Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 27*x^14 - 78*x^13 + 280*x^12 - 578*x^11 + 1404*x^10 - 2304*x^9 + 3384*x^8 - 5082*x^7 + 5100*x^6 - 3542*x^5 + 6427*x^4 + 36*x^3 + 2469*x^2 - 1600*x + 268)
gp: K = bnfinit(y^16 - 4*y^15 + 27*y^14 - 78*y^13 + 280*y^12 - 578*y^11 + 1404*y^10 - 2304*y^9 + 3384*y^8 - 5082*y^7 + 5100*y^6 - 3542*y^5 + 6427*y^4 + 36*y^3 + 2469*y^2 - 1600*y + 268, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 27*x^14 - 78*x^13 + 280*x^12 - 578*x^11 + 1404*x^10 - 2304*x^9 + 3384*x^8 - 5082*x^7 + 5100*x^6 - 3542*x^5 + 6427*x^4 + 36*x^3 + 2469*x^2 - 1600*x + 268);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 27*x^14 - 78*x^13 + 280*x^12 - 578*x^11 + 1404*x^10 - 2304*x^9 + 3384*x^8 - 5082*x^7 + 5100*x^6 - 3542*x^5 + 6427*x^4 + 36*x^3 + 2469*x^2 - 1600*x + 268)
\( x^{16} - 4 x^{15} + 27 x^{14} - 78 x^{13} + 280 x^{12} - 578 x^{11} + 1404 x^{10} - 2304 x^{9} + \cdots + 268 \)
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: |
\(28196723068685041735089\)
\(\medspace = 3^{8}\cdot 73^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(25.30\) | 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: | $3^{1/2}73^{3/4}\approx 43.25663903061054$ | ||
| Ramified primes: |
\(3\), \(73\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{3}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{3}-\frac{3}{8}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{3}{8}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{16}a^{11}+\frac{1}{16}a^{8}-\frac{1}{4}a^{6}+\frac{3}{16}a^{5}+\frac{3}{16}a^{2}+\frac{1}{4}$, $\frac{1}{32}a^{12}-\frac{1}{32}a^{11}-\frac{1}{16}a^{10}+\frac{1}{32}a^{9}+\frac{1}{32}a^{8}-\frac{1}{8}a^{7}+\frac{7}{32}a^{6}+\frac{5}{32}a^{5}-\frac{3}{16}a^{4}-\frac{5}{32}a^{3}-\frac{5}{32}a^{2}+\frac{1}{8}a+\frac{1}{8}$, $\frac{1}{64}a^{13}+\frac{1}{64}a^{11}+\frac{3}{64}a^{10}+\frac{1}{32}a^{9}-\frac{3}{64}a^{8}+\frac{3}{64}a^{7}+\frac{3}{16}a^{6}-\frac{5}{64}a^{5}-\frac{15}{64}a^{4}-\frac{5}{32}a^{3}-\frac{17}{64}a^{2}-\frac{1}{8}a+\frac{1}{16}$, $\frac{1}{64}a^{14}-\frac{1}{64}a^{12}+\frac{1}{64}a^{11}-\frac{1}{32}a^{10}+\frac{3}{64}a^{9}-\frac{3}{64}a^{8}+\frac{1}{16}a^{7}-\frac{3}{64}a^{6}-\frac{5}{64}a^{5}+\frac{5}{32}a^{4}-\frac{15}{64}a^{3}+\frac{11}{32}a^{2}-\frac{5}{16}a+\frac{1}{8}$, $\frac{1}{23\!\cdots\!44}a^{15}-\frac{15\!\cdots\!37}{23\!\cdots\!44}a^{14}+\frac{13\!\cdots\!97}{29\!\cdots\!68}a^{13}+\frac{55\!\cdots\!29}{11\!\cdots\!72}a^{12}-\frac{16\!\cdots\!21}{11\!\cdots\!72}a^{11}-\frac{24\!\cdots\!19}{91\!\cdots\!24}a^{10}-\frac{23\!\cdots\!33}{58\!\cdots\!36}a^{9}+\frac{10\!\cdots\!25}{58\!\cdots\!36}a^{8}-\frac{10\!\cdots\!39}{58\!\cdots\!36}a^{7}-\frac{17\!\cdots\!63}{11\!\cdots\!72}a^{6}+\frac{65\!\cdots\!21}{11\!\cdots\!72}a^{5}-\frac{26\!\cdots\!43}{36\!\cdots\!96}a^{4}-\frac{84\!\cdots\!81}{23\!\cdots\!44}a^{3}+\frac{35\!\cdots\!25}{23\!\cdots\!44}a^{2}-\frac{16\!\cdots\!99}{58\!\cdots\!36}a-\frac{18\!\cdots\!49}{58\!\cdots\!36}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{1602690543}{993771917312} a^{15} + \frac{5615017511}{993771917312} a^{14} - \frac{5032347911}{124221489664} a^{13} + \frac{52419930421}{496885958656} a^{12} - \frac{197225627325}{496885958656} a^{11} + \frac{5748155779}{7763843104} a^{10} - \frac{474902863985}{248442979328} a^{9} + \frac{717905793173}{248442979328} a^{8} - \frac{1065623117819}{248442979328} a^{7} + \frac{3368238515281}{496885958656} a^{6} - \frac{3085337478591}{496885958656} a^{5} + \frac{65593966495}{15527686208} a^{4} - \frac{10497194745973}{993771917312} a^{3} - \frac{2598294039187}{993771917312} a^{2} - \frac{1382385421115}{248442979328} a + \frac{537934268375}{248442979328} \)
(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: |
$\frac{69\!\cdots\!21}{23\!\cdots\!44}a^{15}-\frac{14\!\cdots\!97}{23\!\cdots\!44}a^{14}+\frac{17\!\cdots\!93}{29\!\cdots\!68}a^{13}-\frac{11\!\cdots\!11}{11\!\cdots\!72}a^{12}+\frac{55\!\cdots\!59}{11\!\cdots\!72}a^{11}-\frac{17\!\cdots\!33}{36\!\cdots\!96}a^{10}+\frac{11\!\cdots\!15}{58\!\cdots\!36}a^{9}-\frac{10\!\cdots\!07}{58\!\cdots\!36}a^{8}+\frac{18\!\cdots\!01}{58\!\cdots\!36}a^{7}-\frac{84\!\cdots\!83}{11\!\cdots\!72}a^{6}+\frac{31\!\cdots\!69}{11\!\cdots\!72}a^{5}+\frac{21\!\cdots\!21}{18\!\cdots\!48}a^{4}+\frac{38\!\cdots\!19}{23\!\cdots\!44}a^{3}+\frac{67\!\cdots\!41}{23\!\cdots\!44}a^{2}+\frac{43\!\cdots\!65}{58\!\cdots\!36}a-\frac{14\!\cdots\!29}{58\!\cdots\!36}$, $\frac{30\!\cdots\!47}{11\!\cdots\!72}a^{15}-\frac{468262052266051}{11\!\cdots\!72}a^{14}+\frac{41\!\cdots\!83}{14\!\cdots\!84}a^{13}+\frac{41\!\cdots\!47}{58\!\cdots\!36}a^{12}-\frac{50\!\cdots\!87}{58\!\cdots\!36}a^{11}+\frac{24\!\cdots\!91}{18\!\cdots\!48}a^{10}-\frac{66\!\cdots\!03}{29\!\cdots\!68}a^{9}+\frac{23\!\cdots\!75}{29\!\cdots\!68}a^{8}-\frac{41\!\cdots\!17}{29\!\cdots\!68}a^{7}+\frac{10\!\cdots\!79}{58\!\cdots\!36}a^{6}-\frac{19\!\cdots\!33}{58\!\cdots\!36}a^{5}+\frac{15\!\cdots\!19}{45\!\cdots\!12}a^{4}-\frac{54\!\cdots\!91}{11\!\cdots\!72}a^{3}+\frac{50\!\cdots\!27}{11\!\cdots\!72}a^{2}+\frac{60\!\cdots\!87}{29\!\cdots\!68}a-\frac{23\!\cdots\!51}{29\!\cdots\!68}$, $\frac{291783959384911}{23\!\cdots\!44}a^{15}+\frac{11\!\cdots\!81}{23\!\cdots\!44}a^{14}-\frac{50\!\cdots\!05}{29\!\cdots\!68}a^{13}+\frac{15\!\cdots\!75}{11\!\cdots\!72}a^{12}-\frac{42\!\cdots\!03}{11\!\cdots\!72}a^{11}+\frac{50\!\cdots\!07}{36\!\cdots\!96}a^{10}-\frac{16\!\cdots\!11}{58\!\cdots\!36}a^{9}+\frac{40\!\cdots\!31}{58\!\cdots\!36}a^{8}-\frac{67\!\cdots\!09}{58\!\cdots\!36}a^{7}+\frac{19\!\cdots\!59}{11\!\cdots\!72}a^{6}-\frac{30\!\cdots\!21}{11\!\cdots\!72}a^{5}+\frac{11\!\cdots\!83}{45\!\cdots\!12}a^{4}-\frac{42\!\cdots\!71}{23\!\cdots\!44}a^{3}+\frac{69\!\cdots\!67}{23\!\cdots\!44}a^{2}+\frac{35\!\cdots\!19}{58\!\cdots\!36}a+\frac{51\!\cdots\!61}{58\!\cdots\!36}$, $\frac{31\!\cdots\!79}{23\!\cdots\!44}a^{15}-\frac{32\!\cdots\!23}{23\!\cdots\!44}a^{14}+\frac{22\!\cdots\!91}{29\!\cdots\!68}a^{13}-\frac{41\!\cdots\!29}{11\!\cdots\!72}a^{12}+\frac{13\!\cdots\!53}{11\!\cdots\!72}a^{11}-\frac{12\!\cdots\!65}{36\!\cdots\!96}a^{10}+\frac{44\!\cdots\!69}{58\!\cdots\!36}a^{9}-\frac{92\!\cdots\!53}{58\!\cdots\!36}a^{8}+\frac{14\!\cdots\!19}{58\!\cdots\!36}a^{7}-\frac{39\!\cdots\!57}{11\!\cdots\!72}a^{6}+\frac{47\!\cdots\!55}{11\!\cdots\!72}a^{5}-\frac{19\!\cdots\!38}{57\!\cdots\!89}a^{4}+\frac{40\!\cdots\!09}{23\!\cdots\!44}a^{3}-\frac{40\!\cdots\!57}{23\!\cdots\!44}a^{2}-\frac{76\!\cdots\!29}{58\!\cdots\!36}a+\frac{68\!\cdots\!65}{58\!\cdots\!36}$, $\frac{10\!\cdots\!07}{17\!\cdots\!88}a^{15}-\frac{40\!\cdots\!71}{17\!\cdots\!88}a^{14}+\frac{34\!\cdots\!23}{22\!\cdots\!36}a^{13}-\frac{38\!\cdots\!05}{89\!\cdots\!44}a^{12}+\frac{14\!\cdots\!37}{89\!\cdots\!44}a^{11}-\frac{88\!\cdots\!23}{28\!\cdots\!92}a^{10}+\frac{34\!\cdots\!85}{44\!\cdots\!72}a^{9}-\frac{55\!\cdots\!29}{44\!\cdots\!72}a^{8}+\frac{74\!\cdots\!19}{44\!\cdots\!72}a^{7}-\frac{24\!\cdots\!81}{89\!\cdots\!44}a^{6}+\frac{19\!\cdots\!47}{89\!\cdots\!44}a^{5}-\frac{23\!\cdots\!39}{14\!\cdots\!96}a^{4}+\frac{60\!\cdots\!13}{17\!\cdots\!88}a^{3}+\frac{18\!\cdots\!67}{17\!\cdots\!88}a^{2}+\frac{81\!\cdots\!19}{44\!\cdots\!72}a-\frac{20\!\cdots\!95}{44\!\cdots\!72}$, $\frac{20\!\cdots\!41}{36\!\cdots\!96}a^{15}-\frac{17\!\cdots\!25}{36\!\cdots\!96}a^{14}+\frac{55\!\cdots\!29}{91\!\cdots\!24}a^{13}+\frac{92\!\cdots\!07}{91\!\cdots\!24}a^{12}-\frac{51\!\cdots\!79}{22\!\cdots\!56}a^{11}+\frac{58\!\cdots\!89}{22\!\cdots\!56}a^{10}-\frac{97\!\cdots\!69}{18\!\cdots\!48}a^{9}+\frac{27\!\cdots\!89}{18\!\cdots\!48}a^{8}-\frac{16\!\cdots\!07}{57\!\cdots\!89}a^{7}+\frac{19\!\cdots\!19}{57\!\cdots\!89}a^{6}-\frac{10\!\cdots\!39}{22\!\cdots\!56}a^{5}+\frac{13\!\cdots\!79}{22\!\cdots\!56}a^{4}+\frac{31\!\cdots\!29}{36\!\cdots\!96}a^{3}+\frac{11\!\cdots\!87}{36\!\cdots\!96}a^{2}-\frac{18\!\cdots\!65}{91\!\cdots\!24}a+\frac{30\!\cdots\!69}{91\!\cdots\!24}$, $\frac{19\!\cdots\!53}{23\!\cdots\!44}a^{15}-\frac{81\!\cdots\!21}{23\!\cdots\!44}a^{14}+\frac{14\!\cdots\!97}{29\!\cdots\!68}a^{13}+\frac{37\!\cdots\!85}{11\!\cdots\!72}a^{12}-\frac{12\!\cdots\!09}{11\!\cdots\!72}a^{11}+\frac{20\!\cdots\!43}{36\!\cdots\!96}a^{10}-\frac{72\!\cdots\!45}{58\!\cdots\!36}a^{9}+\frac{16\!\cdots\!65}{58\!\cdots\!36}a^{8}-\frac{29\!\cdots\!59}{58\!\cdots\!36}a^{7}+\frac{58\!\cdots\!53}{11\!\cdots\!72}a^{6}-\frac{62\!\cdots\!39}{11\!\cdots\!72}a^{5}+\frac{13\!\cdots\!61}{18\!\cdots\!48}a^{4}+\frac{41\!\cdots\!39}{23\!\cdots\!44}a^{3}+\frac{71\!\cdots\!33}{23\!\cdots\!44}a^{2}-\frac{13\!\cdots\!23}{58\!\cdots\!36}a+\frac{26\!\cdots\!63}{58\!\cdots\!36}$
| 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: | \( 686254.798714 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 686254.798714 \cdot 1}{6\cdot\sqrt{28196723068685041735089}}\cr\approx \mathstrut & 1.65452666694 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 27*x^14 - 78*x^13 + 280*x^12 - 578*x^11 + 1404*x^10 - 2304*x^9 + 3384*x^8 - 5082*x^7 + 5100*x^6 - 3542*x^5 + 6427*x^4 + 36*x^3 + 2469*x^2 - 1600*x + 268)
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 - 4*x^15 + 27*x^14 - 78*x^13 + 280*x^12 - 578*x^11 + 1404*x^10 - 2304*x^9 + 3384*x^8 - 5082*x^7 + 5100*x^6 - 3542*x^5 + 6427*x^4 + 36*x^3 + 2469*x^2 - 1600*x + 268, 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 - 4*x^15 + 27*x^14 - 78*x^13 + 280*x^12 - 578*x^11 + 1404*x^10 - 2304*x^9 + 3384*x^8 - 5082*x^7 + 5100*x^6 - 3542*x^5 + 6427*x^4 + 36*x^3 + 2469*x^2 - 1600*x + 268);
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 - 4*x^15 + 27*x^14 - 78*x^13 + 280*x^12 - 578*x^11 + 1404*x^10 - 2304*x^9 + 3384*x^8 - 5082*x^7 + 5100*x^6 - 3542*x^5 + 6427*x^4 + 36*x^3 + 2469*x^2 - 1600*x + 268);
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
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{73}) \), \(\Q(\sqrt{-219}) \), 4.0.657.1 x2, 4.2.15987.1 x2, \(\Q(\sqrt{-3}, \sqrt{73})\), 8.0.167918799033.1, 8.0.31510377.1, 8.0.2300257521.1 |
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
| Galois closure: | deg 32 |
| Degree 8 siblings: | 8.0.167918799033.1, 8.0.31510377.1 |
| Degree 16 sibling: | 16.4.16695593025891398600698809.1 |
| Minimal sibling: | 8.0.31510377.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(73\)
| 73.8.6.2 | $x^{8} - 5402 x^{4} - 51217019$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 73.8.4.1 | $x^{8} + 15768 x^{7} + 93236508 x^{6} + 245028523288 x^{5} + 241487187554464 x^{4} + 21812691357056 x^{3} + 3877844534238648 x^{2} + 13666761747168624 x + 1459074653762756$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |