Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 42*x^14 - 278*x^13 + 659*x^12 - 405*x^11 - 17202*x^10 + 70727*x^9 - 176053*x^8 + 320302*x^7 + 178713*x^6 - 1112831*x^5 - 1379291*x^4 + 12513782*x^3 - 11860273*x^2 + 60078180*x + 129212428)
gp: K = bnfinit(y^16 - 5*y^15 + 42*y^14 - 278*y^13 + 659*y^12 - 405*y^11 - 17202*y^10 + 70727*y^9 - 176053*y^8 + 320302*y^7 + 178713*y^6 - 1112831*y^5 - 1379291*y^4 + 12513782*y^3 - 11860273*y^2 + 60078180*y + 129212428, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 42*x^14 - 278*x^13 + 659*x^12 - 405*x^11 - 17202*x^10 + 70727*x^9 - 176053*x^8 + 320302*x^7 + 178713*x^6 - 1112831*x^5 - 1379291*x^4 + 12513782*x^3 - 11860273*x^2 + 60078180*x + 129212428);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 + 42*x^14 - 278*x^13 + 659*x^12 - 405*x^11 - 17202*x^10 + 70727*x^9 - 176053*x^8 + 320302*x^7 + 178713*x^6 - 1112831*x^5 - 1379291*x^4 + 12513782*x^3 - 11860273*x^2 + 60078180*x + 129212428)
\( x^{16} - 5 x^{15} + 42 x^{14} - 278 x^{13} + 659 x^{12} - 405 x^{11} - 17202 x^{10} + 70727 x^{9} + \cdots + 129212428 \)
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: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1162337480711184271576898233831313\)
\(\medspace = 17^{15}\cdot 67^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(116.57\) | 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: | $17^{15/16}67^{1/2}\approx 116.56905215178652$ | ||
| Ramified primes: |
\(17\), \(67\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{4}a^{8}+\frac{3}{8}a^{7}-\frac{1}{8}a^{6}-\frac{3}{8}a^{5}-\frac{1}{4}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{8}a^{10}+\frac{1}{8}a^{9}+\frac{1}{8}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}-\frac{3}{8}a^{3}+\frac{1}{8}a^{2}-\frac{3}{8}a-\frac{1}{4}$, $\frac{1}{8}a^{14}+\frac{1}{8}a^{10}-\frac{1}{4}a^{8}+\frac{1}{8}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{3}{8}a^{4}-\frac{3}{8}a^{2}+\frac{3}{8}a+\frac{1}{4}$, $\frac{1}{13\!\cdots\!08}a^{15}-\frac{38\!\cdots\!07}{13\!\cdots\!08}a^{14}-\frac{90\!\cdots\!51}{16\!\cdots\!26}a^{13}+\frac{28\!\cdots\!71}{65\!\cdots\!04}a^{12}-\frac{13\!\cdots\!53}{13\!\cdots\!08}a^{11}-\frac{72\!\cdots\!39}{13\!\cdots\!08}a^{10}+\frac{21\!\cdots\!94}{82\!\cdots\!63}a^{9}-\frac{18\!\cdots\!49}{13\!\cdots\!08}a^{8}+\frac{45\!\cdots\!17}{13\!\cdots\!08}a^{7}+\frac{34\!\cdots\!05}{16\!\cdots\!26}a^{6}+\frac{29\!\cdots\!57}{13\!\cdots\!08}a^{5}-\frac{61\!\cdots\!89}{13\!\cdots\!08}a^{4}+\frac{52\!\cdots\!15}{13\!\cdots\!08}a^{3}-\frac{56\!\cdots\!34}{82\!\cdots\!63}a^{2}-\frac{46\!\cdots\!17}{13\!\cdots\!08}a+\frac{74\!\cdots\!95}{65\!\cdots\!04}$
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
$C_{2}\times C_{2}\times C_{2}\times C_{8}$, which has order $64$ (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: |
$\frac{32\!\cdots\!75}{18\!\cdots\!04}a^{15}-\frac{23\!\cdots\!51}{18\!\cdots\!04}a^{14}+\frac{88\!\cdots\!51}{92\!\cdots\!52}a^{13}-\frac{62\!\cdots\!33}{92\!\cdots\!52}a^{12}+\frac{44\!\cdots\!55}{18\!\cdots\!04}a^{11}-\frac{84\!\cdots\!01}{18\!\cdots\!04}a^{10}-\frac{21\!\cdots\!47}{92\!\cdots\!52}a^{9}+\frac{33\!\cdots\!11}{18\!\cdots\!04}a^{8}-\frac{11\!\cdots\!99}{18\!\cdots\!04}a^{7}+\frac{19\!\cdots\!18}{11\!\cdots\!19}a^{6}-\frac{45\!\cdots\!71}{18\!\cdots\!04}a^{5}+\frac{40\!\cdots\!81}{18\!\cdots\!04}a^{4}-\frac{43\!\cdots\!05}{18\!\cdots\!04}a^{3}+\frac{97\!\cdots\!73}{46\!\cdots\!76}a^{2}-\frac{53\!\cdots\!35}{18\!\cdots\!04}a+\frac{90\!\cdots\!13}{92\!\cdots\!52}$, $\frac{67\!\cdots\!27}{46\!\cdots\!76}a^{15}-\frac{20\!\cdots\!79}{23\!\cdots\!38}a^{14}+\frac{55\!\cdots\!19}{92\!\cdots\!52}a^{13}-\frac{48\!\cdots\!32}{11\!\cdots\!19}a^{12}+\frac{90\!\cdots\!15}{92\!\cdots\!52}a^{11}+\frac{12\!\cdots\!21}{92\!\cdots\!52}a^{10}-\frac{30\!\cdots\!29}{92\!\cdots\!52}a^{9}+\frac{12\!\cdots\!17}{92\!\cdots\!52}a^{8}-\frac{83\!\cdots\!97}{46\!\cdots\!76}a^{7}-\frac{10\!\cdots\!35}{11\!\cdots\!19}a^{6}+\frac{17\!\cdots\!99}{92\!\cdots\!52}a^{5}-\frac{37\!\cdots\!37}{92\!\cdots\!52}a^{4}-\frac{85\!\cdots\!63}{92\!\cdots\!52}a^{3}+\frac{34\!\cdots\!69}{92\!\cdots\!52}a^{2}-\frac{62\!\cdots\!29}{92\!\cdots\!52}a-\frac{47\!\cdots\!63}{46\!\cdots\!76}$, $\frac{96\!\cdots\!67}{18\!\cdots\!04}a^{15}-\frac{78\!\cdots\!21}{18\!\cdots\!04}a^{14}+\frac{14\!\cdots\!87}{46\!\cdots\!76}a^{13}-\frac{24\!\cdots\!03}{11\!\cdots\!19}a^{12}+\frac{15\!\cdots\!19}{18\!\cdots\!04}a^{11}-\frac{29\!\cdots\!13}{18\!\cdots\!04}a^{10}-\frac{62\!\cdots\!73}{92\!\cdots\!52}a^{9}+\frac{10\!\cdots\!81}{18\!\cdots\!04}a^{8}-\frac{33\!\cdots\!63}{18\!\cdots\!04}a^{7}+\frac{44\!\cdots\!65}{92\!\cdots\!52}a^{6}-\frac{27\!\cdots\!63}{18\!\cdots\!04}a^{5}+\frac{55\!\cdots\!81}{18\!\cdots\!04}a^{4}-\frac{56\!\cdots\!53}{18\!\cdots\!04}a^{3}-\frac{17\!\cdots\!09}{46\!\cdots\!76}a^{2}+\frac{46\!\cdots\!59}{18\!\cdots\!04}a-\frac{65\!\cdots\!29}{92\!\cdots\!52}$, $\frac{27\!\cdots\!97}{13\!\cdots\!08}a^{15}-\frac{31\!\cdots\!03}{13\!\cdots\!08}a^{14}+\frac{11\!\cdots\!65}{16\!\cdots\!26}a^{13}-\frac{34\!\cdots\!13}{82\!\cdots\!63}a^{12}+\frac{35\!\cdots\!25}{13\!\cdots\!08}a^{11}+\frac{22\!\cdots\!97}{13\!\cdots\!08}a^{10}-\frac{71\!\cdots\!93}{65\!\cdots\!04}a^{9}+\frac{90\!\cdots\!47}{13\!\cdots\!08}a^{8}+\frac{49\!\cdots\!23}{13\!\cdots\!08}a^{7}-\frac{13\!\cdots\!25}{65\!\cdots\!04}a^{6}+\frac{69\!\cdots\!83}{13\!\cdots\!08}a^{5}+\frac{42\!\cdots\!83}{13\!\cdots\!08}a^{4}-\frac{42\!\cdots\!03}{13\!\cdots\!08}a^{3}+\frac{36\!\cdots\!69}{32\!\cdots\!52}a^{2}+\frac{33\!\cdots\!61}{13\!\cdots\!08}a-\frac{20\!\cdots\!07}{65\!\cdots\!04}$, $\frac{21\!\cdots\!95}{32\!\cdots\!52}a^{15}-\frac{76\!\cdots\!91}{65\!\cdots\!04}a^{14}+\frac{15\!\cdots\!95}{65\!\cdots\!04}a^{13}-\frac{34\!\cdots\!69}{32\!\cdots\!52}a^{12}+\frac{59\!\cdots\!59}{65\!\cdots\!04}a^{11}+\frac{95\!\cdots\!17}{32\!\cdots\!52}a^{10}-\frac{72\!\cdots\!27}{65\!\cdots\!04}a^{9}+\frac{67\!\cdots\!85}{65\!\cdots\!04}a^{8}-\frac{53\!\cdots\!65}{65\!\cdots\!04}a^{7}-\frac{85\!\cdots\!01}{16\!\cdots\!26}a^{6}-\frac{33\!\cdots\!25}{65\!\cdots\!04}a^{5}-\frac{29\!\cdots\!61}{32\!\cdots\!52}a^{4}-\frac{24\!\cdots\!81}{65\!\cdots\!04}a^{3}-\frac{64\!\cdots\!79}{16\!\cdots\!26}a^{2}-\frac{66\!\cdots\!85}{32\!\cdots\!52}a-\frac{42\!\cdots\!01}{16\!\cdots\!26}$, $\frac{11\!\cdots\!79}{13\!\cdots\!08}a^{15}-\frac{65\!\cdots\!83}{13\!\cdots\!08}a^{14}+\frac{52\!\cdots\!19}{16\!\cdots\!26}a^{13}-\frac{14\!\cdots\!31}{65\!\cdots\!04}a^{12}+\frac{62\!\cdots\!09}{13\!\cdots\!08}a^{11}+\frac{16\!\cdots\!21}{13\!\cdots\!08}a^{10}-\frac{31\!\cdots\!63}{16\!\cdots\!26}a^{9}+\frac{83\!\cdots\!21}{13\!\cdots\!08}a^{8}-\frac{43\!\cdots\!55}{13\!\cdots\!08}a^{7}-\frac{63\!\cdots\!83}{32\!\cdots\!52}a^{6}+\frac{39\!\cdots\!03}{13\!\cdots\!08}a^{5}+\frac{82\!\cdots\!79}{13\!\cdots\!08}a^{4}-\frac{44\!\cdots\!95}{13\!\cdots\!08}a^{3}+\frac{53\!\cdots\!21}{65\!\cdots\!04}a^{2}+\frac{10\!\cdots\!59}{13\!\cdots\!08}a-\frac{78\!\cdots\!81}{65\!\cdots\!04}$, $\frac{30\!\cdots\!55}{32\!\cdots\!52}a^{15}-\frac{39\!\cdots\!47}{65\!\cdots\!04}a^{14}+\frac{15\!\cdots\!57}{32\!\cdots\!52}a^{13}-\frac{26\!\cdots\!46}{82\!\cdots\!63}a^{12}+\frac{86\!\cdots\!28}{82\!\cdots\!63}a^{11}-\frac{12\!\cdots\!33}{65\!\cdots\!04}a^{10}-\frac{36\!\cdots\!19}{32\!\cdots\!52}a^{9}+\frac{21\!\cdots\!55}{32\!\cdots\!52}a^{8}-\frac{11\!\cdots\!83}{65\!\cdots\!04}a^{7}+\frac{90\!\cdots\!41}{32\!\cdots\!52}a^{6}+\frac{79\!\cdots\!05}{16\!\cdots\!26}a^{5}-\frac{23\!\cdots\!91}{65\!\cdots\!04}a^{4}+\frac{57\!\cdots\!19}{82\!\cdots\!63}a^{3}-\frac{24\!\cdots\!95}{65\!\cdots\!04}a^{2}+\frac{53\!\cdots\!61}{65\!\cdots\!04}a+\frac{13\!\cdots\!23}{32\!\cdots\!52}$, $\frac{46\!\cdots\!49}{13\!\cdots\!08}a^{15}-\frac{17\!\cdots\!87}{13\!\cdots\!08}a^{14}+\frac{21\!\cdots\!81}{32\!\cdots\!52}a^{13}-\frac{92\!\cdots\!19}{16\!\cdots\!26}a^{12}+\frac{42\!\cdots\!85}{13\!\cdots\!08}a^{11}-\frac{81\!\cdots\!11}{13\!\cdots\!08}a^{10}-\frac{67\!\cdots\!43}{65\!\cdots\!04}a^{9}+\frac{27\!\cdots\!87}{13\!\cdots\!08}a^{8}-\frac{10\!\cdots\!93}{13\!\cdots\!08}a^{7}+\frac{11\!\cdots\!55}{65\!\cdots\!04}a^{6}-\frac{62\!\cdots\!17}{13\!\cdots\!08}a^{5}-\frac{50\!\cdots\!53}{13\!\cdots\!08}a^{4}-\frac{13\!\cdots\!55}{13\!\cdots\!08}a^{3}+\frac{10\!\cdots\!23}{32\!\cdots\!52}a^{2}-\frac{46\!\cdots\!79}{13\!\cdots\!08}a+\frac{25\!\cdots\!65}{65\!\cdots\!04}$, $\frac{47\!\cdots\!71}{16\!\cdots\!26}a^{15}-\frac{71\!\cdots\!23}{32\!\cdots\!52}a^{14}+\frac{89\!\cdots\!19}{65\!\cdots\!04}a^{13}-\frac{64\!\cdots\!19}{65\!\cdots\!04}a^{12}+\frac{25\!\cdots\!80}{82\!\cdots\!63}a^{11}+\frac{18\!\cdots\!63}{65\!\cdots\!04}a^{10}-\frac{20\!\cdots\!75}{32\!\cdots\!52}a^{9}+\frac{21\!\cdots\!73}{65\!\cdots\!04}a^{8}-\frac{39\!\cdots\!61}{65\!\cdots\!04}a^{7}+\frac{24\!\cdots\!91}{65\!\cdots\!04}a^{6}+\frac{32\!\cdots\!97}{32\!\cdots\!52}a^{5}-\frac{41\!\cdots\!41}{65\!\cdots\!04}a^{4}-\frac{11\!\cdots\!79}{32\!\cdots\!52}a^{3}+\frac{53\!\cdots\!15}{65\!\cdots\!04}a^{2}-\frac{11\!\cdots\!81}{16\!\cdots\!26}a-\frac{37\!\cdots\!39}{16\!\cdots\!26}$
| 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: | \( 887169888.207 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 887169888.207 \cdot 64}{2\cdot\sqrt{1162337480711184271576898233831313}}\cr\approx \mathstrut & 0.819765888498 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 42*x^14 - 278*x^13 + 659*x^12 - 405*x^11 - 17202*x^10 + 70727*x^9 - 176053*x^8 + 320302*x^7 + 178713*x^6 - 1112831*x^5 - 1379291*x^4 + 12513782*x^3 - 11860273*x^2 + 60078180*x + 129212428)
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 - 5*x^15 + 42*x^14 - 278*x^13 + 659*x^12 - 405*x^11 - 17202*x^10 + 70727*x^9 - 176053*x^8 + 320302*x^7 + 178713*x^6 - 1112831*x^5 - 1379291*x^4 + 12513782*x^3 - 11860273*x^2 + 60078180*x + 129212428, 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 - 5*x^15 + 42*x^14 - 278*x^13 + 659*x^12 - 405*x^11 - 17202*x^10 + 70727*x^9 - 176053*x^8 + 320302*x^7 + 178713*x^6 - 1112831*x^5 - 1379291*x^4 + 12513782*x^3 - 11860273*x^2 + 60078180*x + 129212428);
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 - 5*x^15 + 42*x^14 - 278*x^13 + 659*x^12 - 405*x^11 - 17202*x^10 + 70727*x^9 - 176053*x^8 + 320302*x^7 + 178713*x^6 - 1112831*x^5 - 1379291*x^4 + 12513782*x^3 - 11860273*x^2 + 60078180*x + 129212428);
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
$C_2^7.C_8$ (as 16T1155):
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 1024 |
| The 40 conjugacy class representatives for $C_2^7.C_8$ |
| Character table for $C_2^7.C_8$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.1842010303097.1 |
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 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 16.4.57681033264163530732453953.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.16.15.5 | $x^{16} + 17$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |
|
\(67\)
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.4.2.1 | $x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |