Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^18 + 90*x^16 - 72*x^14 + 4684*x^12 + 1792*x^10 - 56320*x^8 + 538208*x^6 + 489808*x^4 - 2811072*x^2 + 5153632)
gp: K = bnfinit(y^20 - 4*y^18 + 90*y^16 - 72*y^14 + 4684*y^12 + 1792*y^10 - 56320*y^8 + 538208*y^6 + 489808*y^4 - 2811072*y^2 + 5153632, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^18 + 90*x^16 - 72*x^14 + 4684*x^12 + 1792*x^10 - 56320*x^8 + 538208*x^6 + 489808*x^4 - 2811072*x^2 + 5153632);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^18 + 90*x^16 - 72*x^14 + 4684*x^12 + 1792*x^10 - 56320*x^8 + 538208*x^6 + 489808*x^4 - 2811072*x^2 + 5153632)
\( x^{20} - 4 x^{18} + 90 x^{16} - 72 x^{14} + 4684 x^{12} + 1792 x^{10} - 56320 x^{8} + 538208 x^{6} + \cdots + 5153632 \)
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: |
\(1243811788377812389377718878208\)
\(\medspace = 2^{55}\cdot 11^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.97\) | 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: | $2^{11/4}11^{9/10}\approx 58.22183708777889$ | ||
| Ramified primes: |
\(2\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{22}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $10$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{88}a^{12}-\frac{1}{22}a^{10}+\frac{1}{44}a^{8}+\frac{2}{11}a^{6}+\frac{5}{22}a^{4}+\frac{4}{11}a^{2}$, $\frac{1}{88}a^{13}-\frac{1}{22}a^{11}+\frac{1}{44}a^{9}+\frac{2}{11}a^{7}+\frac{5}{22}a^{5}+\frac{4}{11}a^{3}$, $\frac{1}{968}a^{14}-\frac{1}{242}a^{12}+\frac{45}{484}a^{10}-\frac{9}{121}a^{8}-\frac{39}{242}a^{6}-\frac{18}{121}a^{4}-\frac{2}{11}a^{2}$, $\frac{1}{968}a^{15}-\frac{1}{242}a^{13}+\frac{45}{484}a^{11}-\frac{9}{121}a^{9}-\frac{39}{242}a^{7}-\frac{18}{121}a^{5}-\frac{2}{11}a^{3}$, $\frac{1}{404624}a^{16}-\frac{23}{101156}a^{14}+\frac{9}{5324}a^{12}+\frac{2873}{50578}a^{10}+\frac{2050}{25289}a^{8}+\frac{11741}{50578}a^{6}-\frac{925}{4598}a^{4}-\frac{5}{11}a^{2}+\frac{8}{19}$, $\frac{1}{404624}a^{17}-\frac{23}{101156}a^{15}+\frac{9}{5324}a^{13}+\frac{2873}{50578}a^{11}+\frac{2050}{25289}a^{9}+\frac{11741}{50578}a^{7}-\frac{925}{4598}a^{5}-\frac{5}{11}a^{3}+\frac{8}{19}a$, $\frac{1}{31\!\cdots\!92}a^{18}-\frac{22\!\cdots\!59}{31\!\cdots\!92}a^{16}+\frac{35\!\cdots\!21}{78\!\cdots\!48}a^{14}+\frac{16\!\cdots\!49}{78\!\cdots\!48}a^{12}+\frac{38\!\cdots\!67}{39\!\cdots\!74}a^{10}-\frac{80\!\cdots\!32}{19\!\cdots\!87}a^{8}+\frac{64\!\cdots\!17}{35\!\cdots\!34}a^{6}-\frac{31\!\cdots\!29}{14\!\cdots\!77}a^{4}-\frac{60\!\cdots\!28}{14\!\cdots\!77}a^{2}+\frac{37\!\cdots\!89}{13\!\cdots\!07}$, $\frac{1}{31\!\cdots\!92}a^{19}-\frac{22\!\cdots\!59}{31\!\cdots\!92}a^{17}+\frac{35\!\cdots\!21}{78\!\cdots\!48}a^{15}+\frac{16\!\cdots\!49}{78\!\cdots\!48}a^{13}+\frac{38\!\cdots\!67}{39\!\cdots\!74}a^{11}-\frac{80\!\cdots\!32}{19\!\cdots\!87}a^{9}+\frac{64\!\cdots\!17}{35\!\cdots\!34}a^{7}-\frac{31\!\cdots\!29}{14\!\cdots\!77}a^{5}-\frac{60\!\cdots\!28}{14\!\cdots\!77}a^{3}+\frac{37\!\cdots\!89}{13\!\cdots\!07}a$
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: | $9$ | 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: |
$\frac{22\!\cdots\!97}{31\!\cdots\!92}a^{18}-\frac{85\!\cdots\!75}{31\!\cdots\!92}a^{16}+\frac{10\!\cdots\!45}{15\!\cdots\!96}a^{14}-\frac{10\!\cdots\!71}{15\!\cdots\!96}a^{12}+\frac{29\!\cdots\!17}{78\!\cdots\!48}a^{10}+\frac{17\!\cdots\!75}{39\!\cdots\!74}a^{8}-\frac{62\!\cdots\!15}{35\!\cdots\!34}a^{6}+\frac{48\!\cdots\!81}{16\!\cdots\!47}a^{4}+\frac{47\!\cdots\!87}{14\!\cdots\!77}a^{2}-\frac{14\!\cdots\!44}{13\!\cdots\!07}$, $\frac{51\!\cdots\!51}{31\!\cdots\!92}a^{18}-\frac{49\!\cdots\!75}{39\!\cdots\!74}a^{16}+\frac{14\!\cdots\!13}{78\!\cdots\!48}a^{14}-\frac{12\!\cdots\!73}{15\!\cdots\!96}a^{12}+\frac{81\!\cdots\!01}{78\!\cdots\!48}a^{10}-\frac{27\!\cdots\!21}{78\!\cdots\!48}a^{8}+\frac{30\!\cdots\!25}{17\!\cdots\!17}a^{6}+\frac{24\!\cdots\!21}{32\!\cdots\!94}a^{4}-\frac{26\!\cdots\!11}{14\!\cdots\!77}a^{2}+\frac{20\!\cdots\!03}{13\!\cdots\!07}$, $\frac{16\!\cdots\!51}{15\!\cdots\!96}a^{18}-\frac{27\!\cdots\!61}{31\!\cdots\!92}a^{16}+\frac{48\!\cdots\!95}{39\!\cdots\!74}a^{14}-\frac{23\!\cdots\!97}{39\!\cdots\!74}a^{12}+\frac{52\!\cdots\!03}{78\!\cdots\!48}a^{10}-\frac{53\!\cdots\!70}{19\!\cdots\!87}a^{8}+\frac{34\!\cdots\!92}{17\!\cdots\!17}a^{6}+\frac{74\!\cdots\!43}{16\!\cdots\!47}a^{4}-\frac{25\!\cdots\!16}{14\!\cdots\!77}a^{2}+\frac{21\!\cdots\!58}{13\!\cdots\!07}$, $\frac{499468967874893}{39\!\cdots\!74}a^{18}-\frac{32\!\cdots\!55}{31\!\cdots\!92}a^{16}+\frac{11\!\cdots\!29}{78\!\cdots\!48}a^{14}-\frac{10\!\cdots\!85}{15\!\cdots\!96}a^{12}+\frac{65\!\cdots\!51}{78\!\cdots\!48}a^{10}-\frac{24\!\cdots\!79}{78\!\cdots\!48}a^{8}+\frac{53\!\cdots\!36}{17\!\cdots\!17}a^{6}+\frac{10\!\cdots\!73}{16\!\cdots\!47}a^{4}-\frac{25\!\cdots\!44}{14\!\cdots\!77}a^{2}+\frac{14\!\cdots\!77}{13\!\cdots\!07}$, $\frac{12\!\cdots\!75}{15\!\cdots\!96}a^{19}+\frac{27\!\cdots\!39}{78\!\cdots\!48}a^{18}-\frac{18\!\cdots\!43}{31\!\cdots\!92}a^{17}-\frac{84\!\cdots\!29}{31\!\cdots\!92}a^{16}+\frac{13\!\cdots\!15}{15\!\cdots\!96}a^{15}+\frac{78\!\cdots\!23}{19\!\cdots\!87}a^{14}-\frac{15\!\cdots\!87}{39\!\cdots\!74}a^{13}-\frac{68\!\cdots\!15}{39\!\cdots\!74}a^{12}+\frac{18\!\cdots\!71}{39\!\cdots\!74}a^{11}+\frac{42\!\cdots\!10}{19\!\cdots\!87}a^{10}-\frac{69\!\cdots\!29}{39\!\cdots\!74}a^{9}-\frac{15\!\cdots\!63}{19\!\cdots\!87}a^{8}+\frac{93\!\cdots\!02}{17\!\cdots\!17}a^{7}+\frac{11\!\cdots\!13}{35\!\cdots\!34}a^{6}+\frac{94\!\cdots\!57}{32\!\cdots\!94}a^{5}+\frac{25\!\cdots\!18}{16\!\cdots\!47}a^{4}-\frac{10\!\cdots\!28}{14\!\cdots\!77}a^{3}-\frac{55\!\cdots\!76}{14\!\cdots\!77}a^{2}+\frac{11\!\cdots\!26}{13\!\cdots\!07}a+\frac{57\!\cdots\!61}{13\!\cdots\!07}$, $\frac{10\!\cdots\!07}{15\!\cdots\!96}a^{19}+\frac{15\!\cdots\!97}{15\!\cdots\!96}a^{18}+\frac{14\!\cdots\!33}{15\!\cdots\!96}a^{17}-\frac{29\!\cdots\!51}{78\!\cdots\!48}a^{16}+\frac{63\!\cdots\!45}{78\!\cdots\!48}a^{15}+\frac{14\!\cdots\!95}{15\!\cdots\!96}a^{14}+\frac{18\!\cdots\!77}{15\!\cdots\!96}a^{13}-\frac{17\!\cdots\!40}{19\!\cdots\!87}a^{12}+\frac{25\!\cdots\!08}{19\!\cdots\!87}a^{11}+\frac{39\!\cdots\!21}{78\!\cdots\!48}a^{10}+\frac{56\!\cdots\!99}{78\!\cdots\!48}a^{9}-\frac{39\!\cdots\!65}{78\!\cdots\!48}a^{8}-\frac{23\!\cdots\!79}{32\!\cdots\!94}a^{7}-\frac{10\!\cdots\!21}{35\!\cdots\!34}a^{6}+\frac{11\!\cdots\!87}{32\!\cdots\!94}a^{5}+\frac{60\!\cdots\!48}{16\!\cdots\!47}a^{4}+\frac{97\!\cdots\!45}{14\!\cdots\!77}a^{3}+\frac{70\!\cdots\!27}{14\!\cdots\!77}a^{2}+\frac{23\!\cdots\!69}{13\!\cdots\!07}a+\frac{82\!\cdots\!93}{13\!\cdots\!07}$, $\frac{91\!\cdots\!57}{15\!\cdots\!96}a^{19}+\frac{677401135072985}{19\!\cdots\!87}a^{18}-\frac{69\!\cdots\!65}{15\!\cdots\!96}a^{17}-\frac{26\!\cdots\!97}{78\!\cdots\!48}a^{16}+\frac{10\!\cdots\!23}{15\!\cdots\!96}a^{15}+\frac{75\!\cdots\!99}{15\!\cdots\!96}a^{14}-\frac{11\!\cdots\!41}{39\!\cdots\!74}a^{13}-\frac{11\!\cdots\!51}{39\!\cdots\!74}a^{12}+\frac{74\!\cdots\!01}{19\!\cdots\!87}a^{11}+\frac{55\!\cdots\!71}{19\!\cdots\!87}a^{10}-\frac{26\!\cdots\!63}{20\!\cdots\!46}a^{9}-\frac{28\!\cdots\!21}{19\!\cdots\!87}a^{8}+\frac{24\!\cdots\!29}{17\!\cdots\!17}a^{7}+\frac{14\!\cdots\!89}{35\!\cdots\!34}a^{6}+\frac{42\!\cdots\!77}{16\!\cdots\!47}a^{5}+\frac{12\!\cdots\!01}{32\!\cdots\!94}a^{4}-\frac{98\!\cdots\!28}{14\!\cdots\!77}a^{3}-\frac{47\!\cdots\!52}{14\!\cdots\!77}a^{2}+\frac{11\!\cdots\!59}{13\!\cdots\!07}a+\frac{42\!\cdots\!15}{13\!\cdots\!07}$, $\frac{11\!\cdots\!85}{28\!\cdots\!72}a^{19}+\frac{15\!\cdots\!93}{31\!\cdots\!92}a^{18}-\frac{19\!\cdots\!93}{28\!\cdots\!72}a^{17}-\frac{54\!\cdots\!65}{78\!\cdots\!48}a^{16}+\frac{45\!\cdots\!07}{14\!\cdots\!36}a^{15}+\frac{27\!\cdots\!43}{78\!\cdots\!48}a^{14}+\frac{76\!\cdots\!57}{12\!\cdots\!76}a^{13}+\frac{13\!\cdots\!79}{15\!\cdots\!96}a^{12}+\frac{31\!\cdots\!96}{17\!\cdots\!17}a^{11}+\frac{14\!\cdots\!21}{78\!\cdots\!48}a^{10}+\frac{38\!\cdots\!81}{71\!\cdots\!68}a^{9}+\frac{55\!\cdots\!53}{78\!\cdots\!48}a^{8}-\frac{45\!\cdots\!82}{17\!\cdots\!17}a^{7}-\frac{14\!\cdots\!35}{35\!\cdots\!34}a^{6}+\frac{56\!\cdots\!39}{32\!\cdots\!94}a^{5}+\frac{63\!\cdots\!43}{29\!\cdots\!54}a^{4}+\frac{92\!\cdots\!40}{14\!\cdots\!77}a^{3}+\frac{11\!\cdots\!32}{14\!\cdots\!77}a^{2}-\frac{12\!\cdots\!59}{13\!\cdots\!07}a-\frac{23\!\cdots\!60}{13\!\cdots\!07}$, $\frac{394163071663839}{71\!\cdots\!68}a^{19}+\frac{4679070617}{705429159519853}a^{18}-\frac{35\!\cdots\!11}{71\!\cdots\!68}a^{17}-\frac{32921768866}{705429159519853}a^{16}+\frac{51\!\cdots\!19}{71\!\cdots\!68}a^{15}+\frac{2128789542141}{28\!\cdots\!12}a^{14}-\frac{26\!\cdots\!73}{71\!\cdots\!68}a^{13}-\frac{2021181166562}{705429159519853}a^{12}+\frac{71\!\cdots\!02}{17\!\cdots\!17}a^{11}+\frac{58693209620891}{14\!\cdots\!06}a^{10}-\frac{30\!\cdots\!80}{17\!\cdots\!17}a^{9}-\frac{84275218997288}{705429159519853}a^{8}+\frac{56\!\cdots\!99}{17\!\cdots\!17}a^{7}+\frac{66402638348552}{705429159519853}a^{6}+\frac{41\!\cdots\!58}{16\!\cdots\!47}a^{5}+\frac{21\!\cdots\!68}{705429159519853}a^{4}-\frac{15\!\cdots\!08}{14\!\cdots\!77}a^{3}-\frac{38\!\cdots\!80}{705429159519853}a^{2}+\frac{22\!\cdots\!46}{13\!\cdots\!07}a-\frac{309726197988943}{705429159519853}$
| 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: | \( 5058464.3856691625 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 5058464.3856691625 \cdot 1}{2\cdot\sqrt{1243811788377812389377718878208}}\cr\approx \mathstrut & 0.217475355929669 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^18 + 90*x^16 - 72*x^14 + 4684*x^12 + 1792*x^10 - 56320*x^8 + 538208*x^6 + 489808*x^4 - 2811072*x^2 + 5153632)
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 - 4*x^18 + 90*x^16 - 72*x^14 + 4684*x^12 + 1792*x^10 - 56320*x^8 + 538208*x^6 + 489808*x^4 - 2811072*x^2 + 5153632, 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 - 4*x^18 + 90*x^16 - 72*x^14 + 4684*x^12 + 1792*x^10 - 56320*x^8 + 538208*x^6 + 489808*x^4 - 2811072*x^2 + 5153632);
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 - 4*x^18 + 90*x^16 - 72*x^14 + 4684*x^12 + 1792*x^10 - 56320*x^8 + 538208*x^6 + 489808*x^4 - 2811072*x^2 + 5153632);
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
$C_{10}\wr C_2$ (as 20T53):
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 200 |
| The 65 conjugacy class representatives for $C_{10}\wr C_2$ |
| Character table for $C_{10}\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 4.0.22528.1, 10.0.479756288.1 |
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 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 20.0.84954018740373771557797888.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{5}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ | $20$ | $20$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{10}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $20$ | $4$ | $5$ | $55$ | |||
|
\(11\)
| 11.5.4.5 | $x^{5} + 22$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.5 | $x^{5} + 22$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.10.5.2 | $x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |