Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^18 - 4*x^16 + 4*x^14 + 9*x^12 - 11*x^10 - 9*x^8 + 12*x^6 + x^4 - 4*x^2 + 1)
gp: K = bnfinit(y^20 - y^18 - 4*y^16 + 4*y^14 + 9*y^12 - 11*y^10 - 9*y^8 + 12*y^6 + y^4 - 4*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^18 - 4*x^16 + 4*x^14 + 9*x^12 - 11*x^10 - 9*x^8 + 12*x^6 + x^4 - 4*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^18 - 4*x^16 + 4*x^14 + 9*x^12 - 11*x^10 - 9*x^8 + 12*x^6 + x^4 - 4*x^2 + 1)
\( x^{20} - x^{18} - 4x^{16} + 4x^{14} + 9x^{12} - 11x^{10} - 9x^{8} + 12x^{6} + x^{4} - 4x^{2} + 1 \)
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: | $[4, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(12646126110965776516096\)
\(\medspace = 2^{10}\cdot 47^{2}\cdot 8647^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.74\) | 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: | $2^{15/8}47^{1/2}8647^{1/2}\approx 2338.36822156596$ | ||
| Ramified primes: |
\(2\), \(47\), \(8647\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$
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: | $11$ | 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: |
$a$, $13a^{18}-5a^{16}-55a^{14}+18a^{12}+128a^{10}-64a^{8}-156a^{6}+59a^{4}+50a^{2}-21$, $24a^{19}-13a^{17}-102a^{15}+49a^{13}+239a^{11}-154a^{9}-288a^{7}+155a^{5}+98a^{3}-51a$, $\frac{13}{2}a^{19}+21a^{18}-\frac{7}{2}a^{17}-10a^{16}-28a^{15}-\frac{179}{2}a^{14}+\frac{27}{2}a^{13}+\frac{75}{2}a^{12}+66a^{11}+\frac{419}{2}a^{10}-\frac{85}{2}a^{9}-\frac{245}{2}a^{8}-81a^{7}-255a^{6}+\frac{89}{2}a^{5}+\frac{243}{2}a^{4}+29a^{3}+\frac{171}{2}a^{2}-\frac{31}{2}a-41$, $21a^{18}-11a^{16}-89a^{14}+41a^{12}+208a^{10}-130a^{8}-250a^{6}+129a^{4}+83a^{2}-42$, $\frac{31}{2}a^{19}-\frac{25}{2}a^{18}-\frac{13}{2}a^{17}+7a^{16}-\frac{131}{2}a^{15}+53a^{14}+\frac{47}{2}a^{13}-\frac{53}{2}a^{12}+\frac{305}{2}a^{11}-124a^{10}-\frac{161}{2}a^{9}+82a^{8}-185a^{7}+\frac{297}{2}a^{6}+75a^{5}-\frac{165}{2}a^{4}+59a^{3}-\frac{99}{2}a^{2}-\frac{51}{2}a+\frac{53}{2}$, $\frac{9}{2}a^{19}+\frac{9}{2}a^{18}-2a^{17}-a^{16}-\frac{39}{2}a^{15}-19a^{14}+\frac{15}{2}a^{13}+3a^{12}+46a^{11}+44a^{10}-25a^{9}-\frac{29}{2}a^{8}-57a^{7}-\frac{109}{2}a^{6}+25a^{5}+\frac{21}{2}a^{4}+\frac{41}{2}a^{3}+17a^{2}-8a-\frac{7}{2}$, $\frac{9}{2}a^{19}-\frac{9}{2}a^{18}-2a^{17}+a^{16}-\frac{39}{2}a^{15}+19a^{14}+\frac{15}{2}a^{13}-3a^{12}+46a^{11}-44a^{10}-25a^{9}+\frac{29}{2}a^{8}-57a^{7}+\frac{109}{2}a^{6}+25a^{5}-\frac{21}{2}a^{4}+\frac{41}{2}a^{3}-17a^{2}-8a+\frac{7}{2}$, $\frac{9}{2}a^{19}-\frac{31}{2}a^{18}-a^{17}+\frac{13}{2}a^{16}-19a^{15}+\frac{131}{2}a^{14}+3a^{13}-\frac{47}{2}a^{12}+44a^{11}-\frac{305}{2}a^{10}-\frac{29}{2}a^{9}+\frac{161}{2}a^{8}-\frac{109}{2}a^{7}+185a^{6}+\frac{21}{2}a^{5}-75a^{4}+17a^{3}-59a^{2}-\frac{9}{2}a+\frac{51}{2}$, $\frac{57}{2}a^{19}-\frac{15}{2}a^{18}-13a^{17}+3a^{16}-121a^{15}+\frac{63}{2}a^{14}+48a^{13}-\frac{21}{2}a^{12}+\frac{565}{2}a^{11}-73a^{10}-\frac{319}{2}a^{9}+37a^{8}-343a^{7}+88a^{6}+\frac{309}{2}a^{5}-33a^{4}+113a^{3}-\frac{55}{2}a^{2}-52a+11$, $\frac{9}{2}a^{19}+\frac{31}{2}a^{18}-a^{17}-\frac{13}{2}a^{16}-19a^{15}-\frac{131}{2}a^{14}+3a^{13}+\frac{47}{2}a^{12}+44a^{11}+\frac{305}{2}a^{10}-\frac{29}{2}a^{9}-\frac{161}{2}a^{8}-\frac{109}{2}a^{7}-185a^{6}+\frac{21}{2}a^{5}+75a^{4}+17a^{3}+59a^{2}-\frac{9}{2}a-\frac{51}{2}$
| 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: | \( 831.066452115 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 831.066452115 \cdot 1}{2\cdot\sqrt{12646126110965776516096}}\cr\approx \mathstrut & 0.143610412956 \end{aligned}\] (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^16 + 4*x^14 + 9*x^12 - 11*x^10 - 9*x^8 + 12*x^6 + x^4 - 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^16 + 4*x^14 + 9*x^12 - 11*x^10 - 9*x^8 + 12*x^6 + x^4 - 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^16 + 4*x^14 + 9*x^12 - 11*x^10 - 9*x^8 + 12*x^6 + x^4 - 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^16 + 4*x^14 + 9*x^12 - 11*x^10 - 9*x^8 + 12*x^6 + x^4 - 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
$C_2^9.C_2^5.S_5$ (as 20T994):
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 1966080 |
| The 280 conjugacy class representatives for $C_2^9.C_2^5.S_5$ |
| Character table for $C_2^9.C_2^5.S_5$ |
Intermediate fields
| 5.3.8647.1, 10.4.3514218623.1 |
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: | 20.2.275524109311254364946432.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.10.5 | $x^{10} + 34 x^{8} - 24 x^{7} + 368 x^{6} - 496 x^{5} + 1568 x^{4} - 1760 x^{3} + 2992 x^{2} - 1856 x + 352$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2]^{10}$ |
| 2.10.0.1 | $x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(47\)
| 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.6.0.1 | $x^{6} + 2 x^{4} + 35 x^{3} + 9 x^{2} + 41 x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 47.6.0.1 | $x^{6} + 2 x^{4} + 35 x^{3} + 9 x^{2} + 41 x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(8647\)
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $8$ | $2$ | $4$ | $4$ |