Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^10 - 13*x^8 - 19*x^6 - 18*x^4 - 9*x^2 - 1)
gp: K = bnfinit(y^20 - 2*y^10 - 13*y^8 - 19*y^6 - 18*y^4 - 9*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^10 - 13*x^8 - 19*x^6 - 18*x^4 - 9*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^10 - 13*x^8 - 19*x^6 - 18*x^4 - 9*x^2 - 1)
\( x^{20} - 2x^{10} - 13x^{8} - 19x^{6} - 18x^{4} - 9x^{2} - 1 \)
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: | $[2, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-11138046967442830397538304\)
\(\medspace = -\,2^{20}\cdot 107^{2}\cdot 5519^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.88\) | 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: |
\(2\), \(107\), \(5519\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{761}a^{18}+\frac{109}{761}a^{16}-\frac{295}{761}a^{14}-\frac{193}{761}a^{12}+\frac{271}{761}a^{10}-\frac{142}{761}a^{8}-\frac{271}{761}a^{6}+\frac{121}{761}a^{4}+\frac{234}{761}a^{2}-\frac{377}{761}$, $\frac{1}{761}a^{19}+\frac{109}{761}a^{17}-\frac{295}{761}a^{15}-\frac{193}{761}a^{13}+\frac{271}{761}a^{11}-\frac{142}{761}a^{9}-\frac{271}{761}a^{7}+\frac{121}{761}a^{5}+\frac{234}{761}a^{3}-\frac{377}{761}a$
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
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: | $10$ | 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: |
$a$, $\frac{1254}{761}a^{19}-\frac{294}{761}a^{17}-\frac{84}{761}a^{15}-\frac{24}{761}a^{13}+\frac{428}{761}a^{11}-\frac{3038}{761}a^{9}-\frac{15648}{761}a^{7}-\frac{19491}{761}a^{5}-\frac{16291}{761}a^{3}-\frac{5504}{761}a$, $\frac{333}{761}a^{18}+\frac{231}{761}a^{16}+\frac{66}{761}a^{14}-\frac{416}{761}a^{12}+\frac{316}{761}a^{10}+\frac{865}{761}a^{8}+\frac{3489}{761}a^{6}+\frac{3084}{761}a^{4}+\frac{1983}{761}a^{2}+\frac{1498}{761}$, $\frac{46}{761}a^{18}+\frac{448}{761}a^{16}-\frac{633}{761}a^{14}+\frac{254}{761}a^{12}+\frac{290}{761}a^{10}-\frac{444}{761}a^{8}-\frac{1812}{761}a^{6}-\frac{4327}{761}a^{4}-\frac{2173}{761}a^{2}-\frac{600}{761}$, $\frac{769}{761}a^{19}-\frac{111}{761}a^{17}+\frac{77}{761}a^{15}+\frac{22}{761}a^{13}+\frac{115}{761}a^{11}+\frac{1136}{761}a^{9}+\frac{10539}{761}a^{7}+\frac{15774}{761}a^{5}+\frac{14870}{761}a^{3}+\frac{6821}{761}a$, $\frac{396}{761}a^{19}+\frac{213}{761}a^{17}-\frac{374}{761}a^{15}+\frac{328}{761}a^{13}-\frac{15}{761}a^{11}+\frac{679}{761}a^{9}+\frac{4581}{761}a^{7}+\frac{6115}{761}a^{5}+\frac{7027}{761}a^{3}+\frac{2419}{761}a$, $\frac{294}{761}a^{19}-\frac{406}{761}a^{18}-\frac{84}{761}a^{17}-\frac{116}{761}a^{16}-\frac{24}{761}a^{15}+\frac{293}{761}a^{14}+\frac{428}{761}a^{13}-\frac{25}{761}a^{12}-\frac{530}{761}a^{11}-\frac{442}{761}a^{10}+\frac{654}{761}a^{9}+\frac{1338}{761}a^{8}+\frac{4335}{761}a^{7}+\frac{5769}{761}a^{6}+\frac{6281}{761}a^{5}+\frac{7949}{761}a^{4}+\frac{5782}{761}a^{3}+\frac{6209}{761}a^{2}+\frac{1254}{761}a+\frac{1623}{761}$, $\frac{406}{761}a^{19}+\frac{333}{761}a^{18}-\frac{116}{761}a^{17}-\frac{231}{761}a^{16}+\frac{293}{761}a^{15}-\frac{66}{761}a^{14}-\frac{25}{761}a^{13}+\frac{416}{761}a^{12}-\frac{442}{761}a^{11}-\frac{316}{761}a^{10}+\frac{1338}{761}a^{9}-\frac{865}{761}a^{8}+\frac{5769}{761}a^{7}-\frac{3489}{761}a^{6}+\frac{7949}{761}a^{5}-\frac{3084}{761}a^{4}+\frac{6209}{761}a^{3}-\frac{1983}{761}a^{2}+\frac{1623}{761}a-\frac{737}{761}$, $\frac{599}{761}a^{19}-\frac{725}{761}a^{18}+\frac{606}{761}a^{17}+\frac{119}{761}a^{16}-\frac{914}{761}a^{15}+\frac{795}{761}a^{14}+\frac{65}{761}a^{13}-\frac{860}{761}a^{12}+\frac{997}{761}a^{11}-\frac{137}{761}a^{10}-\frac{2109}{761}a^{9}+\frac{2498}{761}a^{8}-\frac{9368}{761}a^{7}+\frac{8508}{761}a^{6}-\frac{15797}{761}a^{5}+\frac{9683}{761}a^{4}-\frac{11273}{761}a^{3}+\frac{3858}{761}a^{2}-\frac{2850}{761}a+\frac{126}{761}$, $\frac{28}{761}a^{19}-\frac{466}{761}a^{18}+\frac{753}{761}a^{17}+\frac{193}{761}a^{16}-\frac{872}{761}a^{15}+\frac{490}{761}a^{14}+\frac{77}{761}a^{13}-\frac{621}{761}a^{12}+\frac{783}{761}a^{11}+\frac{40}{761}a^{10}-\frac{590}{761}a^{9}+\frac{1487}{761}a^{8}-\frac{1544}{761}a^{7}+\frac{5287}{761}a^{6}-\frac{6432}{761}a^{5}+\frac{5255}{761}a^{4}-\frac{2747}{761}a^{3}+\frac{540}{761}a^{2}-\frac{98}{761}a-\frac{870}{761}$
| 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: | \( 19951.314381307206 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 19951.314381307206 \cdot 1}{2\cdot\sqrt{11138046967442830397538304}}\cr\approx \mathstrut & 0.182480221938198 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^10 - 13*x^8 - 19*x^6 - 18*x^4 - 9*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 - 2*x^10 - 13*x^8 - 19*x^6 - 18*x^4 - 9*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 - 2*x^10 - 13*x^8 - 19*x^6 - 18*x^4 - 9*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 - 2*x^10 - 13*x^8 - 19*x^6 - 18*x^4 - 9*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
$C_2^{10}.C_2\wr S_5$ (as 20T1015):
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 3932160 |
| The 506 conjugacy class representatives for $C_2^{10}.C_2\wr S_5$ |
| Character table for $C_2^{10}.C_2\wr S_5$ |
Intermediate fields
| 5.3.5519.1, 10.4.3259151627.1 |
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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}$ | $16{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }$ | $16{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | $20$ |
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 $20$ | $2$ | $10$ | $20$ | |||
|
\(107\)
| 107.4.2.1 | $x^{4} + 206 x^{3} + 10827 x^{2} + 22454 x + 1146188$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 107.16.0.1 | $x^{16} - x + 7$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | |
|
\(5519\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $8$ | $2$ | $4$ | $4$ |