Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 16*x^14 + 42*x^13 + 85*x^12 - 256*x^11 + 244*x^10 + 1253*x^9 - 1877*x^8 - 2124*x^7 + 7286*x^6 - 1283*x^5 - 17162*x^4 + 2622*x^3 + 16878*x^2 + 375*x + 813)
gp: K = bnfinit(y^16 - 2*y^15 - 16*y^14 + 42*y^13 + 85*y^12 - 256*y^11 + 244*y^10 + 1253*y^9 - 1877*y^8 - 2124*y^7 + 7286*y^6 - 1283*y^5 - 17162*y^4 + 2622*y^3 + 16878*y^2 + 375*y + 813, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 16*x^14 + 42*x^13 + 85*x^12 - 256*x^11 + 244*x^10 + 1253*x^9 - 1877*x^8 - 2124*x^7 + 7286*x^6 - 1283*x^5 - 17162*x^4 + 2622*x^3 + 16878*x^2 + 375*x + 813);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 16*x^14 + 42*x^13 + 85*x^12 - 256*x^11 + 244*x^10 + 1253*x^9 - 1877*x^8 - 2124*x^7 + 7286*x^6 - 1283*x^5 - 17162*x^4 + 2622*x^3 + 16878*x^2 + 375*x + 813)
\( x^{16} - 2 x^{15} - 16 x^{14} + 42 x^{13} + 85 x^{12} - 256 x^{11} + 244 x^{10} + 1253 x^{9} + \cdots + 813 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(379099670317033992358041\)
\(\medspace = 3^{12}\cdot 61^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.76\) | 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: | $3^{3/4}61^{3/4}\approx 49.755181783405924$ | ||
| Ramified primes: |
\(3\), \(61\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5}a^{11}+\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{2}{5}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{15}a^{12}-\frac{4}{15}a^{10}-\frac{1}{5}a^{8}-\frac{1}{15}a^{6}-\frac{2}{15}a^{4}+\frac{2}{5}a^{2}+\frac{1}{5}$, $\frac{1}{45}a^{13}+\frac{1}{45}a^{12}+\frac{2}{45}a^{11}-\frac{13}{45}a^{10}-\frac{7}{15}a^{9}+\frac{1}{5}a^{8}-\frac{7}{45}a^{7}-\frac{7}{45}a^{6}-\frac{14}{45}a^{5}+\frac{16}{45}a^{4}-\frac{2}{5}a^{3}+\frac{4}{15}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{22995}a^{14}+\frac{26}{22995}a^{13}-\frac{1}{7665}a^{12}+\frac{1297}{22995}a^{11}+\frac{1754}{22995}a^{10}-\frac{3607}{7665}a^{9}-\frac{2392}{22995}a^{8}+\frac{9358}{22995}a^{7}-\frac{848}{7665}a^{6}-\frac{73}{315}a^{5}+\frac{7597}{22995}a^{4}-\frac{263}{1095}a^{3}+\frac{8}{73}a^{2}+\frac{160}{511}a-\frac{391}{1533}$, $\frac{1}{67\!\cdots\!75}a^{15}-\frac{48\!\cdots\!38}{67\!\cdots\!75}a^{14}+\frac{20\!\cdots\!64}{22\!\cdots\!25}a^{13}-\frac{30\!\cdots\!97}{90\!\cdots\!33}a^{12}-\frac{10\!\cdots\!33}{27\!\cdots\!99}a^{11}+\frac{29\!\cdots\!89}{67\!\cdots\!75}a^{10}-\frac{55\!\cdots\!96}{13\!\cdots\!95}a^{9}-\frac{14\!\cdots\!77}{67\!\cdots\!75}a^{8}-\frac{11\!\cdots\!53}{45\!\cdots\!65}a^{7}-\frac{11\!\cdots\!58}{22\!\cdots\!25}a^{6}-\frac{32\!\cdots\!88}{19\!\cdots\!85}a^{5}-\frac{70\!\cdots\!68}{67\!\cdots\!75}a^{4}-\frac{95\!\cdots\!19}{32\!\cdots\!75}a^{3}-\frac{87\!\cdots\!23}{22\!\cdots\!25}a^{2}+\frac{79\!\cdots\!94}{22\!\cdots\!25}a-\frac{43\!\cdots\!64}{22\!\cdots\!25}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
$C_{4}$, which has order $4$ (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: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{36921365106}{188280668324725} a^{15} + \frac{118310705419}{564842004974175} a^{14} + \frac{2022846387779}{564842004974175} a^{13} - \frac{219281853057}{37656133664945} a^{12} - \frac{2807885611939}{112968400994835} a^{11} + \frac{24458951411648}{564842004974175} a^{10} - \frac{73655926932}{7531226732989} a^{9} - \frac{188538992666614}{564842004974175} a^{8} + \frac{27749547249101}{112968400994835} a^{7} + \frac{135873862258019}{188280668324725} a^{6} - \frac{174735924804509}{112968400994835} a^{5} - \frac{328815252097976}{564842004974175} a^{4} + \frac{786609814496154}{188280668324725} a^{3} + \frac{76533010158734}{188280668324725} a^{2} - \frac{722082103992477}{188280668324725} a + \frac{92221951777917}{188280668324725} \)
(order $6$)
| 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: |
$\frac{13\!\cdots\!12}{67\!\cdots\!75}a^{15}-\frac{20\!\cdots\!47}{22\!\cdots\!25}a^{14}-\frac{51\!\cdots\!77}{22\!\cdots\!25}a^{13}+\frac{21\!\cdots\!56}{13\!\cdots\!95}a^{12}-\frac{42\!\cdots\!43}{13\!\cdots\!95}a^{11}-\frac{58\!\cdots\!92}{67\!\cdots\!75}a^{10}+\frac{21\!\cdots\!37}{13\!\cdots\!95}a^{9}+\frac{22\!\cdots\!36}{32\!\cdots\!75}a^{8}-\frac{11\!\cdots\!59}{12\!\cdots\!19}a^{7}+\frac{39\!\cdots\!21}{96\!\cdots\!25}a^{6}+\frac{23\!\cdots\!04}{13\!\cdots\!95}a^{5}-\frac{21\!\cdots\!96}{67\!\cdots\!75}a^{4}-\frac{57\!\cdots\!68}{32\!\cdots\!75}a^{3}+\frac{12\!\cdots\!84}{22\!\cdots\!25}a^{2}+\frac{18\!\cdots\!66}{75\!\cdots\!75}a+\frac{69\!\cdots\!27}{22\!\cdots\!25}$, $\frac{95\!\cdots\!66}{13\!\cdots\!95}a^{15}-\frac{15\!\cdots\!53}{13\!\cdots\!95}a^{14}-\frac{30\!\cdots\!71}{27\!\cdots\!99}a^{13}+\frac{66\!\cdots\!03}{27\!\cdots\!99}a^{12}+\frac{27\!\cdots\!27}{45\!\cdots\!65}a^{11}-\frac{58\!\cdots\!31}{45\!\cdots\!65}a^{10}+\frac{19\!\cdots\!27}{13\!\cdots\!95}a^{9}+\frac{14\!\cdots\!47}{19\!\cdots\!85}a^{8}-\frac{12\!\cdots\!33}{19\!\cdots\!85}a^{7}-\frac{22\!\cdots\!91}{19\!\cdots\!85}a^{6}+\frac{17\!\cdots\!01}{45\!\cdots\!65}a^{5}+\frac{87\!\cdots\!52}{45\!\cdots\!65}a^{4}-\frac{68\!\cdots\!33}{64\!\cdots\!95}a^{3}-\frac{18\!\cdots\!17}{45\!\cdots\!65}a^{2}+\frac{12\!\cdots\!58}{15\!\cdots\!55}a+\frac{31\!\cdots\!96}{15\!\cdots\!55}$, $\frac{168243556457876}{56\!\cdots\!75}a^{15}+\frac{12\!\cdots\!14}{37\!\cdots\!75}a^{14}-\frac{53\!\cdots\!76}{37\!\cdots\!75}a^{13}-\frac{26\!\cdots\!07}{74\!\cdots\!95}a^{12}+\frac{14\!\cdots\!56}{74\!\cdots\!95}a^{11}+\frac{35\!\cdots\!31}{12\!\cdots\!25}a^{10}-\frac{11\!\cdots\!26}{16\!\cdots\!31}a^{9}+\frac{12\!\cdots\!68}{53\!\cdots\!25}a^{8}+\frac{16\!\cdots\!88}{10\!\cdots\!85}a^{7}-\frac{35\!\cdots\!34}{53\!\cdots\!25}a^{6}+\frac{71\!\cdots\!87}{10\!\cdots\!15}a^{5}+\frac{57\!\cdots\!51}{41\!\cdots\!75}a^{4}-\frac{41\!\cdots\!03}{17\!\cdots\!75}a^{3}-\frac{75\!\cdots\!86}{12\!\cdots\!25}a^{2}+\frac{14\!\cdots\!13}{12\!\cdots\!25}a-\frac{93\!\cdots\!96}{41\!\cdots\!75}$, $\frac{17\!\cdots\!83}{10\!\cdots\!25}a^{15}-\frac{67\!\cdots\!11}{96\!\cdots\!25}a^{14}-\frac{12\!\cdots\!91}{96\!\cdots\!25}a^{13}+\frac{21\!\cdots\!68}{19\!\cdots\!85}a^{12}-\frac{15\!\cdots\!48}{19\!\cdots\!85}a^{11}-\frac{14\!\cdots\!34}{32\!\cdots\!75}a^{10}+\frac{30\!\cdots\!83}{21\!\cdots\!65}a^{9}-\frac{36\!\cdots\!64}{96\!\cdots\!25}a^{8}-\frac{18\!\cdots\!27}{38\!\cdots\!57}a^{7}+\frac{59\!\cdots\!32}{96\!\cdots\!25}a^{6}+\frac{11\!\cdots\!74}{19\!\cdots\!85}a^{5}-\frac{24\!\cdots\!64}{10\!\cdots\!25}a^{4}+\frac{14\!\cdots\!69}{32\!\cdots\!75}a^{3}+\frac{79\!\cdots\!49}{32\!\cdots\!75}a^{2}-\frac{40\!\cdots\!22}{32\!\cdots\!75}a-\frac{10\!\cdots\!81}{10\!\cdots\!25}$, $\frac{25\!\cdots\!49}{10\!\cdots\!25}a^{15}+\frac{26\!\cdots\!34}{67\!\cdots\!75}a^{14}-\frac{22\!\cdots\!86}{67\!\cdots\!75}a^{13}-\frac{11\!\cdots\!08}{13\!\cdots\!95}a^{12}+\frac{17\!\cdots\!88}{13\!\cdots\!95}a^{11}+\frac{47\!\cdots\!72}{75\!\cdots\!75}a^{10}-\frac{18\!\cdots\!04}{15\!\cdots\!55}a^{9}+\frac{85\!\cdots\!36}{67\!\cdots\!75}a^{8}+\frac{13\!\cdots\!14}{13\!\cdots\!95}a^{7}-\frac{73\!\cdots\!18}{67\!\cdots\!75}a^{6}+\frac{10\!\cdots\!92}{13\!\cdots\!95}a^{5}+\frac{95\!\cdots\!08}{22\!\cdots\!25}a^{4}-\frac{60\!\cdots\!38}{32\!\cdots\!75}a^{3}-\frac{17\!\cdots\!98}{32\!\cdots\!75}a^{2}+\frac{38\!\cdots\!58}{22\!\cdots\!25}a+\frac{61\!\cdots\!74}{75\!\cdots\!75}$, $\frac{10\!\cdots\!74}{67\!\cdots\!75}a^{15}-\frac{83\!\cdots\!57}{67\!\cdots\!75}a^{14}-\frac{23\!\cdots\!53}{75\!\cdots\!75}a^{13}+\frac{26\!\cdots\!54}{13\!\cdots\!95}a^{12}-\frac{38\!\cdots\!63}{13\!\cdots\!95}a^{11}-\frac{19\!\cdots\!23}{22\!\cdots\!25}a^{10}+\frac{66\!\cdots\!31}{27\!\cdots\!99}a^{9}-\frac{16\!\cdots\!83}{67\!\cdots\!75}a^{8}-\frac{15\!\cdots\!47}{15\!\cdots\!55}a^{7}+\frac{87\!\cdots\!29}{67\!\cdots\!75}a^{6}+\frac{87\!\cdots\!97}{13\!\cdots\!95}a^{5}-\frac{13\!\cdots\!07}{32\!\cdots\!75}a^{4}+\frac{80\!\cdots\!93}{10\!\cdots\!25}a^{3}+\frac{52\!\cdots\!91}{75\!\cdots\!75}a^{2}+\frac{73\!\cdots\!76}{22\!\cdots\!25}a+\frac{10\!\cdots\!54}{10\!\cdots\!25}$, $\frac{87\!\cdots\!42}{22\!\cdots\!25}a^{15}-\frac{96\!\cdots\!63}{32\!\cdots\!75}a^{14}-\frac{20\!\cdots\!08}{32\!\cdots\!75}a^{13}+\frac{35\!\cdots\!41}{45\!\cdots\!65}a^{12}+\frac{17\!\cdots\!98}{45\!\cdots\!65}a^{11}-\frac{84\!\cdots\!67}{22\!\cdots\!25}a^{10}+\frac{35\!\cdots\!13}{61\!\cdots\!05}a^{9}+\frac{10\!\cdots\!56}{22\!\cdots\!25}a^{8}-\frac{14\!\cdots\!71}{45\!\cdots\!65}a^{7}-\frac{15\!\cdots\!53}{22\!\cdots\!25}a^{6}+\frac{68\!\cdots\!17}{45\!\cdots\!65}a^{5}+\frac{32\!\cdots\!54}{22\!\cdots\!25}a^{4}-\frac{37\!\cdots\!58}{10\!\cdots\!25}a^{3}-\frac{29\!\cdots\!41}{75\!\cdots\!75}a^{2}+\frac{26\!\cdots\!18}{75\!\cdots\!75}a-\frac{11\!\cdots\!23}{75\!\cdots\!75}$
| 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: | \( 435198.818124 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 435198.818124 \cdot 4}{6\cdot\sqrt{379099670317033992358041}}\cr\approx \mathstrut & 1.14461304635 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 16*x^14 + 42*x^13 + 85*x^12 - 256*x^11 + 244*x^10 + 1253*x^9 - 1877*x^8 - 2124*x^7 + 7286*x^6 - 1283*x^5 - 17162*x^4 + 2622*x^3 + 16878*x^2 + 375*x + 813)
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^16 - 2*x^15 - 16*x^14 + 42*x^13 + 85*x^12 - 256*x^11 + 244*x^10 + 1253*x^9 - 1877*x^8 - 2124*x^7 + 7286*x^6 - 1283*x^5 - 17162*x^4 + 2622*x^3 + 16878*x^2 + 375*x + 813, 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^16 - 2*x^15 - 16*x^14 + 42*x^13 + 85*x^12 - 256*x^11 + 244*x^10 + 1253*x^9 - 1877*x^8 - 2124*x^7 + 7286*x^6 - 1283*x^5 - 17162*x^4 + 2622*x^3 + 16878*x^2 + 375*x + 813);
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^16 - 2*x^15 - 16*x^14 + 42*x^13 + 85*x^12 - 256*x^11 + 244*x^10 + 1253*x^9 - 1877*x^8 - 2124*x^7 + 7286*x^6 - 1283*x^5 - 17162*x^4 + 2622*x^3 + 16878*x^2 + 375*x + 813);
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
$C_4\wr C_2$ (as 16T42):
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 32 |
| The 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{61}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-183}) \), 4.2.11163.1 x2, 4.0.549.1 x2, \(\Q(\sqrt{-3}, \sqrt{61})\), 8.0.615710703429.2, 8.0.165469149.1, 8.0.1121513121.2 |
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 32 |
| Degree 8 siblings: | 8.0.165469149.1, 8.0.615710703429.2 |
| Degree 16 sibling: | 16.4.1410629873249683485564270561.1 |
| Minimal sibling: | 8.0.165469149.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
| 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
|
\(61\)
| 61.4.3.1 | $x^{4} + 183$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 61.4.3.1 | $x^{4} + 183$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |