Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 33*x^13 - 40*x^12 - 90*x^11 + 54*x^10 + 1791*x^9 - 954*x^8 + 6819*x^7 + 25096*x^6 - 2367*x^5 - 52062*x^4 - 63600*x^3 - 29592*x^2 + 6576*x + 4384)
gp: K = bnfinit(y^15 - 33*y^13 - 40*y^12 - 90*y^11 + 54*y^10 + 1791*y^9 - 954*y^8 + 6819*y^7 + 25096*y^6 - 2367*y^5 - 52062*y^4 - 63600*y^3 - 29592*y^2 + 6576*y + 4384, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 33*x^13 - 40*x^12 - 90*x^11 + 54*x^10 + 1791*x^9 - 954*x^8 + 6819*x^7 + 25096*x^6 - 2367*x^5 - 52062*x^4 - 63600*x^3 - 29592*x^2 + 6576*x + 4384);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 33*x^13 - 40*x^12 - 90*x^11 + 54*x^10 + 1791*x^9 - 954*x^8 + 6819*x^7 + 25096*x^6 - 2367*x^5 - 52062*x^4 - 63600*x^3 - 29592*x^2 + 6576*x + 4384)
\( x^{15} - 33 x^{13} - 40 x^{12} - 90 x^{11} + 54 x^{10} + 1791 x^{9} - 954 x^{8} + 6819 x^{7} + \cdots + 4384 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[9, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-773560756544273759852688384\)
\(\medspace = -\,2^{10}\cdot 3^{21}\cdot 137^{2}\cdot 1567^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(62.02\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(137\), \(1567\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-4701}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{6}a^{11}-\frac{1}{6}a^{9}+\frac{1}{6}a^{5}-\frac{1}{6}a^{3}+\frac{1}{3}a^{2}-\frac{1}{2}a-\frac{1}{3}$, $\frac{1}{12}a^{12}-\frac{1}{12}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{5}{12}a^{6}-\frac{1}{2}a^{5}-\frac{1}{12}a^{4}-\frac{1}{3}a^{3}-\frac{1}{4}a^{2}-\frac{1}{6}a$, $\frac{1}{72}a^{13}-\frac{1}{36}a^{12}-\frac{1}{72}a^{11}+\frac{1}{36}a^{10}-\frac{1}{36}a^{9}+\frac{1}{4}a^{8}+\frac{19}{72}a^{7}+\frac{2}{9}a^{6}-\frac{25}{72}a^{5}-\frac{7}{36}a^{4}+\frac{11}{24}a^{3}+\frac{2}{9}a^{2}+\frac{7}{18}a-\frac{2}{9}$, $\frac{1}{26\!\cdots\!56}a^{14}-\frac{39\!\cdots\!27}{65\!\cdots\!64}a^{13}+\frac{19\!\cdots\!05}{87\!\cdots\!52}a^{12}-\frac{25\!\cdots\!17}{65\!\cdots\!64}a^{11}+\frac{52\!\cdots\!37}{43\!\cdots\!76}a^{10}-\frac{11\!\cdots\!97}{13\!\cdots\!28}a^{9}+\frac{28\!\cdots\!95}{26\!\cdots\!56}a^{8}-\frac{48\!\cdots\!67}{13\!\cdots\!28}a^{7}-\frac{37\!\cdots\!87}{87\!\cdots\!52}a^{6}-\frac{89\!\cdots\!39}{21\!\cdots\!88}a^{5}+\frac{52\!\cdots\!85}{26\!\cdots\!56}a^{4}+\frac{11\!\cdots\!23}{13\!\cdots\!28}a^{3}-\frac{13\!\cdots\!19}{32\!\cdots\!82}a^{2}-\frac{51\!\cdots\!90}{18\!\cdots\!49}a-\frac{34\!\cdots\!13}{16\!\cdots\!41}$
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: | $11$ | 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: |
$\frac{88\!\cdots\!33}{21\!\cdots\!88}a^{14}-\frac{18\!\cdots\!35}{36\!\cdots\!98}a^{13}-\frac{27\!\cdots\!01}{21\!\cdots\!88}a^{12}-\frac{20\!\cdots\!39}{54\!\cdots\!47}a^{11}-\frac{39\!\cdots\!47}{10\!\cdots\!94}a^{10}+\frac{12\!\cdots\!30}{18\!\cdots\!49}a^{9}+\frac{14\!\cdots\!59}{21\!\cdots\!88}a^{8}-\frac{21\!\cdots\!83}{18\!\cdots\!49}a^{7}+\frac{92\!\cdots\!31}{21\!\cdots\!88}a^{6}+\frac{26\!\cdots\!52}{54\!\cdots\!47}a^{5}-\frac{15\!\cdots\!31}{21\!\cdots\!88}a^{4}-\frac{13\!\cdots\!85}{10\!\cdots\!94}a^{3}-\frac{57\!\cdots\!46}{54\!\cdots\!47}a^{2}+\frac{11\!\cdots\!29}{10\!\cdots\!94}a+\frac{26\!\cdots\!81}{18\!\cdots\!49}$, $\frac{88\!\cdots\!33}{21\!\cdots\!88}a^{14}-\frac{18\!\cdots\!35}{36\!\cdots\!98}a^{13}-\frac{27\!\cdots\!01}{21\!\cdots\!88}a^{12}-\frac{20\!\cdots\!39}{54\!\cdots\!47}a^{11}-\frac{39\!\cdots\!47}{10\!\cdots\!94}a^{10}+\frac{12\!\cdots\!30}{18\!\cdots\!49}a^{9}+\frac{14\!\cdots\!59}{21\!\cdots\!88}a^{8}-\frac{21\!\cdots\!83}{18\!\cdots\!49}a^{7}+\frac{92\!\cdots\!31}{21\!\cdots\!88}a^{6}+\frac{26\!\cdots\!52}{54\!\cdots\!47}a^{5}-\frac{15\!\cdots\!31}{21\!\cdots\!88}a^{4}-\frac{13\!\cdots\!85}{10\!\cdots\!94}a^{3}-\frac{57\!\cdots\!46}{54\!\cdots\!47}a^{2}+\frac{11\!\cdots\!29}{10\!\cdots\!94}a+\frac{26\!\cdots\!32}{18\!\cdots\!49}$, $\frac{72\!\cdots\!65}{87\!\cdots\!52}a^{14}-\frac{11\!\cdots\!01}{10\!\cdots\!94}a^{13}-\frac{75\!\cdots\!03}{29\!\cdots\!84}a^{12}-\frac{92\!\cdots\!15}{10\!\cdots\!94}a^{11}-\frac{32\!\cdots\!17}{43\!\cdots\!76}a^{10}+\frac{59\!\cdots\!51}{43\!\cdots\!76}a^{9}+\frac{11\!\cdots\!51}{87\!\cdots\!52}a^{8}-\frac{10\!\cdots\!49}{43\!\cdots\!76}a^{7}+\frac{25\!\cdots\!69}{29\!\cdots\!84}a^{6}+\frac{36\!\cdots\!59}{36\!\cdots\!98}a^{5}-\frac{12\!\cdots\!75}{87\!\cdots\!52}a^{4}-\frac{11\!\cdots\!83}{43\!\cdots\!76}a^{3}-\frac{11\!\cdots\!56}{54\!\cdots\!47}a^{2}+\frac{10\!\cdots\!81}{54\!\cdots\!47}a+\frac{16\!\cdots\!78}{54\!\cdots\!47}$, $\frac{26\!\cdots\!87}{36\!\cdots\!98}a^{14}-\frac{20\!\cdots\!65}{21\!\cdots\!88}a^{13}-\frac{16\!\cdots\!19}{72\!\cdots\!96}a^{12}+\frac{15\!\cdots\!75}{21\!\cdots\!88}a^{11}-\frac{13\!\cdots\!85}{21\!\cdots\!88}a^{10}+\frac{21\!\cdots\!00}{18\!\cdots\!49}a^{9}+\frac{41\!\cdots\!85}{36\!\cdots\!98}a^{8}-\frac{47\!\cdots\!29}{21\!\cdots\!88}a^{7}+\frac{54\!\cdots\!41}{72\!\cdots\!96}a^{6}+\frac{19\!\cdots\!97}{21\!\cdots\!88}a^{5}-\frac{31\!\cdots\!55}{21\!\cdots\!88}a^{4}-\frac{15\!\cdots\!09}{72\!\cdots\!96}a^{3}-\frac{31\!\cdots\!49}{21\!\cdots\!88}a^{2}+\frac{10\!\cdots\!99}{54\!\cdots\!47}a+\frac{37\!\cdots\!26}{18\!\cdots\!49}$, $\frac{56\!\cdots\!65}{21\!\cdots\!88}a^{14}-\frac{45\!\cdots\!27}{13\!\cdots\!28}a^{13}-\frac{53\!\cdots\!71}{65\!\cdots\!64}a^{12}+\frac{64\!\cdots\!15}{13\!\cdots\!28}a^{11}-\frac{74\!\cdots\!37}{32\!\cdots\!82}a^{10}+\frac{29\!\cdots\!11}{65\!\cdots\!64}a^{9}+\frac{22\!\cdots\!85}{54\!\cdots\!47}a^{8}-\frac{10\!\cdots\!05}{13\!\cdots\!28}a^{7}+\frac{45\!\cdots\!24}{16\!\cdots\!41}a^{6}+\frac{38\!\cdots\!91}{13\!\cdots\!28}a^{5}-\frac{31\!\cdots\!41}{65\!\cdots\!64}a^{4}-\frac{33\!\cdots\!73}{43\!\cdots\!76}a^{3}-\frac{38\!\cdots\!15}{65\!\cdots\!64}a^{2}+\frac{46\!\cdots\!61}{32\!\cdots\!82}a+\frac{17\!\cdots\!61}{16\!\cdots\!41}$, $\frac{12\!\cdots\!17}{26\!\cdots\!56}a^{14}-\frac{78\!\cdots\!67}{13\!\cdots\!28}a^{13}-\frac{39\!\cdots\!13}{26\!\cdots\!56}a^{12}-\frac{57\!\cdots\!77}{13\!\cdots\!28}a^{11}-\frac{55\!\cdots\!85}{13\!\cdots\!28}a^{10}+\frac{34\!\cdots\!75}{43\!\cdots\!76}a^{9}+\frac{19\!\cdots\!59}{26\!\cdots\!56}a^{8}-\frac{92\!\cdots\!95}{65\!\cdots\!64}a^{7}+\frac{13\!\cdots\!35}{26\!\cdots\!56}a^{6}+\frac{76\!\cdots\!59}{13\!\cdots\!28}a^{5}-\frac{73\!\cdots\!25}{87\!\cdots\!52}a^{4}-\frac{95\!\cdots\!83}{65\!\cdots\!64}a^{3}-\frac{81\!\cdots\!29}{65\!\cdots\!64}a^{2}+\frac{38\!\cdots\!79}{32\!\cdots\!82}a+\frac{92\!\cdots\!45}{54\!\cdots\!47}$, $\frac{31\!\cdots\!55}{26\!\cdots\!56}a^{14}-\frac{80\!\cdots\!64}{54\!\cdots\!47}a^{13}-\frac{97\!\cdots\!91}{26\!\cdots\!56}a^{12}-\frac{39\!\cdots\!89}{36\!\cdots\!98}a^{11}-\frac{13\!\cdots\!35}{13\!\cdots\!28}a^{10}+\frac{25\!\cdots\!57}{13\!\cdots\!28}a^{9}+\frac{49\!\cdots\!61}{26\!\cdots\!56}a^{8}-\frac{15\!\cdots\!49}{43\!\cdots\!76}a^{7}+\frac{32\!\cdots\!89}{26\!\cdots\!56}a^{6}+\frac{46\!\cdots\!23}{32\!\cdots\!82}a^{5}-\frac{54\!\cdots\!13}{26\!\cdots\!56}a^{4}-\frac{47\!\cdots\!17}{13\!\cdots\!28}a^{3}-\frac{16\!\cdots\!77}{54\!\cdots\!47}a^{2}+\frac{95\!\cdots\!91}{32\!\cdots\!82}a+\frac{68\!\cdots\!07}{16\!\cdots\!41}$, $\frac{12\!\cdots\!25}{13\!\cdots\!28}a^{14}-\frac{50\!\cdots\!29}{43\!\cdots\!76}a^{13}-\frac{37\!\cdots\!95}{13\!\cdots\!28}a^{12}-\frac{99\!\cdots\!17}{14\!\cdots\!92}a^{11}-\frac{13\!\cdots\!86}{16\!\cdots\!41}a^{10}+\frac{49\!\cdots\!41}{32\!\cdots\!82}a^{9}+\frac{19\!\cdots\!37}{13\!\cdots\!28}a^{8}-\frac{11\!\cdots\!05}{43\!\cdots\!76}a^{7}+\frac{12\!\cdots\!59}{13\!\cdots\!28}a^{6}+\frac{14\!\cdots\!31}{13\!\cdots\!28}a^{5}-\frac{20\!\cdots\!25}{13\!\cdots\!28}a^{4}-\frac{36\!\cdots\!37}{13\!\cdots\!28}a^{3}-\frac{12\!\cdots\!80}{54\!\cdots\!47}a^{2}+\frac{76\!\cdots\!65}{32\!\cdots\!82}a+\frac{52\!\cdots\!31}{16\!\cdots\!41}$, $\frac{45\!\cdots\!29}{26\!\cdots\!56}a^{14}-\frac{94\!\cdots\!69}{43\!\cdots\!76}a^{13}-\frac{14\!\cdots\!05}{26\!\cdots\!56}a^{12}-\frac{24\!\cdots\!57}{14\!\cdots\!92}a^{11}-\frac{20\!\cdots\!09}{13\!\cdots\!28}a^{10}+\frac{37\!\cdots\!85}{13\!\cdots\!28}a^{9}+\frac{72\!\cdots\!43}{26\!\cdots\!56}a^{8}-\frac{11\!\cdots\!79}{21\!\cdots\!88}a^{7}+\frac{47\!\cdots\!39}{26\!\cdots\!56}a^{6}+\frac{27\!\cdots\!43}{13\!\cdots\!28}a^{5}-\frac{79\!\cdots\!19}{26\!\cdots\!56}a^{4}-\frac{34\!\cdots\!59}{65\!\cdots\!64}a^{3}-\frac{97\!\cdots\!41}{21\!\cdots\!88}a^{2}+\frac{75\!\cdots\!31}{16\!\cdots\!41}a+\frac{10\!\cdots\!85}{16\!\cdots\!41}$, $\frac{34\!\cdots\!63}{26\!\cdots\!56}a^{14}-\frac{10\!\cdots\!99}{65\!\cdots\!64}a^{13}-\frac{10\!\cdots\!55}{26\!\cdots\!56}a^{12}-\frac{81\!\cdots\!75}{65\!\cdots\!64}a^{11}-\frac{15\!\cdots\!45}{13\!\cdots\!28}a^{10}+\frac{31\!\cdots\!89}{14\!\cdots\!92}a^{9}+\frac{54\!\cdots\!85}{26\!\cdots\!56}a^{8}-\frac{50\!\cdots\!21}{13\!\cdots\!28}a^{7}+\frac{35\!\cdots\!69}{26\!\cdots\!56}a^{6}+\frac{10\!\cdots\!27}{65\!\cdots\!64}a^{5}-\frac{19\!\cdots\!31}{87\!\cdots\!52}a^{4}-\frac{52\!\cdots\!67}{13\!\cdots\!28}a^{3}-\frac{22\!\cdots\!11}{65\!\cdots\!64}a^{2}+\frac{52\!\cdots\!09}{16\!\cdots\!41}a+\frac{83\!\cdots\!92}{18\!\cdots\!49}$, $\frac{11\!\cdots\!39}{26\!\cdots\!56}a^{14}-\frac{38\!\cdots\!45}{72\!\cdots\!96}a^{13}-\frac{34\!\cdots\!91}{26\!\cdots\!56}a^{12}-\frac{12\!\cdots\!29}{72\!\cdots\!96}a^{11}-\frac{49\!\cdots\!93}{13\!\cdots\!28}a^{10}+\frac{92\!\cdots\!13}{13\!\cdots\!28}a^{9}+\frac{17\!\cdots\!73}{26\!\cdots\!56}a^{8}-\frac{18\!\cdots\!05}{14\!\cdots\!92}a^{7}+\frac{11\!\cdots\!73}{26\!\cdots\!56}a^{6}+\frac{33\!\cdots\!01}{65\!\cdots\!64}a^{5}-\frac{19\!\cdots\!97}{26\!\cdots\!56}a^{4}-\frac{16\!\cdots\!47}{13\!\cdots\!28}a^{3}-\frac{23\!\cdots\!99}{21\!\cdots\!88}a^{2}+\frac{19\!\cdots\!27}{16\!\cdots\!41}a+\frac{23\!\cdots\!17}{16\!\cdots\!41}$
| 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: | \( 197659874.612 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{3}\cdot 197659874.612 \cdot 1}{2\cdot\sqrt{773560756544273759852688384}}\cr\approx \mathstrut & 0.451284971613 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 33*x^13 - 40*x^12 - 90*x^11 + 54*x^10 + 1791*x^9 - 954*x^8 + 6819*x^7 + 25096*x^6 - 2367*x^5 - 52062*x^4 - 63600*x^3 - 29592*x^2 + 6576*x + 4384)
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^15 - 33*x^13 - 40*x^12 - 90*x^11 + 54*x^10 + 1791*x^9 - 954*x^8 + 6819*x^7 + 25096*x^6 - 2367*x^5 - 52062*x^4 - 63600*x^3 - 29592*x^2 + 6576*x + 4384, 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^15 - 33*x^13 - 40*x^12 - 90*x^11 + 54*x^10 + 1791*x^9 - 954*x^8 + 6819*x^7 + 25096*x^6 - 2367*x^5 - 52062*x^4 - 63600*x^3 - 29592*x^2 + 6576*x + 4384);
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^15 - 33*x^13 - 40*x^12 - 90*x^11 + 54*x^10 + 1791*x^9 - 954*x^8 + 6819*x^7 + 25096*x^6 - 2367*x^5 - 52062*x^4 - 63600*x^3 - 29592*x^2 + 6576*x + 4384);
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
$C_3^4:(S_3\times S_5)$ (as 15T83):
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 non-solvable group of order 58320 |
| The 72 conjugacy class representatives for $C_3^4:(S_3\times S_5)$ |
| Character table for $C_3^4:(S_3\times S_5)$ |
Intermediate fields
| 5.3.14103.1 |
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 15 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 45 siblings: | data not computed |
| 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 | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 2.10.10.11 | $x^{10} + 10 x^{9} + 38 x^{8} + 64 x^{7} + 152 x^{6} + 688 x^{5} + 912 x^{4} - 1024 x^{3} - 1968 x^{2} + 32 x - 32$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
|
\(3\)
| 3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ |
| 3.12.18.49 | $x^{12} + 48 x^{10} - 6 x^{9} + 72 x^{8} - 36 x^{7} - 6 x^{6} + 252 x^{5} - 144 x^{4} + 198 x^{3} + 585$ | $6$ | $2$ | $18$ | 12T119 | $[3/2, 2, 2]_{2}^{4}$ | |
|
\(137\)
| $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 137.3.2.1 | $x^{3} + 137$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 137.3.0.1 | $x^{3} + 6 x + 134$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 137.6.0.1 | $x^{6} + x^{4} + 116 x^{3} + 102 x^{2} + 3 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(1567\)
| $\Q_{1567}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |