Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 15*x^14 - 101*x^13 + 163*x^12 + 173*x^11 + 164*x^10 - 7010*x^9 + 61895*x^8 - 434233*x^7 + 2003583*x^6 - 5981569*x^5 + 11891245*x^4 - 15825337*x^3 + 13679510*x^2 - 6972784*x + 1600672)
gp: K = bnfinit(y^16 - 3*y^15 + 15*y^14 - 101*y^13 + 163*y^12 + 173*y^11 + 164*y^10 - 7010*y^9 + 61895*y^8 - 434233*y^7 + 2003583*y^6 - 5981569*y^5 + 11891245*y^4 - 15825337*y^3 + 13679510*y^2 - 6972784*y + 1600672, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 15*x^14 - 101*x^13 + 163*x^12 + 173*x^11 + 164*x^10 - 7010*x^9 + 61895*x^8 - 434233*x^7 + 2003583*x^6 - 5981569*x^5 + 11891245*x^4 - 15825337*x^3 + 13679510*x^2 - 6972784*x + 1600672);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 15*x^14 - 101*x^13 + 163*x^12 + 173*x^11 + 164*x^10 - 7010*x^9 + 61895*x^8 - 434233*x^7 + 2003583*x^6 - 5981569*x^5 + 11891245*x^4 - 15825337*x^3 + 13679510*x^2 - 6972784*x + 1600672)
\( x^{16} - 3 x^{15} + 15 x^{14} - 101 x^{13} + 163 x^{12} + 173 x^{11} + 164 x^{10} - 7010 x^{9} + \cdots + 1600672 \)
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: |
\(15905028274673815182544659396289\)
\(\medspace = 19^{4}\cdot 73^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(89.14\) | 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: | $19^{1/2}73^{7/8}\approx 186.11625411428943$ | ||
| Ramified primes: |
\(19\), \(73\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{16}a^{10}+\frac{1}{16}a^{8}-\frac{1}{8}a^{5}+\frac{1}{16}a^{4}-\frac{1}{2}a^{3}-\frac{3}{16}a^{2}+\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{32}a^{11}+\frac{1}{32}a^{9}-\frac{1}{8}a^{7}-\frac{1}{16}a^{6}+\frac{5}{32}a^{5}-\frac{1}{8}a^{4}-\frac{3}{32}a^{3}+\frac{3}{16}a^{2}$, $\frac{1}{64}a^{12}-\frac{1}{64}a^{11}+\frac{1}{64}a^{10}-\frac{1}{64}a^{9}-\frac{1}{16}a^{8}+\frac{1}{32}a^{7}+\frac{7}{64}a^{6}-\frac{9}{64}a^{5}+\frac{1}{64}a^{4}-\frac{23}{64}a^{3}+\frac{13}{32}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{512}a^{13}+\frac{1}{64}a^{10}-\frac{5}{512}a^{9}+\frac{19}{256}a^{8}-\frac{55}{512}a^{7}+\frac{15}{256}a^{6}+\frac{9}{64}a^{5}-\frac{7}{256}a^{4}+\frac{163}{512}a^{3}-\frac{15}{256}a^{2}+\frac{5}{32}a+\frac{7}{16}$, $\frac{1}{1024}a^{14}-\frac{1}{1024}a^{13}+\frac{1}{128}a^{11}-\frac{13}{1024}a^{10}+\frac{43}{1024}a^{9}-\frac{93}{1024}a^{8}-\frac{43}{1024}a^{7}+\frac{21}{512}a^{6}-\frac{107}{512}a^{5}-\frac{207}{1024}a^{4}+\frac{319}{1024}a^{3}+\frac{247}{512}a^{2}+\frac{25}{64}a+\frac{9}{32}$, $\frac{1}{80\!\cdots\!64}a^{15}-\frac{18\!\cdots\!83}{50\!\cdots\!04}a^{14}+\frac{27\!\cdots\!19}{80\!\cdots\!64}a^{13}+\frac{29\!\cdots\!61}{10\!\cdots\!08}a^{12}+\frac{11\!\cdots\!35}{80\!\cdots\!64}a^{11}+\frac{59\!\cdots\!75}{40\!\cdots\!32}a^{10}+\frac{22\!\cdots\!83}{40\!\cdots\!32}a^{9}+\frac{84\!\cdots\!65}{10\!\cdots\!08}a^{8}+\frac{74\!\cdots\!03}{80\!\cdots\!64}a^{7}+\frac{16\!\cdots\!81}{20\!\cdots\!16}a^{6}-\frac{15\!\cdots\!13}{80\!\cdots\!64}a^{5}+\frac{31\!\cdots\!51}{25\!\cdots\!52}a^{4}-\frac{23\!\cdots\!11}{80\!\cdots\!64}a^{3}-\frac{19\!\cdots\!13}{40\!\cdots\!32}a^{2}-\frac{19\!\cdots\!13}{50\!\cdots\!04}a+\frac{29\!\cdots\!45}{25\!\cdots\!52}$
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
$C_{89}$, which has order $89$ (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: |
\( -1 \)
(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: |
$\frac{17\!\cdots\!25}{10\!\cdots\!08}a^{15}+\frac{43\!\cdots\!19}{10\!\cdots\!08}a^{14}+\frac{10\!\cdots\!97}{50\!\cdots\!04}a^{13}-\frac{13\!\cdots\!35}{12\!\cdots\!76}a^{12}-\frac{16\!\cdots\!73}{10\!\cdots\!08}a^{11}+\frac{27\!\cdots\!07}{10\!\cdots\!08}a^{10}+\frac{17\!\cdots\!69}{10\!\cdots\!08}a^{9}-\frac{80\!\cdots\!83}{10\!\cdots\!08}a^{8}+\frac{18\!\cdots\!27}{25\!\cdots\!52}a^{7}-\frac{24\!\cdots\!25}{50\!\cdots\!04}a^{6}+\frac{16\!\cdots\!93}{10\!\cdots\!08}a^{5}-\frac{31\!\cdots\!81}{10\!\cdots\!08}a^{4}+\frac{83\!\cdots\!57}{25\!\cdots\!52}a^{3}-\frac{41\!\cdots\!67}{25\!\cdots\!52}a^{2}-\frac{91\!\cdots\!01}{15\!\cdots\!72}a+\frac{44\!\cdots\!31}{15\!\cdots\!72}$, $\frac{76\!\cdots\!43}{40\!\cdots\!32}a^{15}+\frac{59\!\cdots\!23}{10\!\cdots\!08}a^{14}+\frac{88\!\cdots\!85}{40\!\cdots\!32}a^{13}-\frac{59\!\cdots\!77}{50\!\cdots\!04}a^{12}-\frac{72\!\cdots\!39}{40\!\cdots\!32}a^{11}+\frac{58\!\cdots\!35}{20\!\cdots\!16}a^{10}+\frac{37\!\cdots\!43}{20\!\cdots\!16}a^{9}-\frac{86\!\cdots\!37}{10\!\cdots\!08}a^{8}+\frac{32\!\cdots\!97}{40\!\cdots\!32}a^{7}-\frac{51\!\cdots\!39}{10\!\cdots\!08}a^{6}+\frac{69\!\cdots\!53}{40\!\cdots\!32}a^{5}-\frac{32\!\cdots\!17}{10\!\cdots\!08}a^{4}+\frac{13\!\cdots\!43}{40\!\cdots\!32}a^{3}-\frac{29\!\cdots\!75}{20\!\cdots\!16}a^{2}-\frac{61\!\cdots\!11}{25\!\cdots\!52}a+\frac{42\!\cdots\!15}{12\!\cdots\!76}$, $\frac{36\!\cdots\!73}{40\!\cdots\!32}a^{15}+\frac{16\!\cdots\!57}{10\!\cdots\!08}a^{14}+\frac{32\!\cdots\!23}{40\!\cdots\!32}a^{13}-\frac{17\!\cdots\!19}{50\!\cdots\!04}a^{12}-\frac{80\!\cdots\!89}{40\!\cdots\!32}a^{11}+\frac{29\!\cdots\!17}{20\!\cdots\!16}a^{10}+\frac{21\!\cdots\!01}{20\!\cdots\!16}a^{9}-\frac{37\!\cdots\!95}{10\!\cdots\!08}a^{8}+\frac{14\!\cdots\!95}{40\!\cdots\!32}a^{7}-\frac{16\!\cdots\!29}{10\!\cdots\!08}a^{6}+\frac{79\!\cdots\!23}{40\!\cdots\!32}a^{5}+\frac{75\!\cdots\!33}{10\!\cdots\!08}a^{4}-\frac{12\!\cdots\!27}{40\!\cdots\!32}a^{3}+\frac{10\!\cdots\!31}{20\!\cdots\!16}a^{2}-\frac{98\!\cdots\!49}{25\!\cdots\!52}a+\frac{15\!\cdots\!29}{12\!\cdots\!76}$, $\frac{35\!\cdots\!45}{20\!\cdots\!16}a^{15}-\frac{77\!\cdots\!43}{10\!\cdots\!08}a^{14}+\frac{43\!\cdots\!29}{20\!\cdots\!16}a^{13}-\frac{31\!\cdots\!39}{25\!\cdots\!52}a^{12}-\frac{12\!\cdots\!09}{20\!\cdots\!16}a^{11}+\frac{19\!\cdots\!29}{50\!\cdots\!04}a^{10}+\frac{19\!\cdots\!67}{12\!\cdots\!76}a^{9}-\frac{94\!\cdots\!57}{10\!\cdots\!08}a^{8}+\frac{16\!\cdots\!77}{20\!\cdots\!16}a^{7}-\frac{34\!\cdots\!43}{63\!\cdots\!88}a^{6}+\frac{40\!\cdots\!91}{20\!\cdots\!16}a^{5}-\frac{45\!\cdots\!63}{10\!\cdots\!08}a^{4}+\frac{12\!\cdots\!59}{20\!\cdots\!16}a^{3}-\frac{55\!\cdots\!79}{10\!\cdots\!08}a^{2}+\frac{35\!\cdots\!37}{12\!\cdots\!76}a-\frac{40\!\cdots\!21}{63\!\cdots\!88}$, $\frac{15\!\cdots\!77}{40\!\cdots\!32}a^{15}-\frac{16\!\cdots\!65}{25\!\cdots\!52}a^{14}+\frac{19\!\cdots\!99}{40\!\cdots\!32}a^{13}-\frac{16\!\cdots\!83}{50\!\cdots\!04}a^{12}+\frac{70\!\cdots\!47}{40\!\cdots\!32}a^{11}+\frac{20\!\cdots\!95}{20\!\cdots\!16}a^{10}+\frac{43\!\cdots\!03}{20\!\cdots\!16}a^{9}-\frac{12\!\cdots\!23}{50\!\cdots\!04}a^{8}+\frac{83\!\cdots\!91}{40\!\cdots\!32}a^{7}-\frac{14\!\cdots\!43}{10\!\cdots\!08}a^{6}+\frac{23\!\cdots\!63}{40\!\cdots\!32}a^{5}-\frac{19\!\cdots\!31}{12\!\cdots\!76}a^{4}+\frac{99\!\cdots\!41}{40\!\cdots\!32}a^{3}-\frac{50\!\cdots\!05}{20\!\cdots\!16}a^{2}+\frac{35\!\cdots\!47}{25\!\cdots\!52}a-\frac{39\!\cdots\!23}{12\!\cdots\!76}$, $\frac{31\!\cdots\!07}{10\!\cdots\!08}a^{15}-\frac{50\!\cdots\!03}{10\!\cdots\!08}a^{14}+\frac{27\!\cdots\!83}{50\!\cdots\!04}a^{13}-\frac{30\!\cdots\!21}{12\!\cdots\!76}a^{12}+\frac{34\!\cdots\!73}{10\!\cdots\!08}a^{11}+\frac{87\!\cdots\!57}{10\!\cdots\!08}a^{10}-\frac{15\!\cdots\!69}{10\!\cdots\!08}a^{9}-\frac{18\!\cdots\!25}{10\!\cdots\!08}a^{8}+\frac{45\!\cdots\!69}{25\!\cdots\!52}a^{7}-\frac{59\!\cdots\!91}{50\!\cdots\!04}a^{6}+\frac{53\!\cdots\!47}{10\!\cdots\!08}a^{5}-\frac{16\!\cdots\!83}{10\!\cdots\!08}a^{4}+\frac{80\!\cdots\!83}{25\!\cdots\!52}a^{3}-\frac{10\!\cdots\!17}{25\!\cdots\!52}a^{2}+\frac{49\!\cdots\!73}{15\!\cdots\!72}a-\frac{17\!\cdots\!95}{15\!\cdots\!72}$, $\frac{52\!\cdots\!15}{40\!\cdots\!32}a^{15}-\frac{42\!\cdots\!25}{10\!\cdots\!08}a^{14}+\frac{74\!\cdots\!77}{40\!\cdots\!32}a^{13}-\frac{41\!\cdots\!85}{50\!\cdots\!04}a^{12}-\frac{35\!\cdots\!83}{40\!\cdots\!32}a^{11}+\frac{40\!\cdots\!11}{20\!\cdots\!16}a^{10}+\frac{15\!\cdots\!91}{20\!\cdots\!16}a^{9}-\frac{71\!\cdots\!09}{10\!\cdots\!08}a^{8}+\frac{24\!\cdots\!25}{40\!\cdots\!32}a^{7}-\frac{40\!\cdots\!67}{10\!\cdots\!08}a^{6}+\frac{61\!\cdots\!09}{40\!\cdots\!32}a^{5}-\frac{36\!\cdots\!73}{10\!\cdots\!08}a^{4}+\frac{22\!\cdots\!23}{40\!\cdots\!32}a^{3}-\frac{11\!\cdots\!51}{20\!\cdots\!16}a^{2}+\frac{77\!\cdots\!21}{25\!\cdots\!52}a-\frac{97\!\cdots\!89}{12\!\cdots\!76}$
| 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: | \( 4629594070.98 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4629594070.98 \cdot 89}{2\cdot\sqrt{15905028274673815182544659396289}}\cr\approx \mathstrut & 125.480039940 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 15*x^14 - 101*x^13 + 163*x^12 + 173*x^11 + 164*x^10 - 7010*x^9 + 61895*x^8 - 434233*x^7 + 2003583*x^6 - 5981569*x^5 + 11891245*x^4 - 15825337*x^3 + 13679510*x^2 - 6972784*x + 1600672)
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 - 3*x^15 + 15*x^14 - 101*x^13 + 163*x^12 + 173*x^11 + 164*x^10 - 7010*x^9 + 61895*x^8 - 434233*x^7 + 2003583*x^6 - 5981569*x^5 + 11891245*x^4 - 15825337*x^3 + 13679510*x^2 - 6972784*x + 1600672, 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 - 3*x^15 + 15*x^14 - 101*x^13 + 163*x^12 + 173*x^11 + 164*x^10 - 7010*x^9 + 61895*x^8 - 434233*x^7 + 2003583*x^6 - 5981569*x^5 + 11891245*x^4 - 15825337*x^3 + 13679510*x^2 - 6972784*x + 1600672);
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 - 3*x^15 + 15*x^14 - 101*x^13 + 163*x^12 + 173*x^11 + 164*x^10 - 7010*x^9 + 61895*x^8 - 434233*x^7 + 2003583*x^6 - 5981569*x^5 + 11891245*x^4 - 15825337*x^3 + 13679510*x^2 - 6972784*x + 1600672);
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
$C_2^2:C_8$ (as 16T24):
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 20 conjugacy class representatives for $C_2^2 : C_8$ |
| Character table for $C_2^2 : C_8$ |
Intermediate fields
| \(\Q(\sqrt{73}) \), 4.4.389017.1, 4.2.7391323.1, 4.2.101251.1, 8.4.3988110865394017.1, 8.0.11047398519097.1, 8.4.54631655690329.1 |
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 16 sibling: | 16.8.2072759189783766268404402557183778769.1 |
| 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 | ${\href{/padicField/2.2.0.1}{2} }^{4}{,}\,{\href{/padicField/2.1.0.1}{1} }^{8}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.8.4.1 | $x^{8} + 80 x^{6} + 22 x^{5} + 2250 x^{4} - 792 x^{3} + 25817 x^{2} - 22946 x + 107924$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(73\)
| 73.16.14.1 | $x^{16} + 560 x^{15} + 137240 x^{14} + 19227600 x^{13} + 1684816700 x^{12} + 94599694000 x^{11} + 3327837457000 x^{10} + 67300032450000 x^{9} + 609674268043896 x^{8} + 336500162290880 x^{7} + 83195946420160 x^{6} + 11826359641600 x^{5} + 1175201013000 x^{4} + 6895769020000 x^{3} + 239006660174000 x^{2} + 4775207729180000 x + 41740387870087204$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |