Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 35*x^16 + 421*x^14 - 2148*x^12 + 5568*x^10 - 7923*x^8 + 6263*x^6 - 2627*x^4 + 518*x^2 - 37)
gp: K = bnfinit(y^18 - 35*y^16 + 421*y^14 - 2148*y^12 + 5568*y^10 - 7923*y^8 + 6263*y^6 - 2627*y^4 + 518*y^2 - 37, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 35*x^16 + 421*x^14 - 2148*x^12 + 5568*x^10 - 7923*x^8 + 6263*x^6 - 2627*x^4 + 518*x^2 - 37);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 35*x^16 + 421*x^14 - 2148*x^12 + 5568*x^10 - 7923*x^8 + 6263*x^6 - 2627*x^4 + 518*x^2 - 37)
\( x^{18} - 35x^{16} + 421x^{14} - 2148x^{12} + 5568x^{10} - 7923x^{8} + 6263x^{6} - 2627x^{4} + 518x^{2} - 37 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1365795913530635497834467328\)
\(\medspace = 2^{12}\cdot 37^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(32.18\) | 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: | $2^{2/3}37^{5/6}\approx 32.17521323820123$ | ||
| Ramified primes: |
\(2\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{12}a^{12}+\frac{1}{12}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{6}a^{2}-\frac{5}{12}$, $\frac{1}{12}a^{13}+\frac{1}{12}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{6}a^{3}-\frac{5}{12}a$, $\frac{1}{12}a^{14}+\frac{1}{12}a^{10}-\frac{1}{2}a^{5}+\frac{1}{3}a^{4}-\frac{1}{2}a^{3}+\frac{1}{12}a^{2}-\frac{1}{2}$, $\frac{1}{12}a^{15}+\frac{1}{12}a^{11}-\frac{1}{2}a^{6}+\frac{1}{3}a^{5}-\frac{1}{2}a^{4}+\frac{1}{12}a^{3}-\frac{1}{2}a$, $\frac{1}{1749036}a^{16}-\frac{56227}{1749036}a^{14}+\frac{5056}{437259}a^{12}-\frac{284521}{1749036}a^{10}-\frac{285747}{583012}a^{8}-\frac{1}{2}a^{7}+\frac{11813}{874518}a^{6}+\frac{268971}{583012}a^{4}-\frac{340165}{1749036}a^{2}-\frac{1}{2}a-\frac{269333}{583012}$, $\frac{1}{1749036}a^{17}-\frac{56227}{1749036}a^{15}+\frac{5056}{437259}a^{13}-\frac{284521}{1749036}a^{11}+\frac{5759}{583012}a^{9}-\frac{212723}{437259}a^{7}+\frac{268971}{583012}a^{5}-\frac{1}{2}a^{4}+\frac{534353}{1749036}a^{3}-\frac{269333}{583012}a-\frac{1}{2}$
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: | $17$ | 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: |
$\frac{37481}{1749036}a^{17}-\frac{90601}{1749036}a^{16}-\frac{729545}{874518}a^{15}+\frac{1023379}{583012}a^{14}+\frac{20651317}{1749036}a^{13}-\frac{34742875}{1749036}a^{12}-\frac{33260023}{437259}a^{11}+\frac{52183421}{583012}a^{10}+\frac{105009701}{437259}a^{9}-\frac{56175491}{291506}a^{8}-\frac{166248577}{437259}a^{7}+\frac{92987747}{437259}a^{6}+\frac{503006503}{1749036}a^{5}-\frac{213134353}{1749036}a^{4}-\frac{37963829}{437259}a^{3}+\frac{21510549}{583012}a^{2}+\frac{2136812}{437259}a-\frac{1740223}{291506}$, $\frac{91779}{583012}a^{17}-\frac{131413}{874518}a^{16}-\frac{3133681}{583012}a^{15}+\frac{8911765}{1749036}a^{14}+\frac{8993758}{145753}a^{13}-\frac{25233424}{437259}a^{12}-\frac{166978317}{583012}a^{11}+\frac{454960291}{1749036}a^{10}+\frac{375041365}{583012}a^{9}-\frac{486988693}{874518}a^{8}-\frac{108767358}{145753}a^{7}+\frac{525797939}{874518}a^{6}+\frac{256511195}{583012}a^{5}-\frac{282813317}{874518}a^{4}-\frac{68506351}{583012}a^{3}+\frac{139967735}{1749036}a^{2}+\frac{5867263}{583012}a-\frac{6484057}{874518}$, $\frac{49263}{291506}a^{17}+\frac{34253}{583012}a^{16}-\frac{9889585}{1749036}a^{15}-\frac{1129313}{583012}a^{14}+\frac{109024927}{1749036}a^{13}+\frac{6033081}{291506}a^{12}-\frac{461141407}{1749036}a^{11}-\frac{47001687}{583012}a^{10}+\frac{870681667}{1749036}a^{9}+\frac{72033883}{583012}a^{8}-\frac{118884515}{291506}a^{7}-\frac{5747063}{145753}a^{6}+\frac{77633401}{874518}a^{5}-\frac{39595217}{583012}a^{4}+\frac{21065227}{583012}a^{3}+\frac{30431205}{583012}a^{2}-\frac{15830783}{1749036}a-\frac{4891191}{583012}$, $\frac{49263}{291506}a^{17}-\frac{34253}{583012}a^{16}-\frac{9889585}{1749036}a^{15}+\frac{1129313}{583012}a^{14}+\frac{109024927}{1749036}a^{13}-\frac{6033081}{291506}a^{12}-\frac{461141407}{1749036}a^{11}+\frac{47001687}{583012}a^{10}+\frac{870681667}{1749036}a^{9}-\frac{72033883}{583012}a^{8}-\frac{118884515}{291506}a^{7}+\frac{5747063}{145753}a^{6}+\frac{77633401}{874518}a^{5}+\frac{39595217}{583012}a^{4}+\frac{21065227}{583012}a^{3}-\frac{30431205}{583012}a^{2}-\frac{15830783}{1749036}a+\frac{4308179}{583012}$, $\frac{149825}{874518}a^{17}-\frac{10596731}{1749036}a^{15}+\frac{32430884}{437259}a^{13}-\frac{681353105}{1749036}a^{11}+\frac{452904070}{437259}a^{9}-\frac{1293258277}{874518}a^{7}+\frac{971046343}{874518}a^{5}-\frac{678310951}{1749036}a^{3}+\frac{38097827}{874518}a-\frac{1}{2}$, $\frac{49263}{291506}a^{17}-\frac{34253}{583012}a^{16}-\frac{9889585}{1749036}a^{15}+\frac{1129313}{583012}a^{14}+\frac{109024927}{1749036}a^{13}-\frac{6033081}{291506}a^{12}-\frac{461141407}{1749036}a^{11}+\frac{47001687}{583012}a^{10}+\frac{870681667}{1749036}a^{9}-\frac{72033883}{583012}a^{8}-\frac{118884515}{291506}a^{7}+\frac{5747063}{145753}a^{6}+\frac{77633401}{874518}a^{5}+\frac{39595217}{583012}a^{4}+\frac{21065227}{583012}a^{3}-\frac{30431205}{583012}a^{2}-\frac{15830783}{1749036}a+\frac{4891191}{583012}$, $\frac{101231}{291506}a^{17}-\frac{28763}{291506}a^{16}-\frac{3467353}{291506}a^{15}+\frac{501092}{145753}a^{14}+\frac{120102037}{874518}a^{13}-\frac{11954435}{291506}a^{12}-\frac{187824885}{291506}a^{11}+\frac{59988315}{291506}a^{10}+\frac{1280673325}{874518}a^{9}-\frac{151503741}{291506}a^{8}-\frac{249736451}{145753}a^{7}+\frac{103020295}{145753}a^{6}+\frac{147119957}{145753}a^{5}-\frac{74026603}{145753}a^{4}-\frac{114822502}{437259}a^{3}+\frac{24734389}{145753}a^{2}+\frac{17187721}{874518}a-\frac{2616737}{145753}$, $\frac{13753}{145753}a^{16}-\frac{2899397}{874518}a^{14}+\frac{35098655}{874518}a^{12}-\frac{180798605}{874518}a^{10}+\frac{470544077}{874518}a^{8}-\frac{110144327}{145753}a^{6}+\frac{247740566}{437259}a^{4}-\frac{59140373}{291506}a^{2}+\frac{21382553}{874518}$, $\frac{10538}{437259}a^{17}-\frac{7045}{583012}a^{16}-\frac{1740007}{1749036}a^{15}+\frac{154112}{437259}a^{14}+\frac{4440925}{291506}a^{13}-\frac{1187635}{437259}a^{12}-\frac{189613993}{1749036}a^{11}-\frac{213676}{437259}a^{10}+\frac{331627133}{874518}a^{9}+\frac{83769845}{1749036}a^{8}-\frac{294542149}{437259}a^{7}-\frac{39788191}{291506}a^{6}+\frac{257564725}{437259}a^{5}+\frac{258143579}{1749036}a^{4}-\frac{128032369}{583012}a^{3}-\frac{51536287}{874518}a^{2}+\frac{18321215}{874518}a+\frac{7633529}{1749036}$, $\frac{211145}{583012}a^{17}-\frac{137}{583012}a^{16}-\frac{10831931}{874518}a^{15}+\frac{123943}{583012}a^{14}+\frac{62398975}{437259}a^{13}-\frac{6050821}{874518}a^{12}-\frac{584509703}{874518}a^{11}+\frac{43934979}{583012}a^{10}+\frac{2679369889}{1749036}a^{9}-\frac{552217067}{1749036}a^{8}-\frac{540321575}{291506}a^{7}+\frac{174451243}{291506}a^{6}+\frac{2075546969}{1749036}a^{5}-\frac{307313631}{583012}a^{4}-\frac{107043371}{291506}a^{3}+\frac{344153521}{1749036}a^{2}+\frac{67357387}{1749036}a-\frac{38125031}{1749036}$, $\frac{34253}{583012}a^{17}+\frac{620681}{1749036}a^{16}-\frac{1129313}{583012}a^{15}-\frac{7070735}{583012}a^{14}+\frac{6033081}{291506}a^{13}+\frac{81319149}{583012}a^{12}-\frac{47001687}{583012}a^{11}-\frac{378897949}{583012}a^{10}+\frac{72033883}{583012}a^{9}+\frac{1284822779}{874518}a^{8}-\frac{5747063}{145753}a^{7}-\frac{754417534}{437259}a^{6}-\frac{39595217}{583012}a^{5}+\frac{1830641351}{1749036}a^{4}+\frac{30431205}{583012}a^{3}-\frac{523782425}{1749036}a^{2}-\frac{4891191}{583012}a+\frac{12988114}{437259}$, $\frac{200905}{583012}a^{17}-\frac{198869}{874518}a^{16}-\frac{6841067}{583012}a^{15}+\frac{13703753}{1749036}a^{14}+\frac{78069947}{583012}a^{13}-\frac{53309263}{583012}a^{12}-\frac{356791803}{583012}a^{11}+\frac{765784265}{1749036}a^{10}+\frac{193177599}{145753}a^{9}-\frac{1784502467}{1749036}a^{8}-\frac{206971927}{145753}a^{7}+\frac{1071153511}{874518}a^{6}+\frac{412729029}{583012}a^{5}-\frac{648027319}{874518}a^{4}-\frac{75682739}{583012}a^{3}+\frac{118258515}{583012}a^{2}+\frac{523355}{145753}a-\frac{34301417}{1749036}$, $\frac{38207}{1749036}a^{17}+\frac{66569}{583012}a^{16}-\frac{588773}{874518}a^{15}-\frac{1741190}{437259}a^{14}+\frac{10847437}{1749036}a^{13}+\frac{27664065}{583012}a^{12}-\frac{13262447}{874518}a^{11}-\frac{102913760}{437259}a^{10}-\frac{1464029}{874518}a^{9}+\frac{164688735}{291506}a^{8}+\frac{28947097}{874518}a^{7}-\frac{198036165}{291506}a^{6}-\frac{21812519}{1749036}a^{5}+\frac{674453539}{1749036}a^{4}-\frac{6786239}{437259}a^{3}-\frac{74647861}{874518}a^{2}+\frac{3058999}{874518}a+\frac{796996}{145753}$, $\frac{31768}{437259}a^{17}-\frac{38207}{1749036}a^{16}-\frac{4583627}{1749036}a^{15}+\frac{588773}{874518}a^{14}+\frac{28998463}{874518}a^{13}-\frac{10847437}{1749036}a^{12}-\frac{321423215}{1749036}a^{11}+\frac{13262447}{874518}a^{10}+\frac{451296917}{874518}a^{9}+\frac{1464029}{874518}a^{8}-\frac{675136507}{874518}a^{7}-\frac{28947097}{874518}a^{6}+\frac{526886725}{874518}a^{5}+\frac{21812519}{1749036}a^{4}-\frac{381498283}{1749036}a^{3}+\frac{6786239}{437259}a^{2}+\frac{11417332}{437259}a-\frac{3058999}{874518}$, $\frac{200905}{583012}a^{17}+\frac{198869}{874518}a^{16}-\frac{6841067}{583012}a^{15}-\frac{13703753}{1749036}a^{14}+\frac{78069947}{583012}a^{13}+\frac{53309263}{583012}a^{12}-\frac{356791803}{583012}a^{11}-\frac{765784265}{1749036}a^{10}+\frac{193177599}{145753}a^{9}+\frac{1784502467}{1749036}a^{8}-\frac{206971927}{145753}a^{7}-\frac{1071153511}{874518}a^{6}+\frac{412729029}{583012}a^{5}+\frac{648027319}{874518}a^{4}-\frac{75682739}{583012}a^{3}-\frac{118258515}{583012}a^{2}+\frac{523355}{145753}a+\frac{34301417}{1749036}$, $\frac{16138}{437259}a^{17}+\frac{88063}{583012}a^{16}-\frac{2210393}{1749036}a^{15}-\frac{9028841}{1749036}a^{14}+\frac{25351049}{1749036}a^{13}+\frac{103866751}{1749036}a^{12}-\frac{114480167}{1749036}a^{11}-\frac{484647293}{1749036}a^{10}+\frac{219029707}{1749036}a^{9}+\frac{275143327}{437259}a^{8}-\frac{63429475}{874518}a^{7}-\frac{215811135}{291506}a^{6}-\frac{24243283}{437259}a^{5}+\frac{763820995}{1749036}a^{4}+\frac{118438571}{1749036}a^{3}-\frac{196232995}{1749036}a^{2}-\frac{19917053}{1749036}a+\frac{8892497}{874518}$, $\frac{180065}{1749036}a^{17}+\frac{44892}{145753}a^{16}-\frac{2016663}{583012}a^{15}-\frac{3044873}{291506}a^{14}+\frac{5598999}{145753}a^{13}+\frac{51743428}{437259}a^{12}-\frac{95936929}{583012}a^{11}-\frac{77776185}{145753}a^{10}+\frac{544855687}{1749036}a^{9}+\frac{998838029}{874518}a^{8}-\frac{105020755}{437259}a^{7}-\frac{179448132}{145753}a^{6}+\frac{29146835}{1749036}a^{5}+\frac{96288039}{145753}a^{4}+\frac{91254385}{1749036}a^{3}-\frac{145313443}{874518}a^{2}-\frac{23558285}{1749036}a+\frac{14621957}{874518}$
| 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: | \( 46590724.5415 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 46590724.5415 \cdot 1}{2\cdot\sqrt{1365795913530635497834467328}}\cr\approx \mathstrut & 0.165240494546 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 35*x^16 + 421*x^14 - 2148*x^12 + 5568*x^10 - 7923*x^8 + 6263*x^6 - 2627*x^4 + 518*x^2 - 37)
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^18 - 35*x^16 + 421*x^14 - 2148*x^12 + 5568*x^10 - 7923*x^8 + 6263*x^6 - 2627*x^4 + 518*x^2 - 37, 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^18 - 35*x^16 + 421*x^14 - 2148*x^12 + 5568*x^10 - 7923*x^8 + 6263*x^6 - 2627*x^4 + 518*x^2 - 37);
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^18 - 35*x^16 + 421*x^14 - 2148*x^12 + 5568*x^10 - 7923*x^8 + 6263*x^6 - 2627*x^4 + 518*x^2 - 37);
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
$C_3\times S_3$ (as 18T3):
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 18 |
| The 9 conjugacy class representatives for $S_3 \times C_3$ |
| Character table for $S_3 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{37}) \), 3.3.148.1 x3, 3.3.1369.1, 6.6.810448.1, 6.6.1109503312.1 x2, 6.6.69343957.1, 9.9.6075640136512.1 x3 |
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 6 sibling: | 6.6.1109503312.1 |
| Degree 9 sibling: | 9.9.6075640136512.1 |
| Minimal sibling: | 6.6.1109503312.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{6}$ | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.2.0.1}{2} }^{9}$ | R | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{9}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(37\)
| 37.6.5.1 | $x^{6} + 37$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 37.6.5.1 | $x^{6} + 37$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 37.6.5.1 | $x^{6} + 37$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |