Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^16 + 11*x^12 + 3*x^10 - 17*x^8 + 6*x^6 + 2*x^2 - 1)
gp: K = bnfinit(y^20 - 6*y^16 + 11*y^12 + 3*y^10 - 17*y^8 + 6*y^6 + 2*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 6*x^16 + 11*x^12 + 3*x^10 - 17*x^8 + 6*x^6 + 2*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 6*x^16 + 11*x^12 + 3*x^10 - 17*x^8 + 6*x^6 + 2*x^2 - 1)
\( x^{20} - 6x^{16} + 11x^{12} + 3x^{10} - 17x^{8} + 6x^{6} + 2x^{2} - 1 \)
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: |
\(-1816889967552059474944\)
\(\medspace = -\,2^{10}\cdot 36497^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.56\) | 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: | $2^{31/16}36497^{1/2}\approx 731.7693250181059$ | ||
| Ramified primes: |
\(2\), \(36497\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2258}a^{18}-\frac{124}{1129}a^{16}-\frac{597}{2258}a^{14}+\frac{157}{2258}a^{12}-\frac{1}{2}a^{11}+\frac{295}{1129}a^{10}-\frac{1}{2}a^{9}-\frac{338}{1129}a^{8}+\frac{539}{2258}a^{6}-\frac{1}{2}a^{5}-\frac{222}{1129}a^{4}-\frac{1}{2}a^{3}-\frac{265}{1129}a^{2}-\frac{1}{2}a+\frac{239}{1129}$, $\frac{1}{2258}a^{19}-\frac{124}{1129}a^{17}+\frac{266}{1129}a^{15}-\frac{1}{2}a^{14}+\frac{157}{2258}a^{13}+\frac{295}{1129}a^{11}-\frac{338}{1129}a^{9}+\frac{539}{2258}a^{7}+\frac{685}{2258}a^{5}-\frac{1}{2}a^{4}-\frac{265}{1129}a^{3}-\frac{651}{2258}a-\frac{1}{2}$
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: |
$\frac{412}{1129}a^{18}+\frac{563}{1129}a^{16}-\frac{2100}{1129}a^{14}-\frac{3056}{1129}a^{12}+\frac{2603}{1129}a^{10}+\frac{5996}{1129}a^{8}-\frac{2603}{1129}a^{6}-\frac{4546}{1129}a^{4}-\frac{463}{1129}a^{2}+\frac{490}{1129}$, $a$, $\frac{287}{1129}a^{18}-\frac{49}{1129}a^{16}-\frac{1989}{1129}a^{14}-\frac{101}{1129}a^{12}+\frac{4496}{1129}a^{10}+\frac{2434}{1129}a^{8}-\frac{6754}{1129}a^{6}-\frac{980}{1129}a^{4}+\frac{1434}{1129}a^{2}+\frac{1706}{1129}$, $\frac{2646}{1129}a^{19}+\frac{560}{1129}a^{18}+\frac{2869}{2258}a^{17}+\frac{1103}{2258}a^{16}-\frac{14868}{1129}a^{15}-\frac{5917}{2258}a^{14}-\frac{7953}{1129}a^{13}-\frac{5929}{2258}a^{12}+\frac{23442}{1129}a^{11}+\frac{4119}{1129}a^{10}+\frac{19962}{1129}a^{9}+\frac{11729}{2258}a^{8}-\frac{63819}{2258}a^{7}-\frac{5248}{1129}a^{6}+\frac{465}{1129}a^{5}-\frac{3907}{2258}a^{4}-\frac{5969}{2258}a^{3}-\frac{875}{2258}a^{2}+\frac{3695}{1129}a+\frac{107}{1129}$, $\frac{3557}{2258}a^{19}+\frac{950}{1129}a^{18}+\frac{1871}{2258}a^{17}+\frac{1851}{2258}a^{16}-\frac{10101}{1129}a^{15}-\frac{4908}{1129}a^{14}-\frac{10567}{2258}a^{13}-\frac{9917}{2258}a^{12}+\frac{16280}{1129}a^{11}+\frac{6161}{1129}a^{10}+\frac{13667}{1129}a^{9}+\frac{19595}{2258}a^{8}-\frac{21925}{1129}a^{7}-\frac{7290}{1129}a^{6}+\frac{163}{2258}a^{5}-\frac{2941}{1129}a^{4}-\frac{3167}{2258}a^{3}-\frac{3319}{2258}a^{2}+\frac{3359}{2258}a+\frac{1613}{2258}$, $\frac{1079}{1129}a^{19}-\frac{4691}{2258}a^{18}+\frac{1091}{2258}a^{17}-\frac{2889}{2258}a^{16}-\frac{6278}{1129}a^{15}+\frac{13287}{1129}a^{14}-\frac{3334}{1129}a^{13}+\frac{8278}{1129}a^{12}+\frac{21159}{2258}a^{11}-\frac{21142}{1129}a^{10}+\frac{19053}{2258}a^{9}-\frac{40889}{2258}a^{8}-\frac{27933}{2258}a^{7}+\frac{54703}{2258}a^{6}-\frac{4147}{2258}a^{5}+\frac{6573}{2258}a^{4}-\frac{596}{1129}a^{3}+\frac{3559}{2258}a^{2}+\frac{5263}{2258}a-\frac{3439}{1129}$, $\frac{3557}{2258}a^{19}+\frac{3213}{1129}a^{18}+\frac{1871}{2258}a^{17}+\frac{3887}{2258}a^{16}-\frac{10101}{1129}a^{15}-\frac{18054}{1129}a^{14}-\frac{10567}{2258}a^{13}-\frac{21895}{2258}a^{12}+\frac{16280}{1129}a^{11}+\frac{28304}{1129}a^{10}+\frac{13667}{1129}a^{9}+\frac{53479}{2258}a^{8}-\frac{21925}{1129}a^{7}-\frac{37336}{1129}a^{6}+\frac{163}{2258}a^{5}-\frac{2903}{1129}a^{4}-\frac{3167}{2258}a^{3}-\frac{6361}{2258}a^{2}+\frac{3359}{2258}a+\frac{8651}{2258}$, $\frac{1079}{1129}a^{19}+\frac{3577}{2258}a^{18}+\frac{1091}{2258}a^{17}+\frac{1427}{2258}a^{16}-\frac{6278}{1129}a^{15}-\frac{10426}{1129}a^{14}-\frac{3334}{1129}a^{13}-\frac{4278}{1129}a^{12}+\frac{21159}{2258}a^{11}+\frac{17664}{1129}a^{10}+\frac{19053}{2258}a^{9}+\frac{26233}{2258}a^{8}-\frac{27933}{2258}a^{7}-\frac{47747}{2258}a^{6}-\frac{4147}{2258}a^{5}+\frac{315}{2258}a^{4}-\frac{596}{1129}a^{3}-\frac{4735}{2258}a^{2}+\frac{5263}{2258}a+\frac{3637}{1129}$, $\frac{1079}{1129}a^{19}+\frac{4691}{2258}a^{18}+\frac{1091}{2258}a^{17}+\frac{2889}{2258}a^{16}-\frac{6278}{1129}a^{15}-\frac{13287}{1129}a^{14}-\frac{3334}{1129}a^{13}-\frac{8278}{1129}a^{12}+\frac{21159}{2258}a^{11}+\frac{21142}{1129}a^{10}+\frac{19053}{2258}a^{9}+\frac{40889}{2258}a^{8}-\frac{27933}{2258}a^{7}-\frac{54703}{2258}a^{6}-\frac{4147}{2258}a^{5}-\frac{6573}{2258}a^{4}-\frac{596}{1129}a^{3}-\frac{3559}{2258}a^{2}+\frac{5263}{2258}a+\frac{3439}{1129}$, $\frac{3557}{2258}a^{19}-\frac{3213}{1129}a^{18}+\frac{1871}{2258}a^{17}-\frac{3887}{2258}a^{16}-\frac{10101}{1129}a^{15}+\frac{18054}{1129}a^{14}-\frac{10567}{2258}a^{13}+\frac{21895}{2258}a^{12}+\frac{16280}{1129}a^{11}-\frac{28304}{1129}a^{10}+\frac{13667}{1129}a^{9}-\frac{53479}{2258}a^{8}-\frac{21925}{1129}a^{7}+\frac{37336}{1129}a^{6}+\frac{163}{2258}a^{5}+\frac{2903}{1129}a^{4}-\frac{3167}{2258}a^{3}+\frac{6361}{2258}a^{2}+\frac{3359}{2258}a-\frac{8651}{2258}$
| 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: | \( 201.865023128 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 201.865023128 \cdot 1}{2\cdot\sqrt{1816889967552059474944}}\cr\approx \mathstrut & 0.144559145803 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^16 + 11*x^12 + 3*x^10 - 17*x^8 + 6*x^6 + 2*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 - 6*x^16 + 11*x^12 + 3*x^10 - 17*x^8 + 6*x^6 + 2*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 - 6*x^16 + 11*x^12 + 3*x^10 - 17*x^8 + 6*x^6 + 2*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 - 6*x^16 + 11*x^12 + 3*x^10 - 17*x^8 + 6*x^6 + 2*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 20T992):
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
| 5.5.36497.1, 10.2.1332031009.1 |
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 | ${\href{/padicField/3.8.0.1}{8} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.4.0.1}{4} }^{5}$ | $16{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.0.1 | $x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |
| 2.10.10.12 | $x^{10} + 10 x^{9} + 34 x^{8} + 112 x^{7} + 328 x^{6} + 640 x^{5} + 1360 x^{4} + 2176 x^{3} + 1168 x^{2} + 1888 x - 1248$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ | |
|
\(36497\)
| $\Q_{36497}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{36497}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |