Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 33*x^18 - 96*x^17 + 225*x^16 - 428*x^15 + 639*x^14 - 722*x^13 + 605*x^12 - 426*x^11 + 458*x^10 - 746*x^9 + 1001*x^8 - 1010*x^7 + 873*x^6 - 694*x^5 + 455*x^4 - 214*x^3 + 66*x^2 - 12*x + 1)
gp: K = bnfinit(y^20 - 8*y^19 + 33*y^18 - 96*y^17 + 225*y^16 - 428*y^15 + 639*y^14 - 722*y^13 + 605*y^12 - 426*y^11 + 458*y^10 - 746*y^9 + 1001*y^8 - 1010*y^7 + 873*y^6 - 694*y^5 + 455*y^4 - 214*y^3 + 66*y^2 - 12*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 8*x^19 + 33*x^18 - 96*x^17 + 225*x^16 - 428*x^15 + 639*x^14 - 722*x^13 + 605*x^12 - 426*x^11 + 458*x^10 - 746*x^9 + 1001*x^8 - 1010*x^7 + 873*x^6 - 694*x^5 + 455*x^4 - 214*x^3 + 66*x^2 - 12*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 8*x^19 + 33*x^18 - 96*x^17 + 225*x^16 - 428*x^15 + 639*x^14 - 722*x^13 + 605*x^12 - 426*x^11 + 458*x^10 - 746*x^9 + 1001*x^8 - 1010*x^7 + 873*x^6 - 694*x^5 + 455*x^4 - 214*x^3 + 66*x^2 - 12*x + 1)
\( x^{20} - 8 x^{19} + 33 x^{18} - 96 x^{17} + 225 x^{16} - 428 x^{15} + 639 x^{14} - 722 x^{13} + 605 x^{12} + \cdots + 1 \)
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: |
\(6802209109663753569286129\)
\(\medspace = 11^{16}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.44\) | 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: | $11^{4/5}23^{1/2}\approx 32.65713384043754$ | ||
| Ramified primes: |
\(11\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{22}a^{18}-\frac{5}{22}a^{16}-\frac{2}{11}a^{15}-\frac{1}{11}a^{14}+\frac{5}{22}a^{13}-\frac{2}{11}a^{12}+\frac{1}{11}a^{11}+\frac{3}{22}a^{10}-\frac{5}{22}a^{9}+\frac{7}{22}a^{8}-\frac{5}{22}a^{7}-\frac{9}{22}a^{6}+\frac{5}{11}a^{5}+\frac{4}{11}a^{4}-\frac{9}{22}a^{3}+\frac{1}{11}a^{2}-\frac{5}{11}a+\frac{9}{22}$, $\frac{1}{23799180218}a^{19}-\frac{524434697}{23799180218}a^{18}+\frac{20061385}{11899590109}a^{17}-\frac{907596433}{23799180218}a^{16}-\frac{236948879}{1081780919}a^{15}-\frac{721241331}{11899590109}a^{14}-\frac{2040874270}{11899590109}a^{13}-\frac{2776233403}{11899590109}a^{12}-\frac{4958581417}{23799180218}a^{11}-\frac{932558248}{11899590109}a^{10}+\frac{1150444101}{23799180218}a^{9}+\frac{11357637541}{23799180218}a^{8}-\frac{1913419883}{23799180218}a^{7}-\frac{2174157602}{11899590109}a^{6}+\frac{4855882830}{11899590109}a^{5}+\frac{9693032105}{23799180218}a^{4}-\frac{278563587}{11899590109}a^{3}-\frac{977121615}{2163561838}a^{2}-\frac{1317925089}{23799180218}a-\frac{1519328519}{11899590109}$
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
$C_{2}$, which has order $2$
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: |
$\frac{107750325459}{11899590109}a^{19}-\frac{1678675485991}{23799180218}a^{18}+\frac{3371049409088}{11899590109}a^{17}-\frac{1741377605933}{2163561838}a^{16}+\frac{44032497299997}{23799180218}a^{15}-\frac{7439063346519}{2163561838}a^{14}+\frac{117974093360165}{23799180218}a^{13}-\frac{63181679978429}{11899590109}a^{12}+\frac{97946520853975}{23799180218}a^{11}-\frac{65645731896677}{23799180218}a^{10}+\frac{3690722441646}{1081780919}a^{9}-\frac{141136892594501}{23799180218}a^{8}+\frac{90811854898851}{11899590109}a^{7}-\frac{85961493160086}{11899590109}a^{6}+\frac{13040908095957}{2163561838}a^{5}-\frac{10193234975575}{2163561838}a^{4}+\frac{68453317039733}{23799180218}a^{3}-\frac{13633780472127}{11899590109}a^{2}+\frac{6265812955651}{23799180218}a-\frac{632542706965}{23799180218}$, $\frac{134415349599}{23799180218}a^{19}-\frac{1069703330065}{23799180218}a^{18}+\frac{4358825506383}{23799180218}a^{17}-\frac{12494730401427}{23799180218}a^{16}+\frac{1312782655803}{1081780919}a^{15}-\frac{54079230624885}{23799180218}a^{14}+\frac{39352030564455}{11899590109}a^{13}-\frac{85159714307645}{23799180218}a^{12}+\frac{66465630266877}{23799180218}a^{11}-\frac{43927081429113}{23799180218}a^{10}+\frac{52889581149329}{23799180218}a^{9}-\frac{46668578881704}{11899590109}a^{8}+\frac{122047607420021}{23799180218}a^{7}-\frac{115928281624359}{23799180218}a^{6}+\frac{48106477872221}{11899590109}a^{5}-\frac{75605692467311}{23799180218}a^{4}+\frac{23342693872934}{11899590109}a^{3}-\frac{1676550370191}{2163561838}a^{2}+\frac{2023289518304}{11899590109}a-\frac{184781058928}{11899590109}$, $\frac{142146568505}{11899590109}a^{19}-\frac{1065666414286}{11899590109}a^{18}+\frac{8297171725091}{23799180218}a^{17}-\frac{23025673710965}{23799180218}a^{16}+\frac{26012322543296}{11899590109}a^{15}-\frac{47259982718703}{11899590109}a^{14}+\frac{5995536962153}{1081780919}a^{13}-\frac{67459089984994}{11899590109}a^{12}+\frac{98646183712103}{23799180218}a^{11}-\frac{66029680183301}{23799180218}a^{10}+\frac{46593883004668}{11899590109}a^{9}-\frac{162722658477089}{23799180218}a^{8}+\frac{198881318027771}{23799180218}a^{7}-\frac{90105885137772}{11899590109}a^{6}+\frac{74675396038465}{11899590109}a^{5}-\frac{57507498107251}{11899590109}a^{4}+\frac{65647520863377}{23799180218}a^{3}-\frac{11658480673703}{11899590109}a^{2}+\frac{2309109803722}{11899590109}a-\frac{370560526431}{23799180218}$, $\frac{252918138597}{23799180218}a^{19}-\frac{979723577421}{11899590109}a^{18}+\frac{3921221431928}{11899590109}a^{17}-\frac{11114580862361}{11899590109}a^{16}+\frac{25506061990221}{11899590109}a^{15}-\frac{47289761675538}{11899590109}a^{14}+\frac{67972771127786}{11899590109}a^{13}-\frac{145006394154271}{23799180218}a^{12}+\frac{111807477486151}{23799180218}a^{11}-\frac{74830317457265}{23799180218}a^{10}+\frac{46813362333751}{11899590109}a^{9}-\frac{14815410056443}{2163561838}a^{8}+\frac{9499700803600}{1081780919}a^{7}-\frac{98503057065652}{11899590109}a^{6}+\frac{82038346356017}{11899590109}a^{5}-\frac{63989416935324}{11899590109}a^{4}+\frac{77837478232485}{23799180218}a^{3}-\frac{30609749670323}{23799180218}a^{2}+\frac{3406568040669}{11899590109}a-\frac{677442475895}{23799180218}$, $\frac{55013480081}{23799180218}a^{19}-\frac{497972786467}{23799180218}a^{18}+\frac{1115355732268}{11899590109}a^{17}-\frac{6833397868987}{23799180218}a^{16}+\frac{8270674117150}{11899590109}a^{15}-\frac{16351584633456}{11899590109}a^{14}+\frac{51193950431497}{23799180218}a^{13}-\frac{60842308845213}{23799180218}a^{12}+\frac{26471116360240}{11899590109}a^{11}-\frac{35828269077855}{23799180218}a^{10}+\frac{16664234682102}{11899590109}a^{9}-\frac{56498818126749}{23799180218}a^{8}+\frac{41314954735322}{11899590109}a^{7}-\frac{86201610350749}{23799180218}a^{6}+\frac{73129573984703}{23799180218}a^{5}-\frac{58505922686647}{23799180218}a^{4}+\frac{40183228442417}{23799180218}a^{3}-\frac{18836363466739}{23799180218}a^{2}+\frac{459586245469}{2163561838}a-\frac{302261012953}{11899590109}$, $\frac{41925336421}{11899590109}a^{19}-\frac{602759889051}{23799180218}a^{18}+\frac{2257572329651}{23799180218}a^{17}-\frac{6068333431955}{23799180218}a^{16}+\frac{1215665064117}{2163561838}a^{15}-\frac{23481447786213}{23799180218}a^{14}+\frac{31056509896509}{23799180218}a^{13}-\frac{29142087149981}{23799180218}a^{12}+\frac{18660015124931}{23799180218}a^{11}-\frac{6155253178900}{11899590109}a^{10}+\frac{22721810923193}{23799180218}a^{9}-\frac{40333441153583}{23799180218}a^{8}+\frac{22584870208141}{11899590109}a^{7}-\frac{37409795436413}{23799180218}a^{6}+\frac{15200085249264}{11899590109}a^{5}-\frac{11346139485694}{11899590109}a^{4}+\frac{5408903170463}{11899590109}a^{3}-\frac{108663753081}{1081780919}a^{2}+\frac{35209631416}{11899590109}a+\frac{55820878555}{23799180218}$, $\frac{13053637533}{23799180218}a^{19}-\frac{112895406165}{11899590109}a^{18}+\frac{637341278162}{11899590109}a^{17}-\frac{394299286657}{2163561838}a^{16}+\frac{11094506693125}{23799180218}a^{15}-\frac{2124878963515}{2163561838}a^{14}+\frac{39046836409807}{23799180218}a^{13}-\frac{49081951017929}{23799180218}a^{12}+\frac{43975213997181}{23799180218}a^{11}-\frac{14025632929377}{11899590109}a^{10}+\frac{1986761960043}{2163561838}a^{9}-\frac{20386606594237}{11899590109}a^{8}+\frac{65630635745467}{23799180218}a^{7}-\frac{34800572791442}{11899590109}a^{6}+\frac{2623770982172}{1081780919}a^{5}-\frac{2161328117480}{1081780919}a^{4}+\frac{16997939480133}{11899590109}a^{3}-\frac{7741493608832}{11899590109}a^{2}+\frac{3820606931615}{23799180218}a-\frac{406414935607}{23799180218}$, $\frac{2816912009}{2163561838}a^{19}-\frac{14220934391}{2163561838}a^{18}+\frac{32836585905}{2163561838}a^{17}-\frac{43659824311}{2163561838}a^{16}+\frac{9976975957}{1081780919}a^{15}+\frac{155560741203}{2163561838}a^{14}-\frac{610476633919}{2163561838}a^{13}+\frac{600584540256}{1081780919}a^{12}-\frac{1414861518797}{2163561838}a^{11}+\frac{901936090001}{2163561838}a^{10}-\frac{129081175645}{2163561838}a^{9}+\frac{279794602019}{2163561838}a^{8}-\frac{1327519609569}{2163561838}a^{7}+\frac{950068296162}{1081780919}a^{6}-\frac{803471443236}{1081780919}a^{5}+\frac{700089911422}{1081780919}a^{4}-\frac{626392362287}{1081780919}a^{3}+\frac{692827877093}{2163561838}a^{2}-\frac{190936662559}{2163561838}a+\frac{11488307564}{1081780919}$, $\frac{11931250211}{1034746966}a^{19}-\frac{92213839053}{1034746966}a^{18}+\frac{367251663543}{1034746966}a^{17}-\frac{1035676923871}{1034746966}a^{16}+\frac{215161783199}{94067906}a^{15}-\frac{4367176793941}{1034746966}a^{14}+\frac{3113439667651}{517373483}a^{13}-\frac{6558363591039}{1034746966}a^{12}+\frac{2482042430660}{517373483}a^{11}-\frac{3299961098501}{1034746966}a^{10}+\frac{4291143548575}{1034746966}a^{9}-\frac{7537670395027}{1034746966}a^{8}+\frac{4761904000661}{517373483}a^{7}-\frac{4420605231318}{517373483}a^{6}+\frac{3669209771168}{517373483}a^{5}-\frac{5723357464035}{1034746966}a^{4}+\frac{3405117073513}{1034746966}a^{3}-\frac{117131703681}{94067906}a^{2}+\frac{275662217735}{1034746966}a-\frac{12346024410}{517373483}$
| 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: | \( 7947.13273942 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 7947.13273942 \cdot 2}{2\cdot\sqrt{6802209109663753569286129}}\cr\approx \mathstrut & 0.292202657416 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 33*x^18 - 96*x^17 + 225*x^16 - 428*x^15 + 639*x^14 - 722*x^13 + 605*x^12 - 426*x^11 + 458*x^10 - 746*x^9 + 1001*x^8 - 1010*x^7 + 873*x^6 - 694*x^5 + 455*x^4 - 214*x^3 + 66*x^2 - 12*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^20 - 8*x^19 + 33*x^18 - 96*x^17 + 225*x^16 - 428*x^15 + 639*x^14 - 722*x^13 + 605*x^12 - 426*x^11 + 458*x^10 - 746*x^9 + 1001*x^8 - 1010*x^7 + 873*x^6 - 694*x^5 + 455*x^4 - 214*x^3 + 66*x^2 - 12*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^20 - 8*x^19 + 33*x^18 - 96*x^17 + 225*x^16 - 428*x^15 + 639*x^14 - 722*x^13 + 605*x^12 - 426*x^11 + 458*x^10 - 746*x^9 + 1001*x^8 - 1010*x^7 + 873*x^6 - 694*x^5 + 455*x^4 - 214*x^3 + 66*x^2 - 12*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^20 - 8*x^19 + 33*x^18 - 96*x^17 + 225*x^16 - 428*x^15 + 639*x^14 - 722*x^13 + 605*x^12 - 426*x^11 + 458*x^10 - 746*x^9 + 1001*x^8 - 1010*x^7 + 873*x^6 - 694*x^5 + 455*x^4 - 214*x^3 + 66*x^2 - 12*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
$C_2^4:C_5$ (as 20T17):
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 80 |
| The 8 conjugacy class representatives for $C_2^4:C_5$ |
| Character table for $C_2^4:C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.2.113395848049.1 x2, 10.6.113395848049.1 |
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 10 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.2.113395848049.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{4}$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |