Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 784*x^14 + 6272*x^12 - 21952*x^10 + 241472*x^8 + 13522432*x^4 + 189314048)
gp: K = bnfinit(y^20 + 784*y^14 + 6272*y^12 - 21952*y^10 + 241472*y^8 + 13522432*y^4 + 189314048, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 784*x^14 + 6272*x^12 - 21952*x^10 + 241472*x^8 + 13522432*x^4 + 189314048);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 784*x^14 + 6272*x^12 - 21952*x^10 + 241472*x^8 + 13522432*x^4 + 189314048)
\( x^{20} + 784x^{14} + 6272x^{12} - 21952x^{10} + 241472x^{8} + 13522432x^{4} + 189314048 \)
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: |
\(12020004674398148709330936922112\)
\(\medspace = 2^{30}\cdot 7^{15}\cdot 11^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(35.81\) | 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: | $2^{3/2}7^{3/4}11^{9/10}\approx 105.34694361957366$ | ||
| Ramified primes: |
\(2\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{77}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $10$ | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{28}a^{4}$, $\frac{1}{28}a^{5}$, $\frac{1}{56}a^{6}$, $\frac{1}{56}a^{7}$, $\frac{1}{784}a^{8}$, $\frac{1}{784}a^{9}$, $\frac{1}{6272}a^{10}+\frac{1}{112}a^{4}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{6272}a^{11}+\frac{1}{112}a^{5}-\frac{1}{4}a^{3}-\frac{1}{4}a$, $\frac{1}{87808}a^{12}-\frac{1}{224}a^{6}-\frac{1}{56}a^{4}+\frac{1}{8}a^{2}$, $\frac{1}{87808}a^{13}-\frac{1}{224}a^{7}-\frac{1}{56}a^{5}+\frac{1}{8}a^{3}$, $\frac{1}{175616}a^{14}+\frac{1}{3136}a^{8}-\frac{1}{112}a^{6}-\frac{1}{112}a^{4}$, $\frac{1}{175616}a^{15}+\frac{1}{3136}a^{9}-\frac{1}{112}a^{7}-\frac{1}{112}a^{5}$, $\frac{1}{2458624}a^{16}-\frac{1}{1568}a^{8}+\frac{1}{224}a^{6}+\frac{1}{112}a^{4}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{2458624}a^{17}-\frac{1}{1568}a^{9}+\frac{1}{224}a^{7}+\frac{1}{112}a^{5}-\frac{1}{4}a^{3}-\frac{1}{4}a$, $\frac{1}{14243757860864}a^{18}-\frac{643105}{3560939465216}a^{16}+\frac{418311}{508705637888}a^{14}-\frac{1136783}{254352818944}a^{12}+\frac{70447}{1297718464}a^{10}-\frac{1758431}{4542014624}a^{8}+\frac{113817}{20276851}a^{6}+\frac{1748983}{324429616}a^{4}-\frac{386215}{5793386}a^{2}+\frac{2321853}{5793386}$, $\frac{1}{14243757860864}a^{19}-\frac{643105}{3560939465216}a^{17}+\frac{418311}{508705637888}a^{15}-\frac{1136783}{254352818944}a^{13}+\frac{70447}{1297718464}a^{11}-\frac{1758431}{4542014624}a^{9}+\frac{113817}{20276851}a^{7}+\frac{1748983}{324429616}a^{5}-\frac{386215}{5793386}a^{3}+\frac{2321853}{5793386}a$
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
$C_{2}\times C_{2}$, which has order $4$ (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: |
$\frac{15707}{1780469732608}a^{18}+\frac{982175}{7121878930432}a^{16}-\frac{52081}{254352818944}a^{14}+\frac{145323}{36336116992}a^{12}+\frac{1457661}{9084029248}a^{10}+\frac{146019}{283875914}a^{8}-\frac{1433689}{324429616}a^{6}+\frac{971431}{46347088}a^{4}+\frac{953235}{23173544}a^{2}+\frac{8373295}{11586772}$, $\frac{411585}{14243757860864}a^{18}+\frac{7885}{145344467968}a^{16}-\frac{104453}{254352818944}a^{14}+\frac{2752885}{127176409472}a^{12}+\frac{955517}{4542014624}a^{10}-\frac{7257477}{9084029248}a^{8}+\frac{50749}{20276851}a^{6}+\frac{540573}{46347088}a^{4}+\frac{3187251}{23173544}a^{2}+\frac{4035601}{11586772}$, $\frac{7233}{7121878930432}a^{18}-\frac{122103}{1017411275776}a^{16}+\frac{13607}{72672233984}a^{14}-\frac{176717}{254352818944}a^{12}-\frac{950745}{9084029248}a^{10}-\frac{871417}{1297718464}a^{8}+\frac{550705}{162214808}a^{6}-\frac{8154371}{162214808}a^{4}+\frac{41735}{23173544}a^{2}-\frac{18736525}{11586772}$, $\frac{41399}{2034822551552}a^{18}+\frac{106969}{3560939465216}a^{16}-\frac{52867}{254352818944}a^{14}+\frac{3345585}{254352818944}a^{12}+\frac{2656363}{18168058496}a^{10}-\frac{4007911}{9084029248}a^{8}-\frac{96109}{324429616}a^{6}-\frac{2930463}{324429616}a^{4}+\frac{2937971}{11586772}a^{2}-\frac{277459}{11586772}$, $\frac{45}{609124096}a^{19}+\frac{41}{1218248192}a^{18}-\frac{95}{304562048}a^{17}-\frac{9}{174035456}a^{16}-\frac{25}{10877216}a^{15}+\frac{19}{87017728}a^{14}+\frac{1361}{21754432}a^{13}+\frac{87}{3107776}a^{12}+\frac{341}{1553888}a^{11}+\frac{619}{3107776}a^{10}-\frac{509}{97118}a^{9}-\frac{99}{221984}a^{8}+\frac{881}{55496}a^{7}+\frac{1111}{110992}a^{6}+\frac{1003}{27748}a^{5}+\frac{11}{7928}a^{4}+\frac{891}{1982}a^{3}+\frac{748}{991}a^{2}-\frac{1881}{991}a+\frac{7963}{991}$, $\frac{2018559}{14243757860864}a^{19}-\frac{2139299}{14243757860864}a^{18}-\frac{1557961}{1017411275776}a^{17}-\frac{6984147}{7121878930432}a^{16}-\frac{575651}{508705637888}a^{15}-\frac{1226017}{254352818944}a^{14}+\frac{17904175}{127176409472}a^{13}-\frac{12971635}{127176409472}a^{12}-\frac{4719353}{18168058496}a^{11}-\frac{4796577}{2595436928}a^{10}-\frac{1875738}{141937957}a^{9}-\frac{10620027}{1297718464}a^{8}+\frac{938267}{11586772}a^{7}-\frac{1476177}{162214808}a^{6}-\frac{69846289}{324429616}a^{5}+\frac{3254694}{20276851}a^{4}+\frac{23515559}{23173544}a^{3}-\frac{113668991}{23173544}a^{2}-\frac{75177717}{5793386}a-\frac{41361539}{2896693}$, $\frac{936839}{14243757860864}a^{19}+\frac{129107}{3560939465216}a^{18}+\frac{57167}{890234866304}a^{17}+\frac{416751}{1017411275776}a^{16}-\frac{160087}{127176409472}a^{15}-\frac{1028723}{254352818944}a^{14}+\frac{6301375}{127176409472}a^{13}-\frac{2490893}{254352818944}a^{12}+\frac{1698063}{4542014624}a^{11}+\frac{2819541}{18168058496}a^{10}-\frac{14584167}{9084029248}a^{9}-\frac{807657}{283875914}a^{8}-\frac{1939367}{648859232}a^{7}-\frac{8055415}{324429616}a^{6}-\frac{4796891}{81107404}a^{5}-\frac{49166277}{162214808}a^{4}+\frac{4109515}{23173544}a^{3}-\frac{16257485}{23173544}a^{2}-\frac{5438437}{5793386}a-\frac{13269573}{5793386}$, $\frac{1907701}{14243757860864}a^{19}-\frac{1803773}{7121878930432}a^{18}-\frac{582649}{508705637888}a^{17}-\frac{14631469}{7121878930432}a^{16}-\frac{134563}{127176409472}a^{15}+\frac{298829}{63588204736}a^{14}+\frac{30875813}{254352818944}a^{13}-\frac{5312165}{31794102368}a^{12}-\frac{734739}{18168058496}a^{11}-\frac{30789265}{9084029248}a^{10}-\frac{110773813}{9084029248}a^{9}-\frac{16334363}{2271007312}a^{8}+\frac{19303563}{324429616}a^{7}+\frac{26524489}{648859232}a^{6}-\frac{42387511}{324429616}a^{5}-\frac{29450289}{46347088}a^{4}+\frac{2522521}{11586772}a^{3}-\frac{35438075}{5793386}a^{2}-\frac{220245933}{11586772}a-\frac{381572605}{11586772}$, $\frac{4161401}{14243757860864}a^{19}-\frac{437501}{1780469732608}a^{18}-\frac{822225}{445117433152}a^{17}-\frac{19785085}{7121878930432}a^{16}+\frac{6605}{3974262796}a^{15}+\frac{3435023}{254352818944}a^{14}+\frac{31714707}{127176409472}a^{13}-\frac{52319667}{254352818944}a^{12}+\frac{3686785}{18168058496}a^{11}-\frac{4529375}{1135503656}a^{10}-\frac{20409055}{1297718464}a^{9}+\frac{2925569}{2271007312}a^{8}+\frac{12394781}{92694176}a^{7}+\frac{21117755}{324429616}a^{6}-\frac{156061267}{324429616}a^{5}-\frac{432236713}{324429616}a^{4}+\frac{77664787}{23173544}a^{3}+\frac{52308387}{23173544}a^{2}-\frac{121619731}{11586772}a-\frac{460317223}{11586772}$
| 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: | \( 3576806.093381649 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 3576806.093381649 \cdot 4}{2\cdot\sqrt{12020004674398148709330936922112}}\cr\approx \mathstrut & 0.197866265794926 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 + 784*x^14 + 6272*x^12 - 21952*x^10 + 241472*x^8 + 13522432*x^4 + 189314048)
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 + 784*x^14 + 6272*x^12 - 21952*x^10 + 241472*x^8 + 13522432*x^4 + 189314048, 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 + 784*x^14 + 6272*x^12 - 21952*x^10 + 241472*x^8 + 13522432*x^4 + 189314048);
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 + 784*x^14 + 6272*x^12 - 21952*x^10 + 241472*x^8 + 13522432*x^4 + 189314048);
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
| \(\Q(\sqrt{-7}) \), 4.0.241472.1, 10.0.246071287.1 |
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 | $20$ | $20$ | R | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}$ | $20$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ | $20$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{10}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.15.1 | $x^{10} + 70 x^{8} + 2 x^{7} + 1960 x^{6} - 26 x^{5} + 27441 x^{4} - 2240 x^{3} + 192110 x^{2} - 14504 x + 537993$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
| 2.10.15.5 | $x^{10} + 46 x^{8} + 808 x^{6} + 6768 x^{4} + 27216 x^{2} + 9568$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ | |
|
\(7\)
| 7.20.15.1 | $x^{20} + 39 x^{16} + 16 x^{15} - 344 x^{12} - 7232 x^{11} + 96 x^{10} + 4344 x^{8} + 131648 x^{7} + 38272 x^{6} + 256 x^{5} + 59536 x^{4} - 212224 x^{3} + 84096 x^{2} - 8704 x + 9328$ | $4$ | $5$ | $15$ | 20T12 | $[\ ]_{4}^{10}$ |
|
\(11\)
| 11.10.9.5 | $x^{10} + 88$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 11.10.0.1 | $x^{10} + 7 x^{5} + 8 x^{4} + 10 x^{3} + 6 x^{2} + 6 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |