a^{2}+\frac{2057}{1745}a+\frac{223}{1745}$, $\frac{177}{349}a^{19}+\frac{401}{1745}a^{18}+\frac{1562}{1745}a^{17}-\frac{2654}{1745}a^{16}+\frac{265}{349}a^{15}-\frac{3612}{1745}a^{14}+\frac{5157}{1745}a^{13}-\frac{4406}{1745}a^{12}+\frac{4819}{1745}a^{11}-\frac{819}{349}a^{10}+\frac{5493}{1745}a^{9}-\frac{1347}{1745}a^{8}+\frac{3688}{1745}a^{7}+\frac{2083}{1745}a^{6}+\frac{2132}{1745}a^{5}-\frac{188}{1745}a^{4}-\frac{1346}{1745}a^{3}-\frac{1886}{1745}a^{2}+\frac{452}{1745}a-\frac{2196}{1745}$, $\frac{529}{1745}a^{19}+\frac{679}{1745}a^{18}+\frac{436}{1745}a^{17}-\frac{1084}{1745}a^{16}-\frac{813}{1745}a^{15}+\frac{756}{1745}a^{14}+\frac{203}{1745}a^{13}+\frac{126}{349}a^{12}-\frac{2291}{1745}a^{11}+\frac{3852}{1745}a^{10}-\frac{290}{349}a^{9}+\frac{881}{349}a^{8}-\frac{393}{349}a^{7}+\frac{5286}{1745}a^{6}-\frac{435}{349}a^{5}+\frac{713}{1745}a^{4}-\frac{205}{349}a^{3}+\frac{622}{1745}a^{2}-\frac{192}{1745}a-\frac{701}{1745}$
| 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: | \( 1171.34610492 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 1171.34610492 \cdot 1}{12\cdot\sqrt{3020631315406201552896}}\cr\approx \mathstrut & 0.170315359279 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 + x^18 - 4*x^17 + 2*x^16 - 4*x^15 + 7*x^14 - 6*x^13 + 8*x^12 - 6*x^11 + 7*x^10 - 6*x^9 + 4*x^8 - 2*x^7 - 2*x^5 - x^4 + x^2 + 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^18 - 4*x^17 + 2*x^16 - 4*x^15 + 7*x^14 - 6*x^13 + 8*x^12 - 6*x^11 + 7*x^10 - 6*x^9 + 4*x^8 - 2*x^7 - 2*x^5 - x^4 + x^2 + 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^18 - 4*x^17 + 2*x^16 - 4*x^15 + 7*x^14 - 6*x^13 + 8*x^12 - 6*x^11 + 7*x^10 - 6*x^9 + 4*x^8 - 2*x^7 - 2*x^5 - x^4 + x^2 + 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^18 - 4*x^17 + 2*x^16 - 4*x^15 + 7*x^14 - 6*x^13 + 8*x^12 - 6*x^11 + 7*x^10 - 6*x^9 + 4*x^8 - 2*x^7 - 2*x^5 - x^4 + x^2 + 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
$S_5^2:C_2^2$ (as 20T656):
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 57600 |
| The 70 conjugacy class representatives for $S_5^2:C_2^2$ |
| Character table for $S_5^2:C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 10.0.226173952.1 |
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 24 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 20.0.32548097666553954221423000898562424832.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{5}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $20$ | $2$ | $10$ | $20$ | |||
|
\(3\)
| 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 | $[\ ]_{2}^{10}$ |
|
\(220873\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |