Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 21*x^14 - 56*x^13 + 81*x^12 - 74*x^11 + 75*x^10 + 222*x^9 - 131*x^8 + 2056*x^7 + 336*x^6 + 3352*x^5 + 3101*x^4 + 1602*x^3 + 5492*x^2 - 3816*x + 16336)
gp: K = bnfinit(y^16 - 6*y^15 + 21*y^14 - 56*y^13 + 81*y^12 - 74*y^11 + 75*y^10 + 222*y^9 - 131*y^8 + 2056*y^7 + 336*y^6 + 3352*y^5 + 3101*y^4 + 1602*y^3 + 5492*y^2 - 3816*y + 16336, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 21*x^14 - 56*x^13 + 81*x^12 - 74*x^11 + 75*x^10 + 222*x^9 - 131*x^8 + 2056*x^7 + 336*x^6 + 3352*x^5 + 3101*x^4 + 1602*x^3 + 5492*x^2 - 3816*x + 16336);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 21*x^14 - 56*x^13 + 81*x^12 - 74*x^11 + 75*x^10 + 222*x^9 - 131*x^8 + 2056*x^7 + 336*x^6 + 3352*x^5 + 3101*x^4 + 1602*x^3 + 5492*x^2 - 3816*x + 16336)
\( x^{16} - 6 x^{15} + 21 x^{14} - 56 x^{13} + 81 x^{12} - 74 x^{11} + 75 x^{10} + 222 x^{9} - 131 x^{8} + \cdots + 16336 \)
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: |
\(48003408671952806969030689\)
\(\medspace = 17^{10}\cdot 47^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(40.28\) | 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: | $17^{3/4}47^{1/2}\approx 57.396527724841604$ | ||
| Ramified primes: |
\(17\), \(47\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{14}a^{11}-\frac{3}{14}a^{10}+\frac{1}{7}a^{8}+\frac{1}{14}a^{7}+\frac{3}{14}a^{6}-\frac{5}{14}a^{5}+\frac{3}{7}a^{4}+\frac{3}{7}a^{3}+\frac{1}{14}a^{2}+\frac{2}{7}a-\frac{2}{7}$, $\frac{1}{28}a^{12}-\frac{1}{14}a^{10}+\frac{1}{14}a^{9}-\frac{1}{4}a^{8}+\frac{3}{14}a^{7}+\frac{1}{7}a^{6}-\frac{1}{14}a^{5}-\frac{11}{28}a^{4}-\frac{1}{14}a^{3}+\frac{1}{4}a^{2}+\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{84}a^{13}+\frac{1}{42}a^{11}-\frac{5}{42}a^{10}+\frac{1}{12}a^{9}-\frac{1}{6}a^{8}-\frac{5}{21}a^{7}-\frac{3}{14}a^{6}+\frac{25}{84}a^{5}+\frac{2}{21}a^{4}+\frac{17}{84}a^{3}-\frac{5}{14}a^{2}+\frac{8}{21}a-\frac{4}{21}$, $\frac{1}{168}a^{14}-\frac{1}{168}a^{13}+\frac{1}{84}a^{12}-\frac{19}{168}a^{10}-\frac{1}{8}a^{9}-\frac{1}{7}a^{8}-\frac{1}{6}a^{7}+\frac{37}{168}a^{6}-\frac{5}{24}a^{5}-\frac{15}{56}a^{4}-\frac{59}{168}a^{3}+\frac{4}{21}a^{2}-\frac{4}{21}$, $\frac{1}{21\!\cdots\!04}a^{15}+\frac{29\!\cdots\!53}{10\!\cdots\!52}a^{14}+\frac{15\!\cdots\!19}{21\!\cdots\!04}a^{13}+\frac{30\!\cdots\!65}{58\!\cdots\!14}a^{12}-\frac{70\!\cdots\!29}{33\!\cdots\!08}a^{11}-\frac{11\!\cdots\!15}{10\!\cdots\!52}a^{10}+\frac{18\!\cdots\!53}{21\!\cdots\!04}a^{9}+\frac{39\!\cdots\!15}{10\!\cdots\!52}a^{8}-\frac{54\!\cdots\!21}{70\!\cdots\!68}a^{7}+\frac{98\!\cdots\!45}{53\!\cdots\!26}a^{6}+\frac{45\!\cdots\!61}{10\!\cdots\!52}a^{5}+\frac{86\!\cdots\!09}{29\!\cdots\!93}a^{4}-\frac{61\!\cdots\!93}{23\!\cdots\!56}a^{3}+\frac{46\!\cdots\!51}{35\!\cdots\!84}a^{2}-\frac{50\!\cdots\!64}{20\!\cdots\!51}a-\frac{99\!\cdots\!46}{26\!\cdots\!63}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
$C_{20}$, which has order $20$ (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: |
$\frac{83\!\cdots\!81}{35\!\cdots\!84}a^{15}-\frac{47\!\cdots\!03}{23\!\cdots\!56}a^{14}+\frac{70\!\cdots\!85}{70\!\cdots\!68}a^{13}-\frac{58\!\cdots\!11}{17\!\cdots\!42}a^{12}+\frac{24\!\cdots\!87}{35\!\cdots\!84}a^{11}-\frac{78\!\cdots\!43}{10\!\cdots\!24}a^{10}-\frac{35\!\cdots\!25}{10\!\cdots\!24}a^{9}+\frac{41\!\cdots\!21}{11\!\cdots\!28}a^{8}-\frac{15\!\cdots\!31}{35\!\cdots\!84}a^{7}+\frac{47\!\cdots\!59}{70\!\cdots\!68}a^{6}-\frac{11\!\cdots\!47}{70\!\cdots\!68}a^{5}-\frac{75\!\cdots\!79}{18\!\cdots\!12}a^{4}-\frac{40\!\cdots\!81}{70\!\cdots\!68}a^{3}-\frac{13\!\cdots\!49}{35\!\cdots\!84}a^{2}+\frac{27\!\cdots\!89}{45\!\cdots\!78}a-\frac{16\!\cdots\!01}{29\!\cdots\!07}$, $\frac{30\!\cdots\!01}{15\!\cdots\!36}a^{15}-\frac{31\!\cdots\!51}{26\!\cdots\!63}a^{14}+\frac{36\!\cdots\!71}{10\!\cdots\!52}a^{13}-\frac{24\!\cdots\!65}{35\!\cdots\!84}a^{12}+\frac{12\!\cdots\!49}{35\!\cdots\!84}a^{11}+\frac{85\!\cdots\!01}{53\!\cdots\!26}a^{10}-\frac{23\!\cdots\!71}{10\!\cdots\!52}a^{9}+\frac{89\!\cdots\!17}{10\!\cdots\!52}a^{8}-\frac{23\!\cdots\!87}{11\!\cdots\!28}a^{7}+\frac{14\!\cdots\!05}{26\!\cdots\!63}a^{6}-\frac{20\!\cdots\!85}{53\!\cdots\!26}a^{5}-\frac{30\!\cdots\!61}{81\!\cdots\!04}a^{4}-\frac{73\!\cdots\!29}{35\!\cdots\!84}a^{3}+\frac{36\!\cdots\!47}{11\!\cdots\!28}a^{2}+\frac{43\!\cdots\!66}{20\!\cdots\!51}a-\frac{12\!\cdots\!78}{37\!\cdots\!09}$, $\frac{15\!\cdots\!03}{21\!\cdots\!04}a^{15}-\frac{16\!\cdots\!13}{53\!\cdots\!26}a^{14}+\frac{11\!\cdots\!49}{21\!\cdots\!04}a^{13}-\frac{12\!\cdots\!90}{88\!\cdots\!21}a^{12}-\frac{32\!\cdots\!27}{70\!\cdots\!68}a^{11}+\frac{61\!\cdots\!55}{53\!\cdots\!26}a^{10}-\frac{13\!\cdots\!67}{30\!\cdots\!72}a^{9}+\frac{10\!\cdots\!09}{10\!\cdots\!52}a^{8}+\frac{17\!\cdots\!53}{70\!\cdots\!68}a^{7}+\frac{68\!\cdots\!33}{10\!\cdots\!52}a^{6}+\frac{34\!\cdots\!73}{10\!\cdots\!52}a^{5}-\frac{13\!\cdots\!87}{81\!\cdots\!04}a^{4}+\frac{11\!\cdots\!03}{70\!\cdots\!68}a^{3}+\frac{21\!\cdots\!71}{35\!\cdots\!84}a^{2}-\frac{47\!\cdots\!63}{20\!\cdots\!51}a+\frac{17\!\cdots\!91}{26\!\cdots\!63}$, $\frac{10\!\cdots\!43}{20\!\cdots\!64}a^{15}-\frac{26\!\cdots\!57}{69\!\cdots\!88}a^{14}+\frac{15\!\cdots\!69}{10\!\cdots\!32}a^{13}-\frac{11\!\cdots\!00}{26\!\cdots\!33}a^{12}+\frac{19\!\cdots\!47}{20\!\cdots\!64}a^{11}-\frac{32\!\cdots\!35}{20\!\cdots\!64}a^{10}+\frac{69\!\cdots\!25}{24\!\cdots\!46}a^{9}-\frac{15\!\cdots\!13}{52\!\cdots\!66}a^{8}+\frac{15\!\cdots\!53}{69\!\cdots\!88}a^{7}+\frac{15\!\cdots\!59}{20\!\cdots\!64}a^{6}+\frac{28\!\cdots\!97}{20\!\cdots\!64}a^{5}-\frac{61\!\cdots\!37}{16\!\cdots\!28}a^{4}-\frac{15\!\cdots\!00}{87\!\cdots\!11}a^{3}+\frac{97\!\cdots\!10}{26\!\cdots\!33}a^{2}+\frac{61\!\cdots\!63}{20\!\cdots\!41}a-\frac{14\!\cdots\!79}{37\!\cdots\!19}$, $\frac{18\!\cdots\!81}{60\!\cdots\!06}a^{15}-\frac{65\!\cdots\!59}{24\!\cdots\!24}a^{14}+\frac{22\!\cdots\!25}{24\!\cdots\!24}a^{13}-\frac{49\!\cdots\!49}{20\!\cdots\!02}a^{12}+\frac{22\!\cdots\!46}{48\!\cdots\!31}a^{11}-\frac{80\!\cdots\!99}{24\!\cdots\!24}a^{10}+\frac{21\!\cdots\!43}{34\!\cdots\!32}a^{9}+\frac{33\!\cdots\!73}{12\!\cdots\!12}a^{8}-\frac{63\!\cdots\!13}{10\!\cdots\!51}a^{7}+\frac{12\!\cdots\!89}{24\!\cdots\!24}a^{6}-\frac{46\!\cdots\!77}{24\!\cdots\!24}a^{5}-\frac{11\!\cdots\!19}{24\!\cdots\!24}a^{4}-\frac{54\!\cdots\!27}{81\!\cdots\!08}a^{3}-\frac{12\!\cdots\!05}{13\!\cdots\!68}a^{2}-\frac{16\!\cdots\!30}{43\!\cdots\!79}a-\frac{51\!\cdots\!05}{30\!\cdots\!53}$, $\frac{33\!\cdots\!45}{58\!\cdots\!14}a^{15}-\frac{11\!\cdots\!55}{23\!\cdots\!56}a^{14}+\frac{42\!\cdots\!61}{23\!\cdots\!56}a^{13}-\frac{13\!\cdots\!78}{29\!\cdots\!07}a^{12}+\frac{57\!\cdots\!53}{84\!\cdots\!02}a^{11}-\frac{21\!\cdots\!47}{23\!\cdots\!56}a^{10}-\frac{20\!\cdots\!91}{23\!\cdots\!56}a^{9}+\frac{17\!\cdots\!29}{11\!\cdots\!28}a^{8}-\frac{76\!\cdots\!78}{29\!\cdots\!07}a^{7}+\frac{99\!\cdots\!41}{23\!\cdots\!56}a^{6}-\frac{12\!\cdots\!49}{23\!\cdots\!56}a^{5}-\frac{53\!\cdots\!03}{18\!\cdots\!12}a^{4}+\frac{60\!\cdots\!75}{23\!\cdots\!56}a^{3}+\frac{80\!\cdots\!55}{11\!\cdots\!28}a^{2}-\frac{18\!\cdots\!35}{45\!\cdots\!78}a-\frac{46\!\cdots\!67}{29\!\cdots\!07}$, $\frac{34\!\cdots\!75}{23\!\cdots\!56}a^{15}-\frac{16\!\cdots\!91}{23\!\cdots\!56}a^{14}+\frac{52\!\cdots\!35}{29\!\cdots\!07}a^{13}-\frac{61\!\cdots\!03}{16\!\cdots\!04}a^{12}+\frac{19\!\cdots\!89}{33\!\cdots\!08}a^{11}+\frac{19\!\cdots\!07}{33\!\cdots\!08}a^{10}+\frac{13\!\cdots\!55}{11\!\cdots\!28}a^{9}+\frac{41\!\cdots\!15}{11\!\cdots\!28}a^{8}+\frac{81\!\cdots\!15}{23\!\cdots\!56}a^{7}+\frac{81\!\cdots\!45}{33\!\cdots\!08}a^{6}+\frac{10\!\cdots\!07}{23\!\cdots\!56}a^{5}+\frac{77\!\cdots\!45}{18\!\cdots\!12}a^{4}+\frac{77\!\cdots\!21}{16\!\cdots\!04}a^{3}+\frac{37\!\cdots\!61}{11\!\cdots\!28}a^{2}+\frac{30\!\cdots\!89}{22\!\cdots\!39}a-\frac{16\!\cdots\!36}{29\!\cdots\!07}$
| 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: | \( 145848.015244 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 145848.015244 \cdot 20}{2\cdot\sqrt{48003408671952806969030689}}\cr\approx \mathstrut & 0.511332540703 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 21*x^14 - 56*x^13 + 81*x^12 - 74*x^11 + 75*x^10 + 222*x^9 - 131*x^8 + 2056*x^7 + 336*x^6 + 3352*x^5 + 3101*x^4 + 1602*x^3 + 5492*x^2 - 3816*x + 16336)
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 - 6*x^15 + 21*x^14 - 56*x^13 + 81*x^12 - 74*x^11 + 75*x^10 + 222*x^9 - 131*x^8 + 2056*x^7 + 336*x^6 + 3352*x^5 + 3101*x^4 + 1602*x^3 + 5492*x^2 - 3816*x + 16336, 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 - 6*x^15 + 21*x^14 - 56*x^13 + 81*x^12 - 74*x^11 + 75*x^10 + 222*x^9 - 131*x^8 + 2056*x^7 + 336*x^6 + 3352*x^5 + 3101*x^4 + 1602*x^3 + 5492*x^2 - 3816*x + 16336);
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 - 6*x^15 + 21*x^14 - 56*x^13 + 81*x^12 - 74*x^11 + 75*x^10 + 222*x^9 - 131*x^8 + 2056*x^7 + 336*x^6 + 3352*x^5 + 3101*x^4 + 1602*x^3 + 5492*x^2 - 3816*x + 16336);
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
| \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-799}) \), 4.0.37553.1 x2, 4.2.13583.1 x2, \(\Q(\sqrt{17}, \sqrt{-47})\), 8.0.6928449225617.2, 8.0.23973872753.2, 8.0.407555836801.2 |
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: | 8.0.23973872753.2, 8.0.6928449225617.2 |
| Degree 16 sibling: | 16.4.6280210550563314266206369.6 |
| Minimal sibling: | 8.0.23973872753.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.4.3.1 | $x^{4} + 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 17.4.3.1 | $x^{4} + 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(47\)
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |