Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 5*x^20 + 2*x^19 + 12*x^18 + 13*x^17 + 14*x^16 + 63*x^15 - 7*x^14 + 68*x^13 + 31*x^12 + 19*x^11 + 40*x^10 - 71*x^9 + 84*x^8 - 79*x^7 + 41*x^6 - 38*x^5 + 21*x^4 - 11*x^3 + 4*x^2 - x + 1)
gp: K = bnfinit(y^22 - y^21 + 5*y^20 + 2*y^19 + 12*y^18 + 13*y^17 + 14*y^16 + 63*y^15 - 7*y^14 + 68*y^13 + 31*y^12 + 19*y^11 + 40*y^10 - 71*y^9 + 84*y^8 - 79*y^7 + 41*y^6 - 38*y^5 + 21*y^4 - 11*y^3 + 4*y^2 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^21 + 5*x^20 + 2*x^19 + 12*x^18 + 13*x^17 + 14*x^16 + 63*x^15 - 7*x^14 + 68*x^13 + 31*x^12 + 19*x^11 + 40*x^10 - 71*x^9 + 84*x^8 - 79*x^7 + 41*x^6 - 38*x^5 + 21*x^4 - 11*x^3 + 4*x^2 - x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^21 + 5*x^20 + 2*x^19 + 12*x^18 + 13*x^17 + 14*x^16 + 63*x^15 - 7*x^14 + 68*x^13 + 31*x^12 + 19*x^11 + 40*x^10 - 71*x^9 + 84*x^8 - 79*x^7 + 41*x^6 - 38*x^5 + 21*x^4 - 11*x^3 + 4*x^2 - x + 1)
\( x^{22} - x^{21} + 5 x^{20} + 2 x^{19} + 12 x^{18} + 13 x^{17} + 14 x^{16} + 63 x^{15} - 7 x^{14} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-66009317547629600105613290523\)
\(\medspace = -\,3^{11}\cdot 610429790897^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.42\) | 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: | $3^{1/2}610429790897^{1/2}\approx 1353251.407791989$ | ||
| Ramified primes: |
\(3\), \(610429790897\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{97}a^{20}+\frac{38}{97}a^{19}-\frac{34}{97}a^{18}+\frac{48}{97}a^{17}-\frac{43}{97}a^{16}+\frac{18}{97}a^{15}-\frac{35}{97}a^{14}+\frac{32}{97}a^{13}-\frac{38}{97}a^{12}-\frac{34}{97}a^{11}-\frac{48}{97}a^{10}+\frac{3}{97}a^{9}+\frac{27}{97}a^{8}+\frac{8}{97}a^{7}-\frac{28}{97}a^{6}+\frac{47}{97}a^{5}+\frac{36}{97}a^{4}+\frac{10}{97}a^{3}-\frac{25}{97}a^{2}+\frac{3}{97}a+\frac{25}{97}$, $\frac{1}{301661882749}a^{21}+\frac{1447621751}{301661882749}a^{20}+\frac{81067943680}{301661882749}a^{19}+\frac{17800206178}{301661882749}a^{18}-\frac{120973458096}{301661882749}a^{17}-\frac{33076074697}{301661882749}a^{16}+\frac{30206130283}{301661882749}a^{15}+\frac{137735226517}{301661882749}a^{14}-\frac{65319261888}{301661882749}a^{13}+\frac{117906113917}{301661882749}a^{12}-\frac{63650470818}{301661882749}a^{11}+\frac{102408670348}{301661882749}a^{10}-\frac{67187759862}{301661882749}a^{9}+\frac{41819160220}{301661882749}a^{8}+\frac{37992322923}{301661882749}a^{7}-\frac{84255858478}{301661882749}a^{6}-\frac{104870333029}{301661882749}a^{5}-\frac{64409543873}{301661882749}a^{4}-\frac{93089564349}{301661882749}a^{3}+\frac{121388096707}{301661882749}a^{2}-\frac{51349740029}{301661882749}a-\frac{13200153952}{301661882749}$
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: | $10$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{1537273841}{3109916317} a^{21} + \frac{1301100135}{3109916317} a^{20} - \frac{8678360179}{3109916317} a^{19} - \frac{3572581270}{3109916317} a^{18} - \frac{24759069500}{3109916317} a^{17} - \frac{28015134095}{3109916317} a^{16} - \frac{41571852646}{3109916317} a^{15} - \frac{124351678724}{3109916317} a^{14} - \frac{32877307951}{3109916317} a^{13} - \frac{194711711503}{3109916317} a^{12} - \frac{99775857337}{3109916317} a^{11} - \frac{138384201560}{3109916317} a^{10} - \frac{162142769333}{3109916317} a^{9} + \frac{31830498362}{3109916317} a^{8} - \frac{204708032640}{3109916317} a^{7} + \frac{135241277261}{3109916317} a^{6} - \frac{131862673205}{3109916317} a^{5} + \frac{88903208434}{3109916317} a^{4} - \frac{40087828929}{3109916317} a^{3} + \frac{32257408564}{3109916317} a^{2} - \frac{4964743754}{3109916317} a + \frac{7142614869}{3109916317} \)
(order $6$)
| 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: |
$\frac{236173706}{3109916317}a^{21}+\frac{991990974}{3109916317}a^{20}+\frac{498033588}{3109916317}a^{19}+\frac{6311783408}{3109916317}a^{18}+\frac{8030574162}{3109916317}a^{17}+\frac{20050018872}{3109916317}a^{16}+\frac{27503426741}{3109916317}a^{15}+\frac{43638224838}{3109916317}a^{14}+\frac{90177090315}{3109916317}a^{13}+\frac{52120368266}{3109916317}a^{12}+\frac{109175998581}{3109916317}a^{11}+\frac{100651815693}{3109916317}a^{10}+\frac{77315944349}{3109916317}a^{9}+\frac{75577029996}{3109916317}a^{8}-\frac{13796643822}{3109916317}a^{7}+\frac{68834445724}{3109916317}a^{6}-\frac{30486802476}{3109916317}a^{5}+\frac{7805078268}{3109916317}a^{4}-\frac{15347396313}{3109916317}a^{3}-\frac{1184351610}{3109916317}a^{2}-\frac{5605341028}{3109916317}a-\frac{1537273841}{3109916317}$, $\frac{193582041397}{301661882749}a^{21}+\frac{95496215614}{301661882749}a^{20}+\frac{876619079538}{301661882749}a^{19}+\frac{1771415662297}{301661882749}a^{18}+\frac{3794321940372}{301661882749}a^{17}+\frac{7013055180402}{301661882749}a^{16}+\frac{9334230310662}{301661882749}a^{15}+\frac{20514880504378}{301661882749}a^{14}+\frac{21869791787475}{301661882749}a^{13}+\frac{26109730019024}{301661882749}a^{12}+\frac{33466035325281}{301661882749}a^{11}+\frac{28180907560693}{301661882749}a^{10}+\frac{28277112242411}{301661882749}a^{9}+\frac{9882617387436}{301661882749}a^{8}+\frac{9024361442768}{301661882749}a^{7}+\frac{2192771072795}{301661882749}a^{6}-\frac{3551585189776}{301661882749}a^{5}-\frac{1223007200372}{301661882749}a^{4}-\frac{2570977684680}{301661882749}a^{3}-\frac{1097173088770}{301661882749}a^{2}-\frac{646810013111}{301661882749}a+\frac{46333347287}{301661882749}$, $\frac{124858618603}{301661882749}a^{21}+\frac{19816654506}{301661882749}a^{20}+\frac{599858870681}{301661882749}a^{19}+\frac{850234397599}{301661882749}a^{18}+\frac{2345011495088}{301661882749}a^{17}+\frac{3621694798946}{301661882749}a^{16}+\frac{4858959071169}{301661882749}a^{15}+\frac{11283414274094}{301661882749}a^{14}+\frac{9361690758845}{301661882749}a^{13}+\frac{14420027639008}{301661882749}a^{12}+\frac{12132392279370}{301661882749}a^{11}+\frac{12493900650570}{301661882749}a^{10}+\frac{11424960179423}{301661882749}a^{9}-\frac{2533206116203}{301661882749}a^{8}+\frac{4079663173603}{301661882749}a^{7}-\frac{6187528456003}{301661882749}a^{6}+\frac{3894891361966}{301661882749}a^{5}-\frac{4535929000369}{301661882749}a^{4}+\frac{835346330569}{301661882749}a^{3}-\frac{982003595171}{301661882749}a^{2}+\frac{1116316610597}{301661882749}a-\frac{164490359325}{301661882749}$, $\frac{231450834873}{301661882749}a^{21}-\frac{182706735489}{301661882749}a^{20}+\frac{1156154531432}{301661882749}a^{19}+\frac{739457230528}{301661882749}a^{18}+\frac{3085866592814}{301661882749}a^{17}+\frac{4072768647983}{301661882749}a^{16}+\frac{4859273096922}{301661882749}a^{15}+\frac{17095141168858}{301661882749}a^{14}+\frac{3987341257951}{301661882749}a^{13}+\frac{20618827294581}{301661882749}a^{12}+\frac{16653822341347}{301661882749}a^{11}+\frac{12543957495246}{301661882749}a^{10}+\frac{19167794237097}{301661882749}a^{9}-\frac{6552041739724}{301661882749}a^{8}+\frac{22864834580889}{301661882749}a^{7}-\frac{10732482431173}{301661882749}a^{6}+\frac{7364883049845}{301661882749}a^{5}-\frac{5434503452351}{301661882749}a^{4}+\frac{1726250765225}{301661882749}a^{3}-\frac{1660968573246}{301661882749}a^{2}+\frac{110545271993}{301661882749}a-\frac{296523341161}{301661882749}$, $\frac{50206761780}{301661882749}a^{21}-\frac{2264154158}{301661882749}a^{20}+\frac{158013472195}{301661882749}a^{19}+\frac{347363194153}{301661882749}a^{18}+\frac{553051529063}{301661882749}a^{17}+\frac{951913004985}{301661882749}a^{16}+\frac{898518785637}{301661882749}a^{15}+\frac{3077635566024}{301661882749}a^{14}+\frac{2255570102589}{301661882749}a^{13}+\frac{823722004398}{301661882749}a^{12}+\frac{4174657443929}{301661882749}a^{11}+\frac{3100558166071}{301661882749}a^{10}+\frac{1712086618601}{301661882749}a^{9}+\frac{35359291676}{301661882749}a^{8}+\frac{2229720182821}{301661882749}a^{7}+\frac{4957178239896}{301661882749}a^{6}+\frac{60124317146}{301661882749}a^{5}-\frac{453485967592}{301661882749}a^{4}+\frac{1288967876723}{301661882749}a^{3}-\frac{194533483362}{301661882749}a^{2}-\frac{68402979841}{301661882749}a-\frac{652846763652}{301661882749}$, $\frac{210522449411}{301661882749}a^{21}-\frac{81310881357}{301661882749}a^{20}+\frac{1057955106187}{301661882749}a^{19}+\frac{1024427715125}{301661882749}a^{18}+\frac{3436796545017}{301661882749}a^{17}+\frac{4958097509452}{301661882749}a^{16}+\frac{6769013644594}{301661882749}a^{15}+\frac{18134531476097}{301661882749}a^{14}+\frac{10647057316758}{301661882749}a^{13}+\frac{24264525152797}{301661882749}a^{12}+\frac{21451944781485}{301661882749}a^{11}+\frac{21249001732693}{301661882749}a^{10}+\frac{22067348185940}{301661882749}a^{9}+\frac{650526841544}{301661882749}a^{8}+\frac{19583402863286}{301661882749}a^{7}-\frac{8916168454950}{301661882749}a^{6}+\frac{7216035115978}{301661882749}a^{5}-\frac{8463219062632}{301661882749}a^{4}+\frac{2955149621986}{301661882749}a^{3}-\frac{3421264882089}{301661882749}a^{2}+\frac{111604358274}{301661882749}a-\frac{513281188339}{301661882749}$, $\frac{395797514908}{301661882749}a^{21}-\frac{244324649049}{301661882749}a^{20}+\frac{1820689957406}{301661882749}a^{19}+\frac{1447811333063}{301661882749}a^{18}+\frac{5072170896934}{301661882749}a^{17}+\frac{6505139276007}{301661882749}a^{16}+\frac{6953587789137}{301661882749}a^{15}+\frac{25742413488824}{301661882749}a^{14}+\frac{4953058486080}{301661882749}a^{13}+\frac{23902090564761}{301661882749}a^{12}+\frac{16330971729635}{301661882749}a^{11}+\frac{10548088263669}{301661882749}a^{10}+\frac{15164509966079}{301661882749}a^{9}-\frac{26106093238045}{301661882749}a^{8}+\frac{22146978247444}{301661882749}a^{7}-\frac{18904382063107}{301661882749}a^{6}+\frac{11646962341102}{301661882749}a^{5}-\frac{10218855774566}{301661882749}a^{4}+\frac{5052187807920}{301661882749}a^{3}-\frac{1308869869108}{301661882749}a^{2}+\frac{860416644854}{301661882749}a-\frac{340109667589}{301661882749}$, $\frac{549046838751}{301661882749}a^{21}-\frac{146220779018}{301661882749}a^{20}+\frac{2589281422121}{301661882749}a^{19}+\frac{2953077206085}{301661882749}a^{18}+\frac{8636047440275}{301661882749}a^{17}+\frac{12877686201618}{301661882749}a^{16}+\frac{16504173292677}{301661882749}a^{15}+\frac{45017831034877}{301661882749}a^{14}+\frac{27584722372157}{301661882749}a^{13}+\frac{53556483648093}{301661882749}a^{12}+\frac{51041453586122}{301661882749}a^{11}+\frac{46901944138368}{301661882749}a^{10}+\frac{48132863460725}{301661882749}a^{9}-\frac{6479460261763}{301661882749}a^{8}+\frac{38803116672900}{301661882749}a^{7}-\frac{15965401844276}{301661882749}a^{6}+\frac{13858084472860}{301661882749}a^{5}-\frac{16944951460952}{301661882749}a^{4}+\frac{5404065291857}{301661882749}a^{3}-\frac{4863849010438}{301661882749}a^{2}+\frac{45078034258}{301661882749}a-\frac{1084624589314}{301661882749}$, $\frac{242460097863}{301661882749}a^{21}-\frac{2884137702}{301661882749}a^{20}+\frac{738313299201}{301661882749}a^{19}+\frac{1903420655909}{301661882749}a^{18}+\frac{2233867090001}{301661882749}a^{17}+\frac{5451964738292}{301661882749}a^{16}+\frac{3751229063271}{301661882749}a^{15}+\frac{15249122702384}{301661882749}a^{14}+\frac{10014592331121}{301661882749}a^{13}-\frac{269929862849}{301661882749}a^{12}+\frac{24438048304812}{301661882749}a^{11}-\frac{3247310411763}{301661882749}a^{10}+\frac{4162310879051}{301661882749}a^{9}-\frac{9648867127413}{301661882749}a^{8}-\frac{7893384939323}{301661882749}a^{7}+\frac{16893192997573}{301661882749}a^{6}-\frac{25342844853775}{301661882749}a^{5}+\frac{15267487842325}{301661882749}a^{4}-\frac{7663472192380}{301661882749}a^{3}+\frac{4143419735017}{301661882749}a^{2}-\frac{1472638790983}{301661882749}a+\frac{588539586350}{301661882749}$, $\frac{391788446750}{301661882749}a^{21}+\frac{121530615641}{301661882749}a^{20}+\frac{1801620498029}{301661882749}a^{19}+\frac{3213503840513}{301661882749}a^{18}+\frac{7362513951701}{301661882749}a^{17}+\frac{13048127674242}{301661882749}a^{16}+\frac{17160541911280}{301661882749}a^{15}+\frac{39685731289858}{301661882749}a^{14}+\frac{38218421228758}{301661882749}a^{13}+\frac{50819320301544}{301661882749}a^{12}+\frac{59710744087881}{301661882749}a^{11}+\frac{53262532902842}{301661882749}a^{10}+\frac{54285898840620}{301661882749}a^{9}+\frac{13357823215382}{301661882749}a^{8}+\frac{25062088028166}{301661882749}a^{7}-\frac{770838035715}{301661882749}a^{6}+\frac{2523499038109}{301661882749}a^{5}-\frac{7959788903634}{301661882749}a^{4}-\frac{1101432329447}{301661882749}a^{3}-\frac{2795402512381}{301661882749}a^{2}-\frac{648383521909}{301661882749}a-\frac{697589682559}{301661882749}$
| 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: | \( 399685.929551 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 399685.929551 \cdot 1}{6\cdot\sqrt{66009317547629600105613290523}}\cr\approx \mathstrut & 0.156222484339 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 5*x^20 + 2*x^19 + 12*x^18 + 13*x^17 + 14*x^16 + 63*x^15 - 7*x^14 + 68*x^13 + 31*x^12 + 19*x^11 + 40*x^10 - 71*x^9 + 84*x^8 - 79*x^7 + 41*x^6 - 38*x^5 + 21*x^4 - 11*x^3 + 4*x^2 - x + 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^22 - x^21 + 5*x^20 + 2*x^19 + 12*x^18 + 13*x^17 + 14*x^16 + 63*x^15 - 7*x^14 + 68*x^13 + 31*x^12 + 19*x^11 + 40*x^10 - 71*x^9 + 84*x^8 - 79*x^7 + 41*x^6 - 38*x^5 + 21*x^4 - 11*x^3 + 4*x^2 - x + 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^22 - x^21 + 5*x^20 + 2*x^19 + 12*x^18 + 13*x^17 + 14*x^16 + 63*x^15 - 7*x^14 + 68*x^13 + 31*x^12 + 19*x^11 + 40*x^10 - 71*x^9 + 84*x^8 - 79*x^7 + 41*x^6 - 38*x^5 + 21*x^4 - 11*x^3 + 4*x^2 - x + 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^22 - x^21 + 5*x^20 + 2*x^19 + 12*x^18 + 13*x^17 + 14*x^16 + 63*x^15 - 7*x^14 + 68*x^13 + 31*x^12 + 19*x^11 + 40*x^10 - 71*x^9 + 84*x^8 - 79*x^7 + 41*x^6 - 38*x^5 + 21*x^4 - 11*x^3 + 4*x^2 - x + 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
$C_2\times S_{11}$ (as 22T47):
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 79833600 |
| The 112 conjugacy class representatives for $C_2\times S_{11}$ |
| Character table for $C_2\times S_{11}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 11.7.610429790897.1 |
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 22 sibling: | data not computed |
| Degree 44 sibling: | 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 | $22$ | R | $18{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | $22$ | ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ | $22$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.11.0.1}{11} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $22$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.16.8.1 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ | |
|
\(610429790897\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |