Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^19 - x^18 - 8*x^17 + 16*x^15 + 16*x^14 + 21*x^13 - 27*x^12 - 29*x^11 - 63*x^10 - 3*x^9 + 86*x^8 + 60*x^7 + 11*x^6 - 69*x^5 - 49*x^4 + 30*x^3 + 16*x^2 - 6*x - 1)
gp: K = bnfinit(y^21 - 2*y^19 - y^18 - 8*y^17 + 16*y^15 + 16*y^14 + 21*y^13 - 27*y^12 - 29*y^11 - 63*y^10 - 3*y^9 + 86*y^8 + 60*y^7 + 11*y^6 - 69*y^5 - 49*y^4 + 30*y^3 + 16*y^2 - 6*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 2*x^19 - x^18 - 8*x^17 + 16*x^15 + 16*x^14 + 21*x^13 - 27*x^12 - 29*x^11 - 63*x^10 - 3*x^9 + 86*x^8 + 60*x^7 + 11*x^6 - 69*x^5 - 49*x^4 + 30*x^3 + 16*x^2 - 6*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 2*x^19 - x^18 - 8*x^17 + 16*x^15 + 16*x^14 + 21*x^13 - 27*x^12 - 29*x^11 - 63*x^10 - 3*x^9 + 86*x^8 + 60*x^7 + 11*x^6 - 69*x^5 - 49*x^4 + 30*x^3 + 16*x^2 - 6*x - 1)
\( x^{21} - 2 x^{19} - x^{18} - 8 x^{17} + 16 x^{15} + 16 x^{14} + 21 x^{13} - 27 x^{12} - 29 x^{11} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[5, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(12329728569841547876503864001\)
\(\medspace = 7^{9}\cdot 12503^{5}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.76\) | 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: | $7^{1/2}12503^{1/2}\approx 295.8394835041462$ | ||
| Ramified primes: |
\(7\), \(12503\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{87521}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3}a^{18}-\frac{1}{3}a^{17}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{19}-\frac{1}{3}a^{17}-\frac{1}{3}a^{15}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{13458429203169}a^{20}+\frac{252438425646}{4486143067723}a^{19}-\frac{1849290991741}{13458429203169}a^{18}-\frac{389171437909}{4486143067723}a^{17}-\frac{4599705812698}{13458429203169}a^{16}-\frac{2152989302937}{4486143067723}a^{15}-\frac{861306887090}{4486143067723}a^{14}+\frac{91273587584}{4486143067723}a^{13}-\frac{1284864523994}{13458429203169}a^{12}-\frac{3977374637917}{13458429203169}a^{11}+\frac{1738759474967}{4486143067723}a^{10}-\frac{925523404990}{13458429203169}a^{9}+\frac{4646323794359}{13458429203169}a^{8}+\frac{3045195796811}{13458429203169}a^{7}-\frac{1063726261439}{4486143067723}a^{6}-\frac{6618751463020}{13458429203169}a^{5}+\frac{777365339824}{13458429203169}a^{4}+\frac{928487349540}{4486143067723}a^{3}-\frac{4721462855510}{13458429203169}a^{2}-\frac{2487512109382}{13458429203169}a-\frac{797789331107}{4486143067723}$
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: | $12$ | 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: |
$\frac{3564166044841}{13458429203169}a^{20}-\frac{2309032631638}{13458429203169}a^{19}-\frac{3987848183800}{13458429203169}a^{18}-\frac{545264596148}{13458429203169}a^{17}-\frac{31050476181073}{13458429203169}a^{16}+\frac{17493247846723}{13458429203169}a^{15}+\frac{10286979463062}{4486143067723}a^{14}+\frac{10869306945797}{4486143067723}a^{13}+\frac{76838172867361}{13458429203169}a^{12}-\frac{111196936537442}{13458429203169}a^{11}+\frac{21152682824314}{13458429203169}a^{10}-\frac{261338132378656}{13458429203169}a^{9}+\frac{36611170796129}{4486143067723}a^{8}+\frac{34149907412314}{4486143067723}a^{7}+\frac{87011197526440}{13458429203169}a^{6}+\frac{81223183070339}{13458429203169}a^{5}-\frac{171191436652336}{13458429203169}a^{4}+\frac{22451679876764}{13458429203169}a^{3}+\frac{38311179490987}{13458429203169}a^{2}-\frac{49893604651988}{13458429203169}a+\frac{8666462474098}{13458429203169}$, $\frac{1826018220280}{4486143067723}a^{20}-\frac{770731019127}{4486143067723}a^{19}-\frac{8324471970958}{13458429203169}a^{18}-\frac{2517833812082}{13458429203169}a^{17}-\frac{15401634941294}{4486143067723}a^{16}+\frac{6039381210413}{4486143067723}a^{15}+\frac{67068987593755}{13458429203169}a^{14}+\frac{65236308984644}{13458429203169}a^{13}+\frac{116484903284224}{13458429203169}a^{12}-\frac{171450814083097}{13458429203169}a^{11}-\frac{18898009269108}{4486143067723}a^{10}-\frac{388957232169590}{13458429203169}a^{9}+\frac{94337692385840}{13458429203169}a^{8}+\frac{314407223880899}{13458429203169}a^{7}+\frac{222024146244782}{13458429203169}a^{6}+\frac{134114788449452}{13458429203169}a^{5}-\frac{320270207626802}{13458429203169}a^{4}-\frac{34397316925830}{4486143067723}a^{3}+\frac{62979474490337}{13458429203169}a^{2}-\frac{33767219714960}{13458429203169}a+\frac{15471543853474}{13458429203169}$, $a$, $\frac{582609588499}{4486143067723}a^{20}-\frac{1999872065328}{4486143067723}a^{19}-\frac{3323363601949}{13458429203169}a^{18}+\frac{5614713461581}{13458429203169}a^{17}-\frac{3140474514692}{4486143067723}a^{16}+\frac{17745378006814}{4486143067723}a^{15}+\frac{31411689939718}{13458429203169}a^{14}-\frac{25627217459518}{13458429203169}a^{13}-\frac{42682184916467}{13458429203169}a^{12}-\frac{208301041924396}{13458429203169}a^{11}+\frac{15031750781188}{4486143067723}a^{10}-\frac{82899929326400}{13458429203169}a^{9}+\frac{404334287610206}{13458429203169}a^{8}+\frac{187253988176543}{13458429203169}a^{7}-\frac{94518429331858}{13458429203169}a^{6}-\frac{217871773811593}{13458429203169}a^{5}-\frac{303517436224754}{13458429203169}a^{4}+\frac{42245021096252}{4486143067723}a^{3}+\frac{144273182387969}{13458429203169}a^{2}-\frac{30549614963513}{13458429203169}a-\frac{2640833854937}{13458429203169}$, $\frac{290961355540}{4486143067723}a^{20}-\frac{1202303038205}{13458429203169}a^{19}-\frac{128440827954}{4486143067723}a^{18}+\frac{105338668676}{13458429203169}a^{17}-\frac{2682711581826}{4486143067723}a^{16}+\frac{9757982582144}{13458429203169}a^{15}+\frac{911368750814}{4486143067723}a^{14}+\frac{2256987145664}{4486143067723}a^{13}+\frac{6350496896915}{4486143067723}a^{12}-\frac{37100175100931}{13458429203169}a^{11}+\frac{29339830985153}{13458429203169}a^{10}-\frac{28276736771872}{4486143067723}a^{9}+\frac{59939779410893}{13458429203169}a^{8}-\frac{24623602467280}{13458429203169}a^{7}+\frac{13576581510143}{13458429203169}a^{6}+\frac{8252851365268}{4486143067723}a^{5}-\frac{28252407717445}{13458429203169}a^{4}+\frac{58798908408976}{13458429203169}a^{3}-\frac{1328074547717}{4486143067723}a^{2}-\frac{33848244137243}{13458429203169}a+\frac{181302859088}{13458429203169}$, $\frac{2671160955401}{13458429203169}a^{20}-\frac{6560820062125}{13458429203169}a^{19}-\frac{8029160271859}{13458429203169}a^{18}+\frac{1462306372807}{4486143067723}a^{17}-\frac{13730515424072}{13458429203169}a^{16}+\frac{61307958323089}{13458429203169}a^{15}+\frac{73255420670354}{13458429203169}a^{14}-\frac{3860948361401}{13458429203169}a^{13}-\frac{46928740462301}{13458429203169}a^{12}-\frac{290877649295006}{13458429203169}a^{11}-\frac{65468924963345}{13458429203169}a^{10}-\frac{44256758833487}{4486143067723}a^{9}+\frac{146396790745668}{4486143067723}a^{8}+\frac{155642095499703}{4486143067723}a^{7}+\frac{21296725074905}{13458429203169}a^{6}-\frac{260216192477419}{13458429203169}a^{5}-\frac{510363110698030}{13458429203169}a^{4}-\frac{24929834167321}{13458429203169}a^{3}+\frac{76698136506534}{4486143067723}a^{2}+\frac{21576621093476}{13458429203169}a-\frac{2892994404910}{4486143067723}$, $\frac{1251972987460}{13458429203169}a^{20}+\frac{322604719084}{13458429203169}a^{19}-\frac{342542445014}{4486143067723}a^{18}-\frac{2925732133310}{13458429203169}a^{17}-\frac{12932332549834}{13458429203169}a^{16}-\frac{3086639132962}{13458429203169}a^{15}+\frac{8448372800390}{13458429203169}a^{14}+\frac{34053676243975}{13458429203169}a^{13}+\frac{17764944875648}{4486143067723}a^{12}-\frac{9027379180406}{13458429203169}a^{11}-\frac{22687105712356}{13458429203169}a^{10}-\frac{150566276010296}{13458429203169}a^{9}-\frac{46871964562954}{13458429203169}a^{8}-\frac{4209411168676}{13458429203169}a^{7}+\frac{53622461568884}{4486143067723}a^{6}+\frac{46312915680499}{4486143067723}a^{5}-\frac{5958709048666}{13458429203169}a^{4}-\frac{78910485458099}{13458429203169}a^{3}-\frac{83104914797413}{13458429203169}a^{2}-\frac{1553732833474}{13458429203169}a+\frac{14144517134938}{13458429203169}$, $\frac{3900543319695}{4486143067723}a^{20}-\frac{6601682650577}{13458429203169}a^{19}-\frac{20969876801432}{13458429203169}a^{18}-\frac{2242547724554}{13458429203169}a^{17}-\frac{30566342473729}{4486143067723}a^{16}+\frac{55549795105781}{13458429203169}a^{15}+\frac{170548350440585}{13458429203169}a^{14}+\frac{114677024944708}{13458429203169}a^{13}+\frac{180660565273139}{13458429203169}a^{12}-\frac{453762577566166}{13458429203169}a^{11}-\frac{159814037310658}{13458429203169}a^{10}-\frac{712018158241141}{13458429203169}a^{9}+\frac{124594764823121}{4486143067723}a^{8}+\frac{299496532998711}{4486143067723}a^{7}+\frac{125409734340977}{4486143067723}a^{6}-\frac{24496798437962}{13458429203169}a^{5}-\frac{885520338088334}{13458429203169}a^{4}-\frac{212135846257601}{13458429203169}a^{3}+\frac{389404984438834}{13458429203169}a^{2}+\frac{8684216946853}{4486143067723}a-\frac{50079172231877}{13458429203169}$, $\frac{2332210300812}{4486143067723}a^{20}+\frac{4479982927660}{13458429203169}a^{19}-\frac{4383599715000}{4486143067723}a^{18}-\frac{15874203173359}{13458429203169}a^{17}-\frac{20805941196773}{4486143067723}a^{16}-\frac{37184375076136}{13458429203169}a^{15}+\frac{35067412726374}{4486143067723}a^{14}+\frac{61071132798381}{4486143067723}a^{13}+\frac{78306805104015}{4486143067723}a^{12}-\frac{76372453776734}{13458429203169}a^{11}-\frac{304231208950462}{13458429203169}a^{10}-\frac{198017339407949}{4486143067723}a^{9}-\frac{341788128655351}{13458429203169}a^{8}+\frac{523337025434729}{13458429203169}a^{7}+\frac{793786567840256}{13458429203169}a^{6}+\frac{143327618038786}{4486143067723}a^{5}-\frac{355836075616825}{13458429203169}a^{4}-\frac{634796265899252}{13458429203169}a^{3}-\frac{24841249556758}{4486143067723}a^{2}+\frac{175563478014247}{13458429203169}a+\frac{41170433224634}{13458429203169}$, $\frac{1029672745922}{13458429203169}a^{20}-\frac{881286839531}{13458429203169}a^{19}+\frac{20409846106}{4486143067723}a^{18}+\frac{570489936458}{4486143067723}a^{17}-\frac{8954487578711}{13458429203169}a^{16}+\frac{4701909290162}{13458429203169}a^{15}-\frac{3661704585737}{13458429203169}a^{14}-\frac{8586328041757}{13458429203169}a^{13}+\frac{5416749201747}{4486143067723}a^{12}-\frac{14507269064240}{13458429203169}a^{11}+\frac{67260230165387}{13458429203169}a^{10}-\frac{36365530410418}{13458429203169}a^{9}+\frac{59108984168891}{13458429203169}a^{8}-\frac{57598844803042}{13458429203169}a^{7}-\frac{92563193769647}{13458429203169}a^{6}-\frac{10493524840243}{4486143067723}a^{5}+\frac{2242098417344}{13458429203169}a^{4}+\frac{126406257614323}{13458429203169}a^{3}+\frac{61311313456468}{13458429203169}a^{2}-\frac{14279703647701}{4486143067723}a-\frac{5404614126418}{4486143067723}$, $\frac{1444055020823}{4486143067723}a^{20}-\frac{2514203448341}{13458429203169}a^{19}-\frac{6038472299768}{13458429203169}a^{18}-\frac{1510712478698}{13458429203169}a^{17}-\frac{12022029862669}{4486143067723}a^{16}+\frac{19952098188410}{13458429203169}a^{15}+\frac{48359111840402}{13458429203169}a^{14}+\frac{47415367531240}{13458429203169}a^{13}+\frac{82546564289774}{13458429203169}a^{12}-\frac{146449316387095}{13458429203169}a^{11}-\frac{19511086129432}{13458429203169}a^{10}-\frac{310077936403249}{13458429203169}a^{9}+\frac{46915638172632}{4486143067723}a^{8}+\frac{66583564386904}{4486143067723}a^{7}+\frac{59038915395009}{4486143067723}a^{6}+\frac{33764788813318}{13458429203169}a^{5}-\frac{245611242604337}{13458429203169}a^{4}-\frac{60111272284862}{13458429203169}a^{3}+\frac{84412937474035}{13458429203169}a^{2}-\frac{3680493628548}{4486143067723}a+\frac{4998453728989}{13458429203169}$, $\frac{4481378219306}{13458429203169}a^{20}-\frac{55549947001}{13458429203169}a^{19}-\frac{6987497659901}{13458429203169}a^{18}-\frac{3131073907202}{13458429203169}a^{17}-\frac{38346723821984}{13458429203169}a^{16}-\frac{2806573862195}{13458429203169}a^{15}+\frac{17822982899752}{4486143067723}a^{14}+\frac{19203384898669}{4486143067723}a^{13}+\frac{109734266502065}{13458429203169}a^{12}-\frac{27500217686923}{4486143067723}a^{11}-\frac{59452632894365}{13458429203169}a^{10}-\frac{272887511765849}{13458429203169}a^{9}-\frac{30456159481372}{13458429203169}a^{8}+\frac{242931785232770}{13458429203169}a^{7}+\frac{157996192691956}{13458429203169}a^{6}+\frac{81067925178175}{13458429203169}a^{5}-\frac{68251786866617}{4486143067723}a^{4}-\frac{128682311152198}{13458429203169}a^{3}+\frac{131324145780194}{13458429203169}a^{2}+\frac{14938807678971}{4486143067723}a-\frac{18674610073115}{13458429203169}$
| 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: | \( 564369.598704 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{8}\cdot 564369.598704 \cdot 1}{2\cdot\sqrt{12329728569841547876503864001}}\cr\approx \mathstrut & 0.197535785082 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^19 - x^18 - 8*x^17 + 16*x^15 + 16*x^14 + 21*x^13 - 27*x^12 - 29*x^11 - 63*x^10 - 3*x^9 + 86*x^8 + 60*x^7 + 11*x^6 - 69*x^5 - 49*x^4 + 30*x^3 + 16*x^2 - 6*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^21 - 2*x^19 - x^18 - 8*x^17 + 16*x^15 + 16*x^14 + 21*x^13 - 27*x^12 - 29*x^11 - 63*x^10 - 3*x^9 + 86*x^8 + 60*x^7 + 11*x^6 - 69*x^5 - 49*x^4 + 30*x^3 + 16*x^2 - 6*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^21 - 2*x^19 - x^18 - 8*x^17 + 16*x^15 + 16*x^14 + 21*x^13 - 27*x^12 - 29*x^11 - 63*x^10 - 3*x^9 + 86*x^8 + 60*x^7 + 11*x^6 - 69*x^5 - 49*x^4 + 30*x^3 + 16*x^2 - 6*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^21 - 2*x^19 - x^18 - 8*x^17 + 16*x^15 + 16*x^14 + 21*x^13 - 27*x^12 - 29*x^11 - 63*x^10 - 3*x^9 + 86*x^8 + 60*x^7 + 11*x^6 - 69*x^5 - 49*x^4 + 30*x^3 + 16*x^2 - 6*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
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 5040 |
| The 15 conjugacy class representatives for $S_7$ |
| Character table for $S_7$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 7 sibling: | 7.3.4288529.1 |
| Degree 14 sibling: | deg 14 |
| Degree 30 sibling: | data not computed |
| Degree 35 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
| Minimal sibling: | 7.3.4288529.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{3}$ | ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.7.0.1}{7} }^{3}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{3}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }^{3}$ | ${\href{/padicField/47.7.0.1}{7} }^{3}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 7.6.3.1 | $x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 7.6.3.1 | $x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 7.6.3.1 | $x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(12503\)
| $\Q_{12503}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{12503}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{12503}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |