Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^18 + 18*x^16 - 40*x^14 + 92*x^12 - 160*x^10 + 352*x^8 - 352*x^6 + 528*x^4 - 704*x^2 + 352)
gp: K = bnfinit(y^20 - 4*y^18 + 18*y^16 - 40*y^14 + 92*y^12 - 160*y^10 + 352*y^8 - 352*y^6 + 528*y^4 - 704*y^2 + 352, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^18 + 18*x^16 - 40*x^14 + 92*x^12 - 160*x^10 + 352*x^8 - 352*x^6 + 528*x^4 - 704*x^2 + 352);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^18 + 18*x^16 - 40*x^14 + 92*x^12 - 160*x^10 + 352*x^8 - 352*x^6 + 528*x^4 - 704*x^2 + 352)
\( x^{20} - 4x^{18} + 18x^{16} - 40x^{14} + 92x^{12} - 160x^{10} + 352x^{8} - 352x^{6} + 528x^{4} - 704x^{2} + 352 \)
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: |
\(84954018740373771557797888\)
\(\medspace = 2^{55}\cdot 11^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.79\) | 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: | $2^{11/4}11^{9/10}\approx 58.22183708777889$ | ||
| Ramified primes: |
\(2\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{22}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $10$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{8}a^{14}$, $\frac{1}{8}a^{15}$, $\frac{1}{16}a^{16}$, $\frac{1}{16}a^{17}$, $\frac{1}{321989456}a^{18}+\frac{837739}{80497364}a^{16}-\frac{1251545}{20124341}a^{14}+\frac{854403}{80497364}a^{12}+\frac{203587}{40248682}a^{10}-\frac{1633200}{20124341}a^{8}-\frac{7047079}{40248682}a^{6}+\frac{5489405}{40248682}a^{4}+\frac{970885}{20124341}a^{2}-\frac{4788209}{20124341}$, $\frac{1}{321989456}a^{19}+\frac{837739}{80497364}a^{17}-\frac{1251545}{20124341}a^{15}+\frac{854403}{80497364}a^{13}+\frac{203587}{40248682}a^{11}-\frac{1633200}{20124341}a^{9}-\frac{7047079}{40248682}a^{7}+\frac{5489405}{40248682}a^{5}+\frac{970885}{20124341}a^{3}-\frac{4788209}{20124341}a$
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 $1$ (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: |
$\frac{2651761}{321989456}a^{18}-\frac{2145929}{80497364}a^{16}+\frac{9776739}{80497364}a^{14}-\frac{32631263}{160994728}a^{12}+\frac{37114423}{80497364}a^{10}-\frac{49414203}{80497364}a^{8}+\frac{36214365}{20124341}a^{6}-\frac{29980689}{40248682}a^{4}+\frac{48034355}{20124341}a^{2}-\frac{54921555}{20124341}$, $\frac{1465}{321989456}a^{18}-\frac{148583}{40248682}a^{16}+\frac{2537189}{160994728}a^{14}-\frac{12141835}{160994728}a^{12}+\frac{12904021}{80497364}a^{10}-\frac{15807183}{40248682}a^{8}+\frac{19940539}{40248682}a^{6}-\frac{48006757}{40248682}a^{4}+\frac{13642655}{20124341}a^{2}-\frac{31579858}{20124341}$, $\frac{1170421}{321989456}a^{18}-\frac{1772507}{160994728}a^{16}+\frac{4550757}{80497364}a^{14}-\frac{7539309}{80497364}a^{12}+\frac{10302687}{40248682}a^{10}-\frac{23816237}{80497364}a^{8}+\frac{32654637}{40248682}a^{6}-\frac{1171248}{20124341}a^{4}+\frac{23278020}{20124341}a^{2}-\frac{14008650}{20124341}$, $\frac{988555}{321989456}a^{18}-\frac{3090407}{321989456}a^{16}+\frac{6218571}{160994728}a^{14}-\frac{8592551}{160994728}a^{12}+\frac{6949229}{80497364}a^{10}-\frac{5165527}{40248682}a^{8}+\frac{17943125}{40248682}a^{6}-\frac{5081949}{20124341}a^{4}+\frac{3150203}{20124341}a^{2}-\frac{22074408}{20124341}$, $\frac{1210631}{160994728}a^{19}-\frac{1787851}{321989456}a^{18}-\frac{8258839}{321989456}a^{17}+\frac{392509}{20124341}a^{16}+\frac{19219579}{160994728}a^{15}-\frac{13605481}{160994728}a^{14}-\frac{37103405}{160994728}a^{13}+\frac{12969993}{80497364}a^{12}+\frac{43841021}{80497364}a^{11}-\frac{14390211}{40248682}a^{10}-\frac{69486669}{80497364}a^{9}+\frac{48853607}{80497364}a^{8}+\frac{40453668}{20124341}a^{7}-\frac{29279809}{20124341}a^{6}-\frac{54652091}{40248682}a^{5}+\frac{40648907}{40248682}a^{4}+\frac{62809001}{20124341}a^{3}-\frac{33058203}{20124341}a^{2}-\frac{76893068}{20124341}a+\frac{51701256}{20124341}$, $\frac{257815}{80497364}a^{19}+\frac{540037}{321989456}a^{18}-\frac{2351639}{160994728}a^{17}-\frac{890371}{321989456}a^{16}+\frac{5128999}{80497364}a^{15}+\frac{1665621}{80497364}a^{14}-\frac{3127141}{20124341}a^{13}-\frac{1657537}{80497364}a^{12}+\frac{7002154}{20124341}a^{11}+\frac{2618918}{20124341}a^{10}-\frac{50064453}{80497364}a^{9}-\frac{15865913}{80497364}a^{8}+\frac{26039213}{20124341}a^{7}+\frac{30724587}{40248682}a^{6}-\frac{26910182}{20124341}a^{5}-\frac{7370192}{20124341}a^{4}+\frac{28776009}{20124341}a^{3}+\frac{34491013}{20124341}a^{2}-\frac{39235193}{20124341}a-\frac{13324302}{20124341}$, $\frac{988555}{321989456}a^{19}-\frac{5530}{20124341}a^{18}-\frac{3090407}{321989456}a^{17}-\frac{670927}{321989456}a^{16}+\frac{6218571}{160994728}a^{15}-\frac{2361}{160994728}a^{14}-\frac{8592551}{160994728}a^{13}-\frac{980947}{160994728}a^{12}+\frac{6949229}{80497364}a^{11}-\frac{2092565}{40248682}a^{10}-\frac{5165527}{40248682}a^{9}+\frac{5410899}{40248682}a^{8}+\frac{17943125}{40248682}a^{7}-\frac{3515812}{20124341}a^{6}-\frac{5081949}{20124341}a^{5}-\frac{1584365}{40248682}a^{4}+\frac{3150203}{20124341}a^{3}-\frac{13217412}{20124341}a^{2}-\frac{22074408}{20124341}a+\frac{3105588}{20124341}$, $\frac{2522083}{321989456}a^{19}-\frac{415253}{321989456}a^{18}-\frac{8960671}{321989456}a^{17}+\frac{3026577}{321989456}a^{16}+\frac{2517615}{20124341}a^{15}-\frac{6199179}{160994728}a^{14}-\frac{9514245}{40248682}a^{13}+\frac{17570083}{160994728}a^{12}+\frac{41875235}{80497364}a^{11}-\frac{15636281}{80497364}a^{10}-\frac{15839720}{20124341}a^{9}+\frac{23755941}{80497364}a^{8}+\frac{38607241}{20124341}a^{7}-\frac{10050923}{20124341}a^{6}-\frac{46852427}{40248682}a^{5}+\frac{32459287}{40248682}a^{4}+\frac{43486621}{20124341}a^{3}-\frac{11985652}{20124341}a^{2}-\frac{45972067}{20124341}a-\frac{6987605}{20124341}$, $\frac{1019237}{321989456}a^{19}-\frac{2715341}{321989456}a^{18}-\frac{539573}{40248682}a^{17}+\frac{11199403}{321989456}a^{16}+\frac{631802}{20124341}a^{15}-\frac{26983531}{160994728}a^{14}-\frac{728791}{40248682}a^{13}+\frac{58187183}{160994728}a^{12}-\frac{17478205}{80497364}a^{11}-\frac{32729579}{40248682}a^{10}+\frac{49108729}{80497364}a^{9}+\frac{78688645}{80497364}a^{8}-\frac{22494036}{20124341}a^{7}-\frac{33736967}{20124341}a^{6}+\frac{75647847}{40248682}a^{5}+\frac{9932411}{40248682}a^{4}-\frac{72678271}{20124341}a^{3}-\frac{15300126}{20124341}a^{2}+\frac{14035036}{20124341}a-\frac{12153896}{20124341}$
| 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: | \( 28703.22707298631 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 28703.22707298631 \cdot 1}{2\cdot\sqrt{84954018740373771557797888}}\cr\approx \mathstrut & 0.149316379403044 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^18 + 18*x^16 - 40*x^14 + 92*x^12 - 160*x^10 + 352*x^8 - 352*x^6 + 528*x^4 - 704*x^2 + 352)
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 - 4*x^18 + 18*x^16 - 40*x^14 + 92*x^12 - 160*x^10 + 352*x^8 - 352*x^6 + 528*x^4 - 704*x^2 + 352, 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 - 4*x^18 + 18*x^16 - 40*x^14 + 92*x^12 - 160*x^10 + 352*x^8 - 352*x^6 + 528*x^4 - 704*x^2 + 352);
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 - 4*x^18 + 18*x^16 - 40*x^14 + 92*x^12 - 160*x^10 + 352*x^8 - 352*x^6 + 528*x^4 - 704*x^2 + 352);
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
$C_{10}\wr C_2$ (as 20T53):
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 solvable group of order 200 |
| The 65 conjugacy class representatives for $C_{10}\wr C_2$ |
| Character table for $C_{10}\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 4.0.22528.2, 10.0.479756288.1 |
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 | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{5}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ | $20$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 $20$ | $4$ | $5$ | $55$ | |||
|
\(11\)
| 11.10.9.4 | $x^{10} + 22$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 11.10.0.1 | $x^{10} + 7 x^{5} + 8 x^{4} + 10 x^{3} + 6 x^{2} + 6 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |