Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 90*x^20 - 515*x^19 + 2404*x^18 - 9153*x^17 + 29609*x^16 - 81730*x^15 + 195276*x^14 - 404718*x^13 + 730608*x^12 - 1147506*x^11 + 1565057*x^10 - 1843981*x^9 + 1862690*x^8 - 1595475*x^7 + 1141091*x^6 - 667142*x^5 + 309476*x^4 - 109065*x^3 + 27307*x^2 - 4313*x + 323)
gp: K = bnfinit(y^22 - 11*y^21 + 90*y^20 - 515*y^19 + 2404*y^18 - 9153*y^17 + 29609*y^16 - 81730*y^15 + 195276*y^14 - 404718*y^13 + 730608*y^12 - 1147506*y^11 + 1565057*y^10 - 1843981*y^9 + 1862690*y^8 - 1595475*y^7 + 1141091*y^6 - 667142*y^5 + 309476*y^4 - 109065*y^3 + 27307*y^2 - 4313*y + 323, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^21 + 90*x^20 - 515*x^19 + 2404*x^18 - 9153*x^17 + 29609*x^16 - 81730*x^15 + 195276*x^14 - 404718*x^13 + 730608*x^12 - 1147506*x^11 + 1565057*x^10 - 1843981*x^9 + 1862690*x^8 - 1595475*x^7 + 1141091*x^6 - 667142*x^5 + 309476*x^4 - 109065*x^3 + 27307*x^2 - 4313*x + 323);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 11*x^21 + 90*x^20 - 515*x^19 + 2404*x^18 - 9153*x^17 + 29609*x^16 - 81730*x^15 + 195276*x^14 - 404718*x^13 + 730608*x^12 - 1147506*x^11 + 1565057*x^10 - 1843981*x^9 + 1862690*x^8 - 1595475*x^7 + 1141091*x^6 - 667142*x^5 + 309476*x^4 - 109065*x^3 + 27307*x^2 - 4313*x + 323)
\( x^{22} - 11 x^{21} + 90 x^{20} - 515 x^{19} + 2404 x^{18} - 9153 x^{17} + 29609 x^{16} - 81730 x^{15} + \cdots + 323 \)
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: |
\(-20375776275370821024303050129100543619\)
\(\medspace = -\,19^{11}\cdot 211^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(49.64\) | 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: | $19^{1/2}211^{1/2}\approx 63.31666447310692$ | ||
| Ramified primes: |
\(19\), \(211\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{1024}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13}a^{18}+\frac{4}{13}a^{17}+\frac{2}{13}a^{16}+\frac{6}{13}a^{15}-\frac{3}{13}a^{14}-\frac{5}{13}a^{13}-\frac{3}{13}a^{12}+\frac{5}{13}a^{11}+\frac{3}{13}a^{10}+\frac{2}{13}a^{9}-\frac{1}{13}a^{8}-\frac{5}{13}a^{7}-\frac{5}{13}a^{6}+\frac{1}{13}a^{5}-\frac{1}{13}a^{4}-\frac{3}{13}a^{3}+\frac{2}{13}a^{2}-\frac{4}{13}$, $\frac{1}{13}a^{19}-\frac{1}{13}a^{17}-\frac{2}{13}a^{16}-\frac{1}{13}a^{15}-\frac{6}{13}a^{14}+\frac{4}{13}a^{13}+\frac{4}{13}a^{12}-\frac{4}{13}a^{11}+\frac{3}{13}a^{10}+\frac{4}{13}a^{9}-\frac{1}{13}a^{8}+\frac{2}{13}a^{7}-\frac{5}{13}a^{6}-\frac{5}{13}a^{5}+\frac{1}{13}a^{4}+\frac{1}{13}a^{3}+\frac{5}{13}a^{2}-\frac{4}{13}a+\frac{3}{13}$, $\frac{1}{8076887}a^{20}-\frac{10}{8076887}a^{19}-\frac{12228}{351169}a^{18}+\frac{2531481}{8076887}a^{17}+\frac{3278487}{8076887}a^{16}-\frac{2840554}{8076887}a^{15}-\frac{1522233}{8076887}a^{14}+\frac{2742228}{8076887}a^{13}+\frac{3337462}{8076887}a^{12}-\frac{2045200}{8076887}a^{11}+\frac{681015}{8076887}a^{10}-\frac{3607920}{8076887}a^{9}-\frac{32253}{621299}a^{8}-\frac{3450623}{8076887}a^{7}-\frac{1629643}{8076887}a^{6}-\frac{455875}{1153841}a^{5}-\frac{2758699}{8076887}a^{4}-\frac{46906}{8076887}a^{3}+\frac{3416279}{8076887}a^{2}-\frac{324342}{1153841}a+\frac{17677}{36547}$, $\frac{1}{2802679789}a^{21}+\frac{163}{2802679789}a^{20}+\frac{1889922}{215590753}a^{19}-\frac{66626598}{2802679789}a^{18}-\frac{171463894}{400382827}a^{17}+\frac{27276714}{400382827}a^{16}+\frac{56147757}{400382827}a^{15}-\frac{1277049245}{2802679789}a^{14}-\frac{717636370}{2802679789}a^{13}+\frac{6458485}{23551931}a^{12}+\frac{236474166}{2802679789}a^{11}+\frac{272638920}{2802679789}a^{10}+\frac{60717226}{121855643}a^{9}+\frac{28532081}{164863517}a^{8}-\frac{187908783}{2802679789}a^{7}-\frac{323018603}{2802679789}a^{6}+\frac{87220566}{215590753}a^{5}+\frac{593817643}{2802679789}a^{4}-\frac{944102547}{2802679789}a^{3}+\frac{62254386}{215590753}a^{2}-\frac{504433159}{2802679789}a-\frac{42731006}{164863517}$
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
$C_{23}$, which has order $23$ (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: |
\( -1 \)
(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: |
$\frac{2453832}{8076887}a^{20}-\frac{24538320}{8076887}a^{19}+\frac{8497513}{351169}a^{18}-\frac{1059643071}{8076887}a^{17}+\frac{4770715851}{8076887}a^{16}-\frac{17317501476}{8076887}a^{15}+\frac{53668023076}{8076887}a^{14}-\frac{140833147948}{8076887}a^{13}+\frac{319629431799}{8076887}a^{12}-\frac{624472336448}{8076887}a^{11}+\frac{1057285916327}{8076887}a^{10}-\frac{1541891023466}{8076887}a^{9}+\frac{1932100263287}{8076887}a^{8}-\frac{2058566106110}{8076887}a^{7}+\frac{141726452696}{621299}a^{6}-\frac{193986302017}{1153841}a^{5}+\frac{800616326282}{8076887}a^{4}-\frac{361533964348}{8076887}a^{3}+\frac{116098224722}{8076887}a^{2}-\frac{3315472531}{1153841}a+\frac{124544029}{475111}$, $\frac{17138874}{8076887}a^{20}-\frac{171388740}{8076887}a^{19}+\frac{59424838}{351169}a^{18}-\frac{7416362376}{8076887}a^{17}+\frac{33439394080}{8076887}a^{16}-\frac{121554363992}{8076887}a^{15}+\frac{377438526826}{8076887}a^{14}-\frac{992515940378}{8076887}a^{13}+\frac{2258603590517}{8076887}a^{12}-\frac{4426079891252}{8076887}a^{11}+\frac{7522625940483}{8076887}a^{10}-\frac{11020755682486}{8076887}a^{9}+\frac{13891923031016}{8076887}a^{8}-\frac{14911180847224}{8076887}a^{7}+\frac{13480695438024}{8076887}a^{6}-\frac{1438500853729}{1153841}a^{5}+\frac{6054024186508}{8076887}a^{4}-\frac{2812530336017}{8076887}a^{3}+\frac{945560979268}{8076887}a^{2}-\frac{29140601186}{1153841}a+\frac{1253731184}{475111}$, $\frac{17138874}{8076887}a^{20}-\frac{171388740}{8076887}a^{19}+\frac{59424838}{351169}a^{18}-\frac{7416362376}{8076887}a^{17}+\frac{33439394080}{8076887}a^{16}-\frac{121554363992}{8076887}a^{15}+\frac{377438526826}{8076887}a^{14}-\frac{992515940378}{8076887}a^{13}+\frac{2258603590517}{8076887}a^{12}-\frac{4426079891252}{8076887}a^{11}+\frac{7522625940483}{8076887}a^{10}-\frac{11020755682486}{8076887}a^{9}+\frac{13891923031016}{8076887}a^{8}-\frac{14911180847224}{8076887}a^{7}+\frac{13480695438024}{8076887}a^{6}-\frac{1438500853729}{1153841}a^{5}+\frac{6054024186508}{8076887}a^{4}-\frac{2812530336017}{8076887}a^{3}+\frac{945560979268}{8076887}a^{2}-\frac{29140601186}{1153841}a+\frac{1254206295}{475111}$, $\frac{131927407}{8076887}a^{20}-\frac{1319274070}{8076887}a^{19}+\frac{457172355}{351169}a^{18}-\frac{57035366490}{8076887}a^{17}+\frac{256991302805}{8076887}a^{16}-\frac{933578991844}{8076887}a^{15}+\frac{2896229546038}{8076887}a^{14}-\frac{7608529157316}{8076887}a^{13}+\frac{17292248348381}{8076887}a^{12}-\frac{33837812940318}{8076887}a^{11}+\frac{57404941703449}{8076887}a^{10}-\frac{83914810564798}{8076887}a^{9}+\frac{105477775374277}{8076887}a^{8}-\frac{112821875575160}{8076887}a^{7}+\frac{101526157354668}{8076887}a^{6}-\frac{10768595872104}{1153841}a^{5}+\frac{44940068157477}{8076887}a^{4}-\frac{20634494614962}{8076887}a^{3}+\frac{6814967935108}{8076887}a^{2}-\frac{204342717727}{1153841}a+\frac{8418219713}{475111}$, $\frac{699702}{621299}a^{20}-\frac{6997020}{621299}a^{19}+\frac{31527619}{351169}a^{18}-\frac{3933821223}{8076887}a^{17}+\frac{17729586904}{8076887}a^{16}-\frac{64422275636}{8076887}a^{15}+\frac{199924504210}{8076887}a^{14}-\frac{525403882238}{8076887}a^{13}+\frac{1194681318937}{8076887}a^{12}-\frac{2339052797366}{8076887}a^{11}+\frac{3970906996990}{8076887}a^{10}-\frac{5809484935615}{8076887}a^{9}+\frac{7310055389226}{8076887}a^{8}-\frac{7829248943114}{8076887}a^{7}+\frac{7057553994347}{8076887}a^{6}-\frac{750248297510}{1153841}a^{5}+\frac{3140766606541}{8076887}a^{4}-\frac{1448410215822}{8076887}a^{3}+\frac{481619583994}{8076887}a^{2}-\frac{1122926348}{88757}a+\frac{612365200}{475111}$, $\frac{25392344}{8076887}a^{20}-\frac{253923440}{8076887}a^{19}+\frac{87977797}{351169}a^{18}-\frac{10974585939}{8076887}a^{17}+\frac{49438990247}{8076887}a^{16}-\frac{179561549140}{8076887}a^{15}+\frac{556887267816}{8076887}a^{14}-\frac{1462499707016}{8076887}a^{13}+\frac{3322465967103}{8076887}a^{12}-\frac{6498240155324}{8076887}a^{11}+\frac{11016881867070}{8076887}a^{10}-\frac{16091731587881}{8076887}a^{9}+\frac{20205174185587}{8076887}a^{8}-\frac{21582629006552}{8076887}a^{7}+\frac{19384941270018}{8076887}a^{6}-\frac{2050805856997}{1153841}a^{5}+\frac{8525496389114}{8076887}a^{4}-\frac{3892060601809}{8076887}a^{3}+\frac{1273074094343}{8076887}a^{2}-\frac{37545256699}{1153841}a+\frac{1502737744}{475111}$, $\frac{59891807}{8076887}a^{20}-\frac{598918070}{8076887}a^{19}+\frac{207558536}{351169}a^{18}-\frac{25895451957}{8076887}a^{17}+\frac{116689674065}{8076887}a^{16}-\frac{423934029472}{8076887}a^{15}+\frac{1315307742850}{8076887}a^{14}-\frac{3455780132892}{8076887}a^{13}+\frac{7855318220483}{8076887}a^{12}-\frac{15374153611840}{8076887}a^{11}+\frac{26087748714458}{8076887}a^{10}-\frac{38145476394875}{8076887}a^{9}+\frac{47964118407238}{8076887}a^{8}-\frac{51325751806262}{8076887}a^{7}+\frac{46213180450483}{8076887}a^{6}-\frac{4905279690693}{1153841}a^{5}+\frac{20491430378821}{8076887}a^{4}-\frac{9421585830339}{8076887}a^{3}+\frac{3117822693227}{8076887}a^{2}-\frac{93759429886}{1153841}a+\frac{3881089388}{475111}$, $\frac{7837972}{1153841}a^{20}-\frac{78379720}{1153841}a^{19}+\frac{27164030}{50167}a^{18}-\frac{3389132190}{1153841}a^{17}+\frac{15272687843}{1153841}a^{16}-\frac{55487832928}{1153841}a^{15}+\frac{172165418922}{1153841}a^{14}-\frac{452358710834}{1153841}a^{13}+\frac{1028297762473}{1153841}a^{12}-\frac{2012625313468}{1153841}a^{11}+\frac{3415246361109}{1153841}a^{10}-\frac{4993881660542}{1153841}a^{9}+\frac{6279354227425}{1153841}a^{8}-\frac{6719385651032}{1153841}a^{7}+\frac{6049863240453}{1153841}a^{6}-\frac{4494856669138}{1153841}a^{5}+\frac{2682204854512}{1153841}a^{4}-\frac{1233121859482}{1153841}a^{3}+\frac{408068672159}{1153841}a^{2}-\frac{85920626224}{1153841}a+\frac{508360677}{67873}$, $\frac{297663644}{8076887}a^{20}-\frac{2976636440}{8076887}a^{19}+\frac{79348803}{27013}a^{18}-\frac{128693490333}{8076887}a^{17}+\frac{579891603266}{8076887}a^{16}-\frac{2106661806628}{8076887}a^{15}+\frac{6535798111680}{8076887}a^{14}-\frac{17170781591948}{8076887}a^{13}+\frac{39027616715216}{8076887}a^{12}-\frac{76376307826358}{8076887}a^{11}+\frac{129584122838809}{8076887}a^{10}-\frac{14573129698815}{621299}a^{9}+\frac{238171522295853}{8076887}a^{8}-\frac{254806511694586}{8076887}a^{7}+\frac{229357232731125}{8076887}a^{6}-\frac{1871985123021}{88757}a^{5}+\frac{101609007227145}{8076887}a^{4}-\frac{46685975162451}{8076887}a^{3}+\frac{1187290915746}{621299}a^{2}-\frac{463536556469}{1153841}a+\frac{19145688279}{475111}$, $\frac{435389056}{8076887}a^{20}-\frac{4353890560}{8076887}a^{19}+\frac{1508713931}{351169}a^{18}-\frac{188217902757}{8076887}a^{17}+\frac{848040301214}{8076887}a^{16}-\frac{3080572607572}{8076887}a^{15}+\frac{9556279969282}{8076887}a^{14}-\frac{25103279469426}{8076887}a^{13}+\frac{57048963745425}{8076887}a^{12}-\frac{111624852879006}{8076887}a^{11}+\frac{189347925491252}{8076887}a^{10}-\frac{276754330782717}{8076887}a^{9}+\frac{347813404617492}{8076887}a^{8}-\frac{371956876854574}{8076887}a^{7}+\frac{334629430046670}{8076887}a^{6}-\frac{35481285372047}{1153841}a^{5}+\frac{148003105281952}{8076887}a^{4}-\frac{67912196198871}{8076887}a^{3}+\frac{22407221547196}{8076887}a^{2}-\frac{670832660020}{1153841}a+\frac{27572040985}{475111}$
| 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: | \( 294726306.155 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 294726306.155 \cdot 23}{2\cdot\sqrt{20375776275370821024303050129100543619}}\cr\approx \mathstrut & 0.452416266662 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 90*x^20 - 515*x^19 + 2404*x^18 - 9153*x^17 + 29609*x^16 - 81730*x^15 + 195276*x^14 - 404718*x^13 + 730608*x^12 - 1147506*x^11 + 1565057*x^10 - 1843981*x^9 + 1862690*x^8 - 1595475*x^7 + 1141091*x^6 - 667142*x^5 + 309476*x^4 - 109065*x^3 + 27307*x^2 - 4313*x + 323)
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 - 11*x^21 + 90*x^20 - 515*x^19 + 2404*x^18 - 9153*x^17 + 29609*x^16 - 81730*x^15 + 195276*x^14 - 404718*x^13 + 730608*x^12 - 1147506*x^11 + 1565057*x^10 - 1843981*x^9 + 1862690*x^8 - 1595475*x^7 + 1141091*x^6 - 667142*x^5 + 309476*x^4 - 109065*x^3 + 27307*x^2 - 4313*x + 323, 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 - 11*x^21 + 90*x^20 - 515*x^19 + 2404*x^18 - 9153*x^17 + 29609*x^16 - 81730*x^15 + 195276*x^14 - 404718*x^13 + 730608*x^12 - 1147506*x^11 + 1565057*x^10 - 1843981*x^9 + 1862690*x^8 - 1595475*x^7 + 1141091*x^6 - 667142*x^5 + 309476*x^4 - 109065*x^3 + 27307*x^2 - 4313*x + 323);
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 - 11*x^21 + 90*x^20 - 515*x^19 + 2404*x^18 - 9153*x^17 + 29609*x^16 - 81730*x^15 + 195276*x^14 - 404718*x^13 + 730608*x^12 - 1147506*x^11 + 1565057*x^10 - 1843981*x^9 + 1862690*x^8 - 1595475*x^7 + 1141091*x^6 - 667142*x^5 + 309476*x^4 - 109065*x^3 + 27307*x^2 - 4313*x + 323);
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 solvable group of order 44 |
| The 14 conjugacy class representatives for $D_{22}$ |
| Character table for $D_{22}$ |
Intermediate fields
| \(\Q(\sqrt{-19}) \), 11.11.1035571956771279049.1 |
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
| Galois closure: | deg 44 |
| Degree 22 sibling: | deg 22 |
| 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 | $22$ | $22$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{10}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{11}$ | ${\href{/padicField/17.2.0.1}{2} }^{10}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/23.2.0.1}{2} }^{10}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $22$ | $22$ | ${\href{/padicField/37.2.0.1}{2} }^{11}$ | $22$ | ${\href{/padicField/43.11.0.1}{11} }^{2}$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{11}$ | ${\href{/padicField/59.2.0.1}{2} }^{11}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(211\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |