Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^18 + 21*x^16 - 22*x^14 + 98*x^12 - 176*x^10 + 205*x^8 - 152*x^6 + 76*x^4 - 24*x^2 + 4)
gp: K = bnfinit(y^20 - 8*y^18 + 21*y^16 - 22*y^14 + 98*y^12 - 176*y^10 + 205*y^8 - 152*y^6 + 76*y^4 - 24*y^2 + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 8*x^18 + 21*x^16 - 22*x^14 + 98*x^12 - 176*x^10 + 205*x^8 - 152*x^6 + 76*x^4 - 24*x^2 + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 8*x^18 + 21*x^16 - 22*x^14 + 98*x^12 - 176*x^10 + 205*x^8 - 152*x^6 + 76*x^4 - 24*x^2 + 4)
\( x^{20} - 8x^{18} + 21x^{16} - 22x^{14} + 98x^{12} - 176x^{10} + 205x^{8} - 152x^{6} + 76x^{4} - 24x^{2} + 4 \)
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: |
\(17313300662569099483282407424\)
\(\medspace = 2^{42}\cdot 89^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(25.82\) | 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: |
\(2\), \(89\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4}a^{16}-\frac{1}{2}a^{15}-\frac{1}{4}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{13}-\frac{1}{2}a^{11}+\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{616684}a^{18}-\frac{31761}{308342}a^{16}-\frac{118741}{616684}a^{14}+\frac{71804}{154171}a^{12}+\frac{50205}{154171}a^{10}-\frac{1621}{154171}a^{8}-\frac{119931}{616684}a^{6}+\frac{8307}{308342}a^{4}+\frac{116573}{308342}a^{2}-\frac{54715}{154171}$, $\frac{1}{616684}a^{19}-\frac{31761}{308342}a^{17}-\frac{118741}{616684}a^{15}+\frac{71804}{154171}a^{13}+\frac{50205}{154171}a^{11}-\frac{1621}{154171}a^{9}-\frac{119931}{616684}a^{7}+\frac{8307}{308342}a^{5}+\frac{116573}{308342}a^{3}-\frac{54715}{154171}a$
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
$C_{2}$, which has order $2$ (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: |
$\frac{227943}{308342}a^{18}-\frac{906660}{154171}a^{16}+\frac{4706127}{308342}a^{14}-\frac{2385967}{154171}a^{12}+\frac{10938624}{154171}a^{10}-\frac{19320978}{154171}a^{8}+\frac{44879377}{308342}a^{6}-\frac{14806137}{154171}a^{4}+\frac{6331816}{154171}a^{2}-\frac{1247255}{154171}$, $\frac{571941}{616684}a^{18}-\frac{2532591}{308342}a^{16}+\frac{15513003}{616684}a^{14}-\frac{4890375}{154171}a^{12}+\frac{14595400}{154171}a^{10}-\frac{35237126}{154171}a^{8}+\frac{165576561}{616684}a^{6}-\frac{58712971}{308342}a^{4}+\frac{24138209}{308342}a^{2}-\frac{2434800}{154171}$, $\frac{185455}{308342}a^{18}-\frac{1345597}{308342}a^{16}+\frac{2831879}{308342}a^{14}-\frac{1516417}{308342}a^{12}+\frac{8009207}{154171}a^{10}-\frac{9999325}{154171}a^{8}+\frac{17713717}{308342}a^{6}-\frac{7358273}{308342}a^{4}+\frac{571411}{154171}a^{2}+\frac{113106}{154171}$, $\frac{166267}{308342}a^{18}-\frac{1053045}{308342}a^{16}+\frac{1353439}{308342}a^{14}+\frac{1462961}{308342}a^{12}+\frac{6167062}{154171}a^{10}-\frac{2368363}{154171}a^{8}-\frac{6873961}{308342}a^{6}+\frac{12103469}{308342}a^{4}-\frac{3681062}{154171}a^{2}+\frac{1280294}{154171}$, $\frac{252636}{154171}a^{19}-\frac{268019}{308342}a^{18}-\frac{8076103}{616684}a^{17}+\frac{4469735}{616684}a^{16}+\frac{10396581}{308342}a^{15}-\frac{6225607}{308342}a^{14}-\frac{19348865}{616684}a^{13}+\frac{13583373}{616684}a^{12}+\frac{23202832}{154171}a^{11}-\frac{25759301}{308342}a^{10}-\frac{43347000}{154171}a^{9}+\frac{26989767}{154171}a^{8}+\frac{43660565}{154171}a^{7}-\frac{59451791}{308342}a^{6}-\frac{105651019}{616684}a^{5}+\frac{77654667}{616684}a^{4}+\frac{19355987}{308342}a^{3}-\frac{7655090}{154171}a^{2}-\frac{3292631}{308342}a+\frac{2893851}{308342}$, $\frac{571941}{616684}a^{19}-\frac{2532591}{308342}a^{17}+\frac{15513003}{616684}a^{15}-\frac{4890375}{154171}a^{13}+\frac{14595400}{154171}a^{11}-\frac{35237126}{154171}a^{9}+\frac{165576561}{616684}a^{7}-\frac{58712971}{308342}a^{5}+\frac{24138209}{308342}a^{3}-\frac{2588971}{154171}a+1$, $\frac{115579}{616684}a^{19}+\frac{145459}{154171}a^{18}-\frac{1265415}{616684}a^{17}-\frac{4443521}{616684}a^{16}+\frac{4961511}{616684}a^{15}+\frac{2605089}{154171}a^{14}-\frac{8132719}{616684}a^{13}-\frac{7676227}{616684}a^{12}+\frac{3655701}{154171}a^{11}+\frac{25723893}{308342}a^{10}-\frac{12369474}{154171}a^{9}-\frac{20596892}{154171}a^{8}+\frac{65686407}{616684}a^{7}+\frac{18368354}{154171}a^{6}-\frac{51771755}{616684}a^{5}-\frac{41678737}{616684}a^{4}+\frac{5666609}{154171}a^{3}+\frac{3272564}{154171}a^{2}-\frac{2858721}{308342}a-\frac{1236413}{308342}$, $\frac{9648}{154171}a^{19}-\frac{29686}{154171}a^{18}+\frac{32047}{616684}a^{17}+\frac{202762}{154171}a^{16}-\frac{430980}{154171}a^{15}-\frac{328760}{154171}a^{14}+\frac{4740957}{616684}a^{13}-\frac{175363}{154171}a^{12}+\frac{242977}{308342}a^{11}-\frac{2062515}{154171}a^{10}+\frac{5585950}{154171}a^{9}+\frac{2082839}{154171}a^{8}-\frac{8520338}{154171}a^{7}+\frac{1234131}{154171}a^{6}+\frac{28337763}{616684}a^{5}-\frac{2476911}{154171}a^{4}-\frac{2737360}{154171}a^{3}+\frac{1414086}{154171}a^{2}+\frac{1008669}{308342}a-\frac{304664}{154171}$, $\frac{590848}{154171}a^{19}+\frac{192239}{308342}a^{18}-\frac{9704579}{308342}a^{17}-\frac{4129339}{616684}a^{16}+\frac{26250735}{308342}a^{15}+\frac{7924053}{308342}a^{14}-\frac{26887933}{308342}a^{13}-\frac{25267137}{616684}a^{12}+\frac{111019577}{308342}a^{11}+\frac{23372375}{308342}a^{10}-\frac{112453912}{154171}a^{9}-\frac{39239090}{154171}a^{8}+\frac{117949373}{154171}a^{7}+\frac{101685375}{308342}a^{6}-\frac{150062863}{308342}a^{5}-\frac{154837235}{616684}a^{4}+\frac{53915849}{308342}a^{3}+\frac{16385026}{154171}a^{2}-\frac{4379595}{154171}a-\frac{6174309}{308342}$
| 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: | \( 582648.285852 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 582648.285852 \cdot 2}{2\cdot\sqrt{17313300662569099483282407424}}\cr\approx \mathstrut & 0.424634446278 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^18 + 21*x^16 - 22*x^14 + 98*x^12 - 176*x^10 + 205*x^8 - 152*x^6 + 76*x^4 - 24*x^2 + 4)
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 - 8*x^18 + 21*x^16 - 22*x^14 + 98*x^12 - 176*x^10 + 205*x^8 - 152*x^6 + 76*x^4 - 24*x^2 + 4, 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 - 8*x^18 + 21*x^16 - 22*x^14 + 98*x^12 - 176*x^10 + 205*x^8 - 152*x^6 + 76*x^4 - 24*x^2 + 4);
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 - 8*x^18 + 21*x^16 - 22*x^14 + 98*x^12 - 176*x^10 + 205*x^8 - 152*x^6 + 76*x^4 - 24*x^2 + 4);
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
$C_2\wr S_5$ (as 20T288):
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 3840 |
| The 36 conjugacy class representatives for $C_2\wr S_5$ |
| Character table for $C_2\wr S_5$ is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 5.3.31684.1, 10.6.8223751012352.1, 10.0.8223751012352.1, 10.0.256992219136.1 |
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 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.6.8223751012352.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.18.53 | $x^{8} + 2 x^{6} + 4 x^{3} + 2$ | $8$ | $1$ | $18$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ |
| 2.12.24.314 | $x^{12} - 12 x^{10} + 62 x^{8} + 72 x^{7} + 136 x^{6} + 16 x^{5} + 1540 x^{4} + 560 x^{3} + 1232 x^{2} + 1216 x + 3576$ | $4$ | $3$ | $24$ | $C_2^2 \times A_4$ | $[2, 2, 2, 3]^{3}$ | |
|
\(89\)
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.6.4.1 | $x^{6} + 246 x^{5} + 20181 x^{4} + 553022 x^{3} + 82437 x^{2} + 1795920 x + 49014018$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 89.6.4.1 | $x^{6} + 246 x^{5} + 20181 x^{4} + 553022 x^{3} + 82437 x^{2} + 1795920 x + 49014018$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |