Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 + 29*x^20 - 38*x^19 - 4*x^18 + 86*x^17 - 143*x^16 + 138*x^15 - 79*x^14 - 37*x^13 + 184*x^12 - 257*x^11 + 184*x^10 - 37*x^9 - 79*x^8 + 138*x^7 - 143*x^6 + 86*x^5 - 4*x^4 - 38*x^3 + 29*x^2 - 9*x + 1)
gp: K = bnfinit(y^22 - 9*y^21 + 29*y^20 - 38*y^19 - 4*y^18 + 86*y^17 - 143*y^16 + 138*y^15 - 79*y^14 - 37*y^13 + 184*y^12 - 257*y^11 + 184*y^10 - 37*y^9 - 79*y^8 + 138*y^7 - 143*y^6 + 86*y^5 - 4*y^4 - 38*y^3 + 29*y^2 - 9*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 9*x^21 + 29*x^20 - 38*x^19 - 4*x^18 + 86*x^17 - 143*x^16 + 138*x^15 - 79*x^14 - 37*x^13 + 184*x^12 - 257*x^11 + 184*x^10 - 37*x^9 - 79*x^8 + 138*x^7 - 143*x^6 + 86*x^5 - 4*x^4 - 38*x^3 + 29*x^2 - 9*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 9*x^21 + 29*x^20 - 38*x^19 - 4*x^18 + 86*x^17 - 143*x^16 + 138*x^15 - 79*x^14 - 37*x^13 + 184*x^12 - 257*x^11 + 184*x^10 - 37*x^9 - 79*x^8 + 138*x^7 - 143*x^6 + 86*x^5 - 4*x^4 - 38*x^3 + 29*x^2 - 9*x + 1)
\( x^{22} - 9 x^{21} + 29 x^{20} - 38 x^{19} - 4 x^{18} + 86 x^{17} - 143 x^{16} + 138 x^{15} - 79 x^{14} + \cdots + 1 \)
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: | $[8, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-80659324072725558117423074447\)
\(\medspace = -\,23^{20}\cdot 47\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.60\) | 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: | $23^{10/11}47^{1/2}\approx 118.57232216797493$ | ||
| Ramified primes: |
\(23\), \(47\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-47}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{47}a^{18}+\frac{5}{47}a^{17}+\frac{13}{47}a^{16}+\frac{10}{47}a^{15}-\frac{20}{47}a^{14}+\frac{16}{47}a^{13}+\frac{1}{47}a^{12}-\frac{10}{47}a^{11}-\frac{14}{47}a^{10}+\frac{14}{47}a^{9}-\frac{14}{47}a^{8}-\frac{10}{47}a^{7}+\frac{1}{47}a^{6}+\frac{16}{47}a^{5}-\frac{20}{47}a^{4}+\frac{10}{47}a^{3}+\frac{13}{47}a^{2}+\frac{5}{47}a+\frac{1}{47}$, $\frac{1}{47}a^{19}-\frac{12}{47}a^{17}-\frac{8}{47}a^{16}-\frac{23}{47}a^{15}+\frac{22}{47}a^{14}+\frac{15}{47}a^{13}-\frac{15}{47}a^{12}-\frac{11}{47}a^{11}-\frac{10}{47}a^{10}+\frac{10}{47}a^{9}+\frac{13}{47}a^{8}+\frac{4}{47}a^{7}+\frac{11}{47}a^{6}-\frac{6}{47}a^{5}+\frac{16}{47}a^{4}+\frac{10}{47}a^{3}-\frac{13}{47}a^{2}+\frac{23}{47}a-\frac{5}{47}$, $\frac{1}{47}a^{20}+\frac{5}{47}a^{17}-\frac{8}{47}a^{16}+\frac{1}{47}a^{15}+\frac{10}{47}a^{14}-\frac{11}{47}a^{13}+\frac{1}{47}a^{12}+\frac{11}{47}a^{11}-\frac{17}{47}a^{10}-\frac{7}{47}a^{9}-\frac{23}{47}a^{8}-\frac{15}{47}a^{7}+\frac{6}{47}a^{6}+\frac{20}{47}a^{5}+\frac{5}{47}a^{4}+\frac{13}{47}a^{3}-\frac{9}{47}a^{2}+\frac{8}{47}a+\frac{12}{47}$, $\frac{1}{47}a^{21}+\frac{14}{47}a^{17}-\frac{17}{47}a^{16}+\frac{7}{47}a^{15}-\frac{5}{47}a^{14}+\frac{15}{47}a^{13}+\frac{6}{47}a^{12}-\frac{14}{47}a^{11}+\frac{16}{47}a^{10}+\frac{1}{47}a^{9}+\frac{8}{47}a^{8}+\frac{9}{47}a^{7}+\frac{15}{47}a^{6}+\frac{19}{47}a^{5}+\frac{19}{47}a^{4}-\frac{12}{47}a^{3}-\frac{10}{47}a^{2}-\frac{13}{47}a-\frac{5}{47}$
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: | $14$ | 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: |
$a$, $\frac{1623}{47}a^{21}-\frac{13769}{47}a^{20}+\frac{39988}{47}a^{19}-\frac{41244}{47}a^{18}-\frac{27340}{47}a^{17}+\frac{125360}{47}a^{16}-\frac{168126}{47}a^{15}+\frac{138424}{47}a^{14}-\frac{57898}{47}a^{13}-\frac{89179}{47}a^{12}+\frac{252693}{47}a^{11}-\frac{288291}{47}a^{10}+\frac{152086}{47}a^{9}+\frac{16978}{47}a^{8}-\frac{119300}{47}a^{7}+\frac{162977}{47}a^{6}-\frac{149103}{47}a^{5}+\frac{63821}{47}a^{4}+\frac{25800}{47}a^{3}-\frac{48510}{47}a^{2}+\frac{22389}{47}a-\frac{3277}{47}$, $\frac{5512}{47}a^{21}-\frac{46387}{47}a^{20}+\frac{132745}{47}a^{19}-\frac{131928}{47}a^{18}-\frac{98989}{47}a^{17}+\frac{416060}{47}a^{16}-\frac{545347}{47}a^{15}+\frac{442485}{47}a^{14}-\frac{177216}{47}a^{13}-\frac{307272}{47}a^{12}+\frac{834598}{47}a^{11}-\frac{929435}{47}a^{10}+\frac{472204}{47}a^{9}+\frac{71329}{47}a^{8}-\frac{393856}{47}a^{7}+\frac{530884}{47}a^{6}-\frac{478447}{47}a^{5}+\frac{195136}{47}a^{4}+\frac{91559}{47}a^{3}-\frac{156067}{47}a^{2}+\frac{68957}{47}a-\frac{9431}{47}$, $\frac{3034}{47}a^{21}-\frac{25665}{47}a^{20}+\frac{74136}{47}a^{19}-\frac{75452}{47}a^{18}-\frac{52251}{47}a^{17}+\frac{232110}{47}a^{16}-\frac{309078}{47}a^{15}+\frac{253649}{47}a^{14}-\frac{104921}{47}a^{13}-\frac{167110}{47}a^{12}+\frac{467330}{47}a^{11}-\frac{528920}{47}a^{10}+\frac{276292}{47}a^{9}+\frac{33307}{47}a^{8}-4684a^{7}+\frac{300261}{47}a^{6}-\frac{273386}{47}a^{5}+\frac{115489}{47}a^{4}+\frac{48482}{47}a^{3}-\frac{88659}{47}a^{2}+\frac{40599}{47}a-\frac{5955}{47}$, $\frac{1589}{47}a^{21}-\frac{13411}{47}a^{20}+\frac{38550}{47}a^{19}-\frac{38638}{47}a^{18}-\frac{28358}{47}a^{17}+\frac{120892}{47}a^{16}-\frac{158786}{47}a^{15}+\frac{129200}{47}a^{14}-\frac{52513}{47}a^{13}-\frac{88443}{47}a^{12}+\frac{242682}{47}a^{11}-\frac{271059}{47}a^{10}+\frac{138515}{47}a^{9}+\frac{19611}{47}a^{8}-\frac{113924}{47}a^{7}+\frac{154740}{47}a^{6}-\frac{139829}{47}a^{5}+\frac{57376}{47}a^{4}+\frac{26186}{47}a^{3}-\frac{45256}{47}a^{2}+\frac{20315}{47}a-\frac{2927}{47}$, $\frac{3925}{47}a^{21}-\frac{33123}{47}a^{20}+\frac{95273}{47}a^{19}-\frac{95935}{47}a^{18}-\frac{68970}{47}a^{17}+\frac{298609}{47}a^{16}-\frac{394604}{47}a^{15}+\frac{321869}{47}a^{14}-\frac{131017}{47}a^{13}-\frac{217663}{47}a^{12}+\frac{600069}{47}a^{11}-\frac{673916}{47}a^{10}+\frac{347104}{47}a^{9}+\frac{47362}{47}a^{8}-\frac{283090}{47}a^{7}+\frac{383518}{47}a^{6}-\frac{347491}{47}a^{5}+\frac{144231}{47}a^{4}+\frac{64218}{47}a^{3}-\frac{113116}{47}a^{2}+\frac{50784}{47}a-\frac{7101}{47}$, $\frac{1044}{47}a^{21}-\frac{8794}{47}a^{20}+\frac{25208}{47}a^{19}-\frac{25131}{47}a^{18}-\frac{18807}{47}a^{17}+\frac{79260}{47}a^{16}-\frac{103516}{47}a^{15}+\frac{83585}{47}a^{14}-\frac{33641}{47}a^{13}-\frac{58122}{47}a^{12}+\frac{158678}{47}a^{11}-\frac{176502}{47}a^{10}+\frac{88996}{47}a^{9}+\frac{13837}{47}a^{8}-\frac{74359}{47}a^{7}+\frac{100514}{47}a^{6}-\frac{90962}{47}a^{5}+783a^{4}+\frac{17599}{47}a^{3}-\frac{29409}{47}a^{2}+\frac{12911}{47}a-\frac{1821}{47}$, $a^{21}-8a^{20}+21a^{19}-17a^{18}-21a^{17}+65a^{16}-78a^{15}+60a^{14}-19a^{13}-56a^{12}+128a^{11}-129a^{10}+55a^{9}+18a^{8}-61a^{7}+77a^{6}-66a^{5}+20a^{4}+16a^{3}-22a^{2}+7a-2$, $\frac{1044}{47}a^{21}-\frac{8794}{47}a^{20}+\frac{25208}{47}a^{19}-\frac{25131}{47}a^{18}-\frac{18807}{47}a^{17}+\frac{79260}{47}a^{16}-\frac{103516}{47}a^{15}+\frac{83585}{47}a^{14}-\frac{33641}{47}a^{13}-\frac{58122}{47}a^{12}+\frac{158678}{47}a^{11}-\frac{176502}{47}a^{10}+\frac{88996}{47}a^{9}+\frac{13837}{47}a^{8}-\frac{74359}{47}a^{7}+\frac{100514}{47}a^{6}-\frac{90962}{47}a^{5}+783a^{4}+\frac{17599}{47}a^{3}-\frac{29409}{47}a^{2}+\frac{12864}{47}a-\frac{1821}{47}$, $\frac{1229}{47}a^{21}-\frac{10345}{47}a^{20}+\frac{29522}{47}a^{19}-\frac{28780}{47}a^{18}-\frac{23438}{47}a^{17}+\frac{92866}{47}a^{16}-\frac{118741}{47}a^{15}+\frac{95119}{47}a^{14}-\frac{37324}{47}a^{13}-\frac{70005}{47}a^{12}+\frac{185497}{47}a^{11}-\frac{201880}{47}a^{10}+\frac{98625}{47}a^{9}+\frac{18380}{47}a^{8}-\frac{86675}{47}a^{7}+\frac{116591}{47}a^{6}-\frac{103753}{47}a^{5}+\frac{40098}{47}a^{4}+\frac{21512}{47}a^{3}-\frac{33634}{47}a^{2}+\frac{14403}{47}a-41$, $\frac{1863}{47}a^{21}-\frac{15665}{47}a^{20}+\frac{44797}{47}a^{19}-\frac{44612}{47}a^{18}-\frac{32902}{47}a^{17}+\frac{139873}{47}a^{16}-\frac{184742}{47}a^{15}+\frac{151174}{47}a^{14}-\frac{61114}{47}a^{13}-\frac{103192}{47}a^{12}+\frac{281450}{47}a^{11}-\frac{315150}{47}a^{10}+\frac{162588}{47}a^{9}+\frac{21822}{47}a^{8}-\frac{133215}{47}a^{7}+3835a^{6}-\frac{162604}{47}a^{5}+\frac{67375}{47}a^{4}+\frac{29790}{47}a^{3}-\frac{52962}{47}a^{2}+\frac{23937}{47}a-\frac{3412}{47}$, $\frac{997}{47}a^{21}-\frac{8371}{47}a^{20}+\frac{23845}{47}a^{19}-\frac{23345}{47}a^{18}-\frac{18619}{47}a^{17}+\frac{75218}{47}a^{16}-\frac{96795}{47}a^{15}+\frac{77099}{47}a^{14}-\frac{29928}{47}a^{13}-\frac{56383}{47}a^{12}+\frac{150030}{47}a^{11}-\frac{164423}{47}a^{10}+\frac{80348}{47}a^{9}+\frac{15576}{47}a^{8}-\frac{70646}{47}a^{7}+\frac{94028}{47}a^{6}-\frac{84241}{47}a^{5}+697a^{4}+\frac{17787}{47}a^{3}-\frac{27623}{47}a^{2}+\frac{11548}{47}a-\frac{1445}{47}$, $\frac{1917}{47}a^{21}-\frac{16044}{47}a^{20}+\frac{45468}{47}a^{19}-\frac{44176}{47}a^{18}-\frac{35260}{47}a^{17}+\frac{141958}{47}a^{16}-\frac{184536}{47}a^{15}+\frac{149396}{47}a^{14}-\frac{58599}{47}a^{13}-\frac{106954}{47}a^{12}+\frac{284383}{47}a^{11}-\frac{313372}{47}a^{10}+\frac{157382}{47}a^{9}+\frac{25553}{47}a^{8}-\frac{134313}{47}a^{7}+\frac{180206}{47}a^{6}-\frac{161038}{47}a^{5}+\frac{64574}{47}a^{4}+\frac{31525}{47}a^{3}-\frac{52488}{47}a^{2}+\frac{22964}{47}a-\frac{3112}{47}$, $\frac{1784}{47}a^{21}-\frac{15078}{47}a^{20}+\frac{43470}{47}a^{19}-\frac{43930}{47}a^{18}-\frac{31432}{47}a^{17}+\frac{136591}{47}a^{16}-\frac{180220}{47}a^{15}+\frac{146659}{47}a^{14}-\frac{59654}{47}a^{13}-\frac{99506}{47}a^{12}+\frac{274559}{47}a^{11}-\frac{308195}{47}a^{10}+\frac{157968}{47}a^{9}+\frac{22485}{47}a^{8}-\frac{129714}{47}a^{7}+\frac{175388}{47}a^{6}-\frac{158965}{47}a^{5}+\frac{65568}{47}a^{4}+\frac{29898}{47}a^{3}-\frac{52018}{47}a^{2}+\frac{23182}{47}a-\frac{3226}{47}$
| 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: | \( 918526.809002 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{7}\cdot 918526.809002 \cdot 1}{2\cdot\sqrt{80659324072725558117423074447}}\cr\approx \mathstrut & 0.160041862020 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 + 29*x^20 - 38*x^19 - 4*x^18 + 86*x^17 - 143*x^16 + 138*x^15 - 79*x^14 - 37*x^13 + 184*x^12 - 257*x^11 + 184*x^10 - 37*x^9 - 79*x^8 + 138*x^7 - 143*x^6 + 86*x^5 - 4*x^4 - 38*x^3 + 29*x^2 - 9*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 - 9*x^21 + 29*x^20 - 38*x^19 - 4*x^18 + 86*x^17 - 143*x^16 + 138*x^15 - 79*x^14 - 37*x^13 + 184*x^12 - 257*x^11 + 184*x^10 - 37*x^9 - 79*x^8 + 138*x^7 - 143*x^6 + 86*x^5 - 4*x^4 - 38*x^3 + 29*x^2 - 9*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 - 9*x^21 + 29*x^20 - 38*x^19 - 4*x^18 + 86*x^17 - 143*x^16 + 138*x^15 - 79*x^14 - 37*x^13 + 184*x^12 - 257*x^11 + 184*x^10 - 37*x^9 - 79*x^8 + 138*x^7 - 143*x^6 + 86*x^5 - 4*x^4 - 38*x^3 + 29*x^2 - 9*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 - 9*x^21 + 29*x^20 - 38*x^19 - 4*x^18 + 86*x^17 - 143*x^16 + 138*x^15 - 79*x^14 - 37*x^13 + 184*x^12 - 257*x^11 + 184*x^10 - 37*x^9 - 79*x^8 + 138*x^7 - 143*x^6 + 86*x^5 - 4*x^4 - 38*x^3 + 29*x^2 - 9*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
$C_{15}\times C_{420}$ (as 22T28):
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 208 conjugacy class representatives for $C_{15}\times C_{420}$ |
| Character table for $C_{15}\times C_{420}$ |
Intermediate fields
| \(\Q(\zeta_{23})^+\) |
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: | 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 | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | $22$ | $22$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | $22$ | R | $22$ | $22$ | ${\href{/padicField/37.11.0.1}{11} }^{2}$ | $22$ | $22$ | R | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.11.0.1}{11} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.22.20.1 | $x^{22} + 231 x^{21} + 24310 x^{20} + 1539615 x^{19} + 65271580 x^{18} + 1948237137 x^{17} + 41892053577 x^{16} + 652037968890 x^{15} + 7265319209115 x^{14} + 56290884204555 x^{13} + 287278948122936 x^{12} + 881069580352567 x^{11} + 1436394740619993 x^{10} + 1407272105659090 x^{9} + 908164935089445 x^{8} + 407525140102740 x^{7} + 130953632793951 x^{6} + 31291611122541 x^{5} + 17709125163290 x^{4} + 131485551467820 x^{3} + 905728303498040 x^{2} + 3760233666129578 x + 7096289029978197$ | $11$ | $2$ | $20$ | 22T1 | $[\ ]_{11}^{2}$ |
|
\(47\)
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |