Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 16*x^20 + 46*x^18 + 155*x^16 - 748*x^14 + 688*x^12 + 579*x^10 - 1382*x^8 + 904*x^6 - 253*x^4 + 28*x^2 - 1)
gp: K = bnfinit(y^22 - 16*y^20 + 46*y^18 + 155*y^16 - 748*y^14 + 688*y^12 + 579*y^10 - 1382*y^8 + 904*y^6 - 253*y^4 + 28*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 16*x^20 + 46*x^18 + 155*x^16 - 748*x^14 + 688*x^12 + 579*x^10 - 1382*x^8 + 904*x^6 - 253*x^4 + 28*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 16*x^20 + 46*x^18 + 155*x^16 - 748*x^14 + 688*x^12 + 579*x^10 - 1382*x^8 + 904*x^6 - 253*x^4 + 28*x^2 - 1)
\( x^{22} - 16 x^{20} + 46 x^{18} + 155 x^{16} - 748 x^{14} + 688 x^{12} + 579 x^{10} - 1382 x^{8} + \cdots - 1 \)
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: | $[18, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(56501459388151144478039723653407440896\)
\(\medspace = 2^{22}\cdot 1297^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(52.00\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(1297\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5}a^{18}-\frac{2}{5}a^{16}-\frac{1}{5}a^{14}-\frac{1}{5}a^{12}+\frac{2}{5}a^{10}+\frac{1}{5}a^{6}+\frac{2}{5}a^{4}-\frac{2}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{19}-\frac{2}{5}a^{17}-\frac{1}{5}a^{15}-\frac{1}{5}a^{13}+\frac{2}{5}a^{11}+\frac{1}{5}a^{7}+\frac{2}{5}a^{5}-\frac{2}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{801652175}a^{20}+\frac{34501246}{801652175}a^{18}+\frac{88915048}{801652175}a^{16}-\frac{209629594}{801652175}a^{14}+\frac{335997249}{801652175}a^{12}+\frac{164789201}{801652175}a^{10}+\frac{330502516}{801652175}a^{8}-\frac{70344383}{160330435}a^{6}+\frac{329723849}{801652175}a^{4}-\frac{75641988}{160330435}a^{2}-\frac{109251552}{801652175}$, $\frac{1}{801652175}a^{21}+\frac{34501246}{801652175}a^{19}+\frac{88915048}{801652175}a^{17}-\frac{209629594}{801652175}a^{15}+\frac{335997249}{801652175}a^{13}+\frac{164789201}{801652175}a^{11}+\frac{330502516}{801652175}a^{9}-\frac{70344383}{160330435}a^{7}+\frac{329723849}{801652175}a^{5}-\frac{75641988}{160330435}a^{3}-\frac{109251552}{801652175}a$
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: | $19$ | 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: |
$a^{21}-16a^{19}+46a^{17}+155a^{15}-748a^{13}+688a^{11}+579a^{9}-1382a^{7}+904a^{5}-253a^{3}+28a$, $\frac{1170615194}{801652175}a^{21}-\frac{18594899341}{801652175}a^{19}+\frac{51646858417}{801652175}a^{17}+\frac{188262123254}{801652175}a^{15}-\frac{855619687979}{801652175}a^{13}+\frac{695628040539}{801652175}a^{11}+\frac{789753984204}{801652175}a^{9}-\frac{61127998662}{32066087}a^{7}+\frac{835754788576}{801652175}a^{5}-\frac{33911201741}{160330435}a^{3}+\frac{9068804722}{801652175}a$, $\frac{1226566237}{801652175}a^{21}-\frac{20662721508}{801652175}a^{19}+\frac{72142473296}{801652175}a^{17}+\frac{155904112132}{801652175}a^{15}-\frac{1109746358502}{801652175}a^{13}+\frac{1461901826692}{801652175}a^{11}+\frac{549708104467}{801652175}a^{9}-\frac{505372082808}{160330435}a^{7}+\frac{1869052848218}{801652175}a^{5}-\frac{96491070737}{160330435}a^{3}+\frac{26224228591}{801652175}a$, $\frac{2054212106}{801652175}a^{20}-\frac{31916429569}{801652175}a^{18}+\frac{79668403353}{801652175}a^{16}+\frac{356038920506}{801652175}a^{14}-\frac{1373307653186}{801652175}a^{12}+\frac{768125872816}{801652175}a^{10}+\frac{1573796374096}{801652175}a^{8}-\frac{423109729447}{160330435}a^{6}+\frac{838669973129}{801652175}a^{4}-\frac{3956807056}{32066087}a^{2}+\frac{4234209493}{801652175}$, $\frac{2732778272}{801652175}a^{21}-\frac{42889121433}{801652175}a^{19}+\frac{112920096546}{801652175}a^{17}+\frac{453211946402}{801652175}a^{15}-\frac{1894880489702}{801652175}a^{13}+\frac{1360074160682}{801652175}a^{11}+\frac{1806253155652}{801652175}a^{9}-\frac{126647999480}{32066087}a^{7}+\frac{1746002169563}{801652175}a^{5}-\frac{80422826538}{160330435}a^{3}+\frac{29742364486}{801652175}a$, $\frac{370436558}{801652175}a^{21}-\frac{6174899532}{801652175}a^{19}+\frac{20867473859}{801652175}a^{17}+\frac{48176993948}{801652175}a^{15}-\frac{320900255658}{801652175}a^{13}+\frac{415996458908}{801652175}a^{11}+\frac{137049812328}{801652175}a^{9}-\frac{140974952239}{160330435}a^{7}+\frac{568894374092}{801652175}a^{5}-\frac{35792639189}{160330435}a^{3}+\frac{18292097009}{801652175}a$, $\frac{741389862}{801652175}a^{20}-\frac{11270331633}{801652175}a^{18}+\frac{25021896596}{801652175}a^{16}+\frac{136196197182}{801652175}a^{14}-\frac{449043897327}{801652175}a^{12}+\frac{136901296142}{801652175}a^{10}+\frac{594229309067}{801652175}a^{8}-\frac{115941954423}{160330435}a^{6}+\frac{142944538993}{801652175}a^{4}+\frac{1653201933}{160330435}a^{2}-\frac{1805158409}{801652175}$, $\frac{694896678}{801652175}a^{20}-\frac{11074733507}{801652175}a^{18}+\frac{31170086359}{801652175}a^{16}+\frac{111134825938}{801652175}a^{14}-\frac{515448853958}{801652175}a^{12}+\frac{426351891888}{801652175}a^{10}+\frac{479087118898}{801652175}a^{8}-\frac{185583643548}{160330435}a^{6}+\frac{502155658282}{801652175}a^{4}-\frac{19608998506}{160330435}a^{2}+\frac{4396349874}{801652175}$, $\frac{441655382}{160330435}a^{20}-\frac{6863065147}{160330435}a^{18}+\frac{17172102604}{160330435}a^{16}+\frac{76107387296}{160330435}a^{14}-\frac{294728788723}{160330435}a^{12}+\frac{171220357459}{160330435}a^{10}+\frac{324299971742}{160330435}a^{8}-\frac{457687387304}{160330435}a^{6}+\frac{40315555372}{32066087}a^{4}-\frac{32287276532}{160330435}a^{2}+\frac{1568447697}{160330435}$, $\frac{206165329}{801652175}a^{20}-\frac{3459335806}{801652175}a^{18}+\frac{11914106922}{801652175}a^{16}+\frac{26683383614}{801652175}a^{14}-\frac{183648292364}{801652175}a^{12}+\frac{236281104024}{801652175}a^{10}+\frac{89945384014}{801652175}a^{8}-\frac{16108188194}{32066087}a^{6}+\frac{310636809016}{801652175}a^{4}-\frac{18439656176}{160330435}a^{2}+\frac{7688508127}{801652175}$, $a-1$, $\frac{2565716621}{801652175}a^{21}+\frac{537498209}{801652175}a^{20}-\frac{40875762549}{801652175}a^{19}-\frac{7386606241}{801652175}a^{18}+\frac{115390630563}{801652175}a^{17}+\frac{6372157217}{801652175}a^{16}+\frac{403200178516}{801652175}a^{15}+\frac{123039979259}{801652175}a^{14}-\frac{1888027813106}{801652175}a^{13}-\frac{178645627704}{801652175}a^{12}+\frac{1669490310166}{801652175}a^{11}-\frac{343948212926}{801652175}a^{10}+\frac{1529087927911}{801652175}a^{9}+\frac{515223214119}{801652175}a^{8}-\frac{694065368471}{160330435}a^{7}+\frac{32883375507}{160330435}a^{6}+\frac{2182962217799}{801652175}a^{5}-\frac{410784196619}{801652175}a^{4}-\frac{108437563887}{160330435}a^{3}+\frac{5723421381}{32066087}a^{2}+\frac{37930997218}{801652175}a-\frac{10739492273}{801652175}$, $\frac{3090309133}{801652175}a^{21}-\frac{878197691}{801652175}a^{20}-\frac{48740815697}{801652175}a^{19}+\frac{13898876584}{801652175}a^{18}+\frac{131130219589}{801652175}a^{17}-\frac{37937940008}{801652175}a^{16}+\frac{507583373463}{801652175}a^{15}-\frac{143400208841}{801652175}a^{14}-\frac{2192393588993}{801652175}a^{13}+\frac{633048318196}{801652175}a^{12}+\frac{1640342855353}{801652175}a^{11}-\frac{484459174626}{801652175}a^{10}+\frac{2103461430978}{801652175}a^{9}-\frac{613262477856}{801652175}a^{8}-\frac{750448219292}{160330435}a^{7}+\frac{219381407347}{160330435}a^{6}+\frac{2008382471337}{801652175}a^{5}-\frac{572712103619}{801652175}a^{4}-\frac{84341597603}{160330435}a^{3}+\frac{4559196939}{32066087}a^{2}+\frac{26807737094}{801652175}a-\frac{7317645973}{801652175}$, $\frac{2028218412}{801652175}a^{21}-\frac{2545642727}{801652175}a^{20}-\frac{33489156308}{801652175}a^{19}+\frac{39538364978}{801652175}a^{18}+\frac{109018473346}{801652175}a^{17}-\frac{98820476336}{801652175}a^{16}+\frac{280160199257}{801652175}a^{15}-\frac{437300335357}{801652175}a^{14}-\frac{1709382185402}{801652175}a^{13}+\frac{1691579698282}{801652175}a^{12}+\frac{2013438523092}{801652175}a^{11}-\frac{1002184984512}{801652175}a^{10}+\frac{1013864713792}{801652175}a^{9}-\frac{1803378702282}{801652175}a^{8}-\frac{726948743978}{160330435}a^{7}+\frac{105212561792}{32066087}a^{6}+\frac{2593746414418}{801652175}a^{5}-\frac{1247664240808}{801652175}a^{4}-\frac{137054670792}{160330435}a^{3}+\frac{47817173538}{160330435}a^{2}+\frac{47868837316}{801652175}a-\frac{14371563576}{801652175}$, $\frac{3090309133}{801652175}a^{21}-\frac{741389862}{801652175}a^{20}-\frac{48740815697}{801652175}a^{19}+\frac{11270331633}{801652175}a^{18}+\frac{131130219589}{801652175}a^{17}-\frac{25021896596}{801652175}a^{16}+\frac{507583373463}{801652175}a^{15}-\frac{136196197182}{801652175}a^{14}-\frac{2192393588993}{801652175}a^{13}+\frac{449043897327}{801652175}a^{12}+\frac{1640342855353}{801652175}a^{11}-\frac{136901296142}{801652175}a^{10}+\frac{2103461430978}{801652175}a^{9}-\frac{594229309067}{801652175}a^{8}-\frac{750448219292}{160330435}a^{7}+\frac{115941954423}{160330435}a^{6}+\frac{2008382471337}{801652175}a^{5}-\frac{142944538993}{801652175}a^{4}-\frac{84341597603}{160330435}a^{3}-\frac{1653201933}{160330435}a^{2}+\frac{26006084919}{801652175}a+\frac{2606810584}{801652175}$, $\frac{2724136536}{801652175}a^{21}+\frac{1378770808}{801652175}a^{20}-\frac{41468494139}{801652175}a^{19}-\frac{21381202707}{801652175}a^{18}+\frac{92910677643}{801652175}a^{17}+\frac{52944529709}{801652175}a^{16}+\frac{496946712311}{801652175}a^{15}+\frac{238958673973}{801652175}a^{14}-\frac{1656903136666}{801652175}a^{13}-\frac{911427760933}{801652175}a^{12}+\frac{556591484496}{801652175}a^{11}+\frac{508722637658}{801652175}a^{10}+\frac{2108933029576}{801652175}a^{9}+\frac{1011281490078}{801652175}a^{8}-\frac{432092548137}{160330435}a^{7}-\frac{277084604324}{160330435}a^{6}+\frac{659385113174}{801652175}a^{5}+\frac{609380540117}{801652175}a^{4}-\frac{1846253362}{32066087}a^{3}-\frac{21557415109}{160330435}a^{2}-\frac{1854941417}{801652175}a+\frac{6029252734}{801652175}$, $\frac{2659093516}{801652175}a^{21}-\frac{2362341714}{801652175}a^{20}-\frac{42089280429}{801652175}a^{19}+\frac{36714221901}{801652175}a^{18}+\frac{115121693398}{801652175}a^{17}-\frac{92052622687}{801652175}a^{16}+\frac{431504280286}{801652175}a^{15}-\frac{405034952454}{801652175}a^{14}-\frac{1913658017751}{801652175}a^{13}+\frac{1573980234044}{801652175}a^{12}+\frac{1504802617861}{801652175}a^{11}-\frac{944077701774}{801652175}a^{10}+\frac{1776354633981}{801652175}a^{9}-\frac{1669203343324}{801652175}a^{8}-\frac{669846510361}{160330435}a^{7}+\frac{492265045161}{160330435}a^{6}+\frac{1852583279229}{801652175}a^{5}-\frac{1177107795471}{801652175}a^{4}-\frac{79570636737}{160330435}a^{3}+\frac{44630187349}{160330435}a^{2}+\frac{21851921028}{801652175}a-\frac{11450267477}{801652175}$, $\frac{4278665083}{801652175}a^{21}+\frac{205393420}{32066087}a^{20}-\frac{66777719497}{801652175}a^{19}-\frac{15761083722}{160330435}a^{18}+\frac{170639678939}{801652175}a^{17}+\frac{36982877159}{160330435}a^{16}+\frac{729290293213}{801652175}a^{15}+\frac{182849314107}{160330435}a^{14}-\frac{2910636891868}{801652175}a^{13}-\frac{647745524403}{160330435}a^{12}+\frac{1807976385453}{801652175}a^{11}+\frac{289361736426}{160330435}a^{10}+\frac{3134833294453}{801652175}a^{9}+\frac{152908860012}{32066087}a^{8}-\frac{924780490387}{160330435}a^{7}-\frac{913659087737}{160330435}a^{6}+\frac{2104141243462}{801652175}a^{5}+\frac{356087943721}{160330435}a^{4}-\frac{73488632238}{160330435}a^{3}-\frac{52306756206}{160330435}a^{2}+\frac{21929055619}{801652175}a+\frac{2429531828}{160330435}$, $\frac{3641190883}{801652175}a^{21}+\frac{168627333}{160330435}a^{20}-\frac{57015864632}{801652175}a^{19}-\frac{2724009747}{160330435}a^{18}+\frac{148133170884}{801652175}a^{17}+\frac{8129373019}{160330435}a^{16}+\frac{613409600948}{801652175}a^{15}+\frac{25594689333}{160330435}a^{14}-\frac{2511294096283}{801652175}a^{13}-\frac{131584807323}{160330435}a^{12}+\frac{1668252567208}{801652175}a^{11}+\frac{127546483968}{160330435}a^{10}+\frac{2622824049778}{801652175}a^{9}+\frac{105301877448}{160330435}a^{8}-\frac{828115331099}{160330435}a^{7}-\frac{51175641344}{32066087}a^{6}+\frac{1962477382867}{801652175}a^{5}+\frac{156120828702}{160330435}a^{4}-\frac{60198371474}{160330435}a^{3}-\frac{6697403151}{32066087}a^{2}-\frac{3594965241}{801652175}a+\frac{935925964}{160330435}$
| 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: | \( 356788225608 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{2}\cdot 356788225608 \cdot 1}{2\cdot\sqrt{56501459388151144478039723653407440896}}\cr\approx \mathstrut & 0.245612431219948 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 16*x^20 + 46*x^18 + 155*x^16 - 748*x^14 + 688*x^12 + 579*x^10 - 1382*x^8 + 904*x^6 - 253*x^4 + 28*x^2 - 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 - 16*x^20 + 46*x^18 + 155*x^16 - 748*x^14 + 688*x^12 + 579*x^10 - 1382*x^8 + 904*x^6 - 253*x^4 + 28*x^2 - 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 - 16*x^20 + 46*x^18 + 155*x^16 - 748*x^14 + 688*x^12 + 579*x^10 - 1382*x^8 + 904*x^6 - 253*x^4 + 28*x^2 - 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 - 16*x^20 + 46*x^18 + 155*x^16 - 748*x^14 + 688*x^12 + 579*x^10 - 1382*x^8 + 904*x^6 - 253*x^4 + 28*x^2 - 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
$C_2^{10}.D_{11}$ (as 22T30):
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 22528 |
| The 100 conjugacy class representatives for $C_2^{10}.D_{11}$ are not computed |
| Character table for $C_2^{10}.D_{11}$ is not computed |
Intermediate fields
| 11.11.3670285774226257.1 |
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 siblings: | data not computed |
| Degree 44 siblings: | data not computed |
| Minimal sibling: | 22.10.73282392826432034388017521578469450842112.6 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{9}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.11.0.1}{11} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{5}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $2$ | $11$ | $22$ | |||
|
\(1297\)
| $\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |