Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 15*x^14 + 30*x^13 + 79*x^12 - 162*x^11 - 272*x^10 + 555*x^9 + 561*x^8 - 597*x^7 - 1582*x^6 + 2064*x^5 + 769*x^4 - 3099*x^3 + 814*x^2 + 1246*x - 499)
gp: K = bnfinit(y^16 - 2*y^15 - 15*y^14 + 30*y^13 + 79*y^12 - 162*y^11 - 272*y^10 + 555*y^9 + 561*y^8 - 597*y^7 - 1582*y^6 + 2064*y^5 + 769*y^4 - 3099*y^3 + 814*y^2 + 1246*y - 499, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 15*x^14 + 30*x^13 + 79*x^12 - 162*x^11 - 272*x^10 + 555*x^9 + 561*x^8 - 597*x^7 - 1582*x^6 + 2064*x^5 + 769*x^4 - 3099*x^3 + 814*x^2 + 1246*x - 499);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 15*x^14 + 30*x^13 + 79*x^12 - 162*x^11 - 272*x^10 + 555*x^9 + 561*x^8 - 597*x^7 - 1582*x^6 + 2064*x^5 + 769*x^4 - 3099*x^3 + 814*x^2 + 1246*x - 499)
\( x^{16} - 2 x^{15} - 15 x^{14} + 30 x^{13} + 79 x^{12} - 162 x^{11} - 272 x^{10} + 555 x^{9} + 561 x^{8} + \cdots - 499 \)
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: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(102711726879931884765625\)
\(\medspace = 5^{12}\cdot 29^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.43\) | 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: | $5^{3/4}29^{3/4}\approx 41.78553833475025$ | ||
| Ramified primes: |
\(5\), \(29\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13}a^{14}+\frac{2}{13}a^{12}-\frac{5}{13}a^{11}-\frac{1}{13}a^{10}-\frac{2}{13}a^{9}+\frac{6}{13}a^{8}+\frac{1}{13}a^{5}+\frac{6}{13}a^{4}-\frac{5}{13}a-\frac{2}{13}$, $\frac{1}{22\!\cdots\!53}a^{15}+\frac{56\!\cdots\!88}{15\!\cdots\!97}a^{14}-\frac{62\!\cdots\!35}{22\!\cdots\!53}a^{13}-\frac{23\!\cdots\!27}{15\!\cdots\!97}a^{12}-\frac{39\!\cdots\!41}{22\!\cdots\!53}a^{11}-\frac{80\!\cdots\!83}{22\!\cdots\!53}a^{10}-\frac{86\!\cdots\!84}{20\!\cdots\!23}a^{9}-\frac{16\!\cdots\!08}{22\!\cdots\!53}a^{8}+\frac{59\!\cdots\!99}{17\!\cdots\!81}a^{7}-\frac{79\!\cdots\!75}{22\!\cdots\!53}a^{6}-\frac{51\!\cdots\!75}{22\!\cdots\!53}a^{5}+\frac{32\!\cdots\!82}{22\!\cdots\!53}a^{4}-\frac{61\!\cdots\!40}{17\!\cdots\!81}a^{3}-\frac{23\!\cdots\!45}{22\!\cdots\!53}a^{2}-\frac{99\!\cdots\!36}{22\!\cdots\!53}a-\frac{50\!\cdots\!10}{22\!\cdots\!53}$
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: |
$\frac{3539903971501}{65\!\cdots\!77}a^{15}-\frac{27159612698085}{65\!\cdots\!77}a^{14}-\frac{51426305453937}{65\!\cdots\!77}a^{13}+\frac{464806063815796}{65\!\cdots\!77}a^{12}+\frac{339653525801652}{65\!\cdots\!77}a^{11}-\frac{30\!\cdots\!21}{65\!\cdots\!77}a^{10}-\frac{153837153062384}{50\!\cdots\!29}a^{9}+\frac{12\!\cdots\!06}{65\!\cdots\!77}a^{8}+\frac{581315993984319}{50\!\cdots\!29}a^{7}-\frac{30\!\cdots\!60}{65\!\cdots\!77}a^{6}-\frac{35\!\cdots\!28}{65\!\cdots\!77}a^{5}+\frac{54\!\cdots\!21}{65\!\cdots\!77}a^{4}+\frac{37\!\cdots\!90}{50\!\cdots\!29}a^{3}-\frac{42\!\cdots\!68}{65\!\cdots\!77}a^{2}-\frac{18\!\cdots\!98}{65\!\cdots\!77}a+\frac{51\!\cdots\!46}{65\!\cdots\!77}$, $\frac{56\!\cdots\!90}{22\!\cdots\!53}a^{15}-\frac{75\!\cdots\!62}{15\!\cdots\!97}a^{14}-\frac{76\!\cdots\!51}{22\!\cdots\!53}a^{13}+\frac{10\!\cdots\!80}{15\!\cdots\!97}a^{12}+\frac{32\!\cdots\!23}{22\!\cdots\!53}a^{11}-\frac{69\!\cdots\!73}{22\!\cdots\!53}a^{10}-\frac{83\!\cdots\!92}{20\!\cdots\!23}a^{9}+\frac{20\!\cdots\!73}{22\!\cdots\!53}a^{8}+\frac{79\!\cdots\!15}{17\!\cdots\!81}a^{7}+\frac{32\!\cdots\!56}{22\!\cdots\!53}a^{6}-\frac{46\!\cdots\!59}{22\!\cdots\!53}a^{5}+\frac{10\!\cdots\!34}{22\!\cdots\!53}a^{4}-\frac{59\!\cdots\!48}{17\!\cdots\!81}a^{3}-\frac{47\!\cdots\!47}{22\!\cdots\!53}a^{2}+\frac{72\!\cdots\!93}{22\!\cdots\!53}a-\frac{50\!\cdots\!23}{22\!\cdots\!53}$, $\frac{86\!\cdots\!81}{13\!\cdots\!27}a^{15}-\frac{16\!\cdots\!54}{13\!\cdots\!27}a^{14}-\frac{12\!\cdots\!93}{13\!\cdots\!27}a^{13}+\frac{24\!\cdots\!53}{13\!\cdots\!27}a^{12}+\frac{63\!\cdots\!72}{13\!\cdots\!27}a^{11}-\frac{11\!\cdots\!15}{13\!\cdots\!27}a^{10}-\frac{21\!\cdots\!63}{13\!\cdots\!27}a^{9}+\frac{38\!\cdots\!31}{13\!\cdots\!27}a^{8}+\frac{31\!\cdots\!82}{10\!\cdots\!79}a^{7}-\frac{22\!\cdots\!60}{13\!\cdots\!27}a^{6}-\frac{11\!\cdots\!14}{13\!\cdots\!27}a^{5}+\frac{14\!\cdots\!64}{13\!\cdots\!27}a^{4}+\frac{21\!\cdots\!11}{10\!\cdots\!79}a^{3}-\frac{16\!\cdots\!68}{13\!\cdots\!27}a^{2}+\frac{45\!\cdots\!29}{13\!\cdots\!27}a+\frac{16\!\cdots\!87}{13\!\cdots\!27}$, $\frac{54\!\cdots\!58}{13\!\cdots\!27}a^{15}-\frac{10\!\cdots\!02}{13\!\cdots\!27}a^{14}-\frac{79\!\cdots\!75}{13\!\cdots\!27}a^{13}+\frac{15\!\cdots\!32}{13\!\cdots\!27}a^{12}+\frac{39\!\cdots\!92}{13\!\cdots\!27}a^{11}-\frac{77\!\cdots\!75}{13\!\cdots\!27}a^{10}-\frac{13\!\cdots\!25}{13\!\cdots\!27}a^{9}+\frac{24\!\cdots\!41}{13\!\cdots\!27}a^{8}+\frac{19\!\cdots\!68}{10\!\cdots\!79}a^{7}-\frac{15\!\cdots\!80}{13\!\cdots\!27}a^{6}-\frac{76\!\cdots\!19}{13\!\cdots\!27}a^{5}+\frac{94\!\cdots\!85}{13\!\cdots\!27}a^{4}+\frac{47\!\cdots\!43}{10\!\cdots\!79}a^{3}-\frac{10\!\cdots\!93}{13\!\cdots\!27}a^{2}+\frac{25\!\cdots\!43}{13\!\cdots\!27}a+\frac{16\!\cdots\!36}{13\!\cdots\!27}$, $\frac{57\!\cdots\!79}{22\!\cdots\!53}a^{15}-\frac{82\!\cdots\!47}{15\!\cdots\!97}a^{14}-\frac{78\!\cdots\!44}{22\!\cdots\!53}a^{13}+\frac{11\!\cdots\!36}{15\!\cdots\!97}a^{12}+\frac{33\!\cdots\!51}{22\!\cdots\!53}a^{11}-\frac{80\!\cdots\!42}{22\!\cdots\!53}a^{10}-\frac{89\!\cdots\!00}{20\!\cdots\!23}a^{9}+\frac{25\!\cdots\!07}{22\!\cdots\!53}a^{8}+\frac{10\!\cdots\!06}{17\!\cdots\!81}a^{7}-\frac{10\!\cdots\!84}{22\!\cdots\!53}a^{6}-\frac{58\!\cdots\!51}{22\!\cdots\!53}a^{5}+\frac{12\!\cdots\!03}{22\!\cdots\!53}a^{4}-\frac{46\!\cdots\!38}{17\!\cdots\!81}a^{3}-\frac{62\!\cdots\!99}{22\!\cdots\!53}a^{2}+\frac{65\!\cdots\!71}{22\!\cdots\!53}a-\frac{55\!\cdots\!82}{22\!\cdots\!53}$, $\frac{60\!\cdots\!84}{22\!\cdots\!53}a^{15}+\frac{55\!\cdots\!41}{15\!\cdots\!97}a^{14}-\frac{12\!\cdots\!04}{22\!\cdots\!53}a^{13}-\frac{86\!\cdots\!54}{15\!\cdots\!97}a^{12}+\frac{12\!\cdots\!67}{17\!\cdots\!81}a^{11}+\frac{71\!\cdots\!81}{22\!\cdots\!53}a^{10}-\frac{64\!\cdots\!65}{20\!\cdots\!23}a^{9}-\frac{26\!\cdots\!05}{22\!\cdots\!53}a^{8}+\frac{14\!\cdots\!82}{17\!\cdots\!81}a^{7}+\frac{57\!\cdots\!29}{22\!\cdots\!53}a^{6}+\frac{26\!\cdots\!95}{22\!\cdots\!53}a^{5}-\frac{11\!\cdots\!30}{22\!\cdots\!53}a^{4}+\frac{18\!\cdots\!73}{17\!\cdots\!81}a^{3}+\frac{25\!\cdots\!16}{22\!\cdots\!53}a^{2}-\frac{33\!\cdots\!76}{22\!\cdots\!53}a+\frac{33\!\cdots\!97}{22\!\cdots\!53}$, $\frac{14\!\cdots\!17}{26\!\cdots\!17}a^{15}-\frac{12\!\cdots\!71}{17\!\cdots\!33}a^{14}-\frac{23\!\cdots\!71}{26\!\cdots\!17}a^{13}+\frac{18\!\cdots\!24}{17\!\cdots\!33}a^{12}+\frac{13\!\cdots\!09}{26\!\cdots\!17}a^{11}-\frac{15\!\cdots\!99}{26\!\cdots\!17}a^{10}-\frac{46\!\cdots\!72}{23\!\cdots\!47}a^{9}+\frac{52\!\cdots\!71}{26\!\cdots\!17}a^{8}+\frac{93\!\cdots\!04}{20\!\cdots\!09}a^{7}-\frac{29\!\cdots\!29}{26\!\cdots\!17}a^{6}-\frac{25\!\cdots\!55}{26\!\cdots\!17}a^{5}+\frac{15\!\cdots\!44}{26\!\cdots\!17}a^{4}+\frac{20\!\cdots\!11}{20\!\cdots\!09}a^{3}-\frac{36\!\cdots\!26}{26\!\cdots\!17}a^{2}-\frac{72\!\cdots\!21}{20\!\cdots\!09}a+\frac{19\!\cdots\!70}{26\!\cdots\!17}$, $\frac{57\!\cdots\!79}{22\!\cdots\!53}a^{15}-\frac{82\!\cdots\!47}{15\!\cdots\!97}a^{14}-\frac{78\!\cdots\!44}{22\!\cdots\!53}a^{13}+\frac{11\!\cdots\!36}{15\!\cdots\!97}a^{12}+\frac{33\!\cdots\!51}{22\!\cdots\!53}a^{11}-\frac{80\!\cdots\!42}{22\!\cdots\!53}a^{10}-\frac{89\!\cdots\!00}{20\!\cdots\!23}a^{9}+\frac{25\!\cdots\!07}{22\!\cdots\!53}a^{8}+\frac{10\!\cdots\!06}{17\!\cdots\!81}a^{7}-\frac{10\!\cdots\!84}{22\!\cdots\!53}a^{6}-\frac{58\!\cdots\!51}{22\!\cdots\!53}a^{5}+\frac{12\!\cdots\!03}{22\!\cdots\!53}a^{4}-\frac{46\!\cdots\!38}{17\!\cdots\!81}a^{3}-\frac{62\!\cdots\!99}{22\!\cdots\!53}a^{2}+\frac{43\!\cdots\!18}{22\!\cdots\!53}a-\frac{94\!\cdots\!76}{22\!\cdots\!53}$, $\frac{22\!\cdots\!25}{17\!\cdots\!81}a^{15}-\frac{99\!\cdots\!90}{15\!\cdots\!97}a^{14}+\frac{18\!\cdots\!44}{17\!\cdots\!81}a^{13}+\frac{14\!\cdots\!61}{15\!\cdots\!97}a^{12}-\frac{38\!\cdots\!19}{22\!\cdots\!53}a^{11}-\frac{10\!\cdots\!18}{22\!\cdots\!53}a^{10}+\frac{17\!\cdots\!81}{20\!\cdots\!23}a^{9}+\frac{32\!\cdots\!68}{22\!\cdots\!53}a^{8}-\frac{48\!\cdots\!87}{17\!\cdots\!81}a^{7}-\frac{42\!\cdots\!36}{17\!\cdots\!81}a^{6}+\frac{36\!\cdots\!63}{22\!\cdots\!53}a^{5}+\frac{18\!\cdots\!41}{22\!\cdots\!53}a^{4}-\frac{21\!\cdots\!65}{17\!\cdots\!81}a^{3}+\frac{58\!\cdots\!43}{17\!\cdots\!81}a^{2}+\frac{23\!\cdots\!70}{22\!\cdots\!53}a-\frac{11\!\cdots\!72}{22\!\cdots\!53}$, $\frac{95\!\cdots\!92}{22\!\cdots\!53}a^{15}-\frac{13\!\cdots\!07}{15\!\cdots\!97}a^{14}-\frac{13\!\cdots\!13}{22\!\cdots\!53}a^{13}+\frac{18\!\cdots\!55}{15\!\cdots\!97}a^{12}+\frac{64\!\cdots\!30}{22\!\cdots\!53}a^{11}-\frac{13\!\cdots\!91}{22\!\cdots\!53}a^{10}-\frac{18\!\cdots\!30}{20\!\cdots\!23}a^{9}+\frac{43\!\cdots\!24}{22\!\cdots\!53}a^{8}+\frac{26\!\cdots\!69}{17\!\cdots\!81}a^{7}-\frac{25\!\cdots\!59}{22\!\cdots\!53}a^{6}-\frac{11\!\cdots\!39}{22\!\cdots\!53}a^{5}+\frac{18\!\cdots\!37}{22\!\cdots\!53}a^{4}-\frac{21\!\cdots\!52}{17\!\cdots\!81}a^{3}-\frac{16\!\cdots\!66}{22\!\cdots\!53}a^{2}+\frac{78\!\cdots\!19}{22\!\cdots\!53}a-\frac{10\!\cdots\!88}{22\!\cdots\!53}$, $\frac{49\!\cdots\!33}{22\!\cdots\!53}a^{15}-\frac{63\!\cdots\!84}{11\!\cdots\!69}a^{14}-\frac{69\!\cdots\!33}{22\!\cdots\!53}a^{13}+\frac{12\!\cdots\!56}{15\!\cdots\!97}a^{12}+\frac{32\!\cdots\!00}{22\!\cdots\!53}a^{11}-\frac{98\!\cdots\!99}{22\!\cdots\!53}a^{10}-\frac{91\!\cdots\!44}{20\!\cdots\!23}a^{9}+\frac{25\!\cdots\!41}{17\!\cdots\!81}a^{8}+\frac{13\!\cdots\!16}{17\!\cdots\!81}a^{7}-\frac{42\!\cdots\!08}{22\!\cdots\!53}a^{6}-\frac{75\!\cdots\!75}{22\!\cdots\!53}a^{5}+\frac{10\!\cdots\!89}{17\!\cdots\!81}a^{4}+\frac{11\!\cdots\!20}{17\!\cdots\!81}a^{3}-\frac{15\!\cdots\!61}{22\!\cdots\!53}a^{2}+\frac{64\!\cdots\!77}{22\!\cdots\!53}a+\frac{11\!\cdots\!67}{17\!\cdots\!81}$
| 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: | \( 215243.090952 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 215243.090952 \cdot 1}{2\cdot\sqrt{102711726879931884765625}}\cr\approx \mathstrut & 0.133982675998 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 15*x^14 + 30*x^13 + 79*x^12 - 162*x^11 - 272*x^10 + 555*x^9 + 561*x^8 - 597*x^7 - 1582*x^6 + 2064*x^5 + 769*x^4 - 3099*x^3 + 814*x^2 + 1246*x - 499)
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 - 2*x^15 - 15*x^14 + 30*x^13 + 79*x^12 - 162*x^11 - 272*x^10 + 555*x^9 + 561*x^8 - 597*x^7 - 1582*x^6 + 2064*x^5 + 769*x^4 - 3099*x^3 + 814*x^2 + 1246*x - 499, 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 - 2*x^15 - 15*x^14 + 30*x^13 + 79*x^12 - 162*x^11 - 272*x^10 + 555*x^9 + 561*x^8 - 597*x^7 - 1582*x^6 + 2064*x^5 + 769*x^4 - 3099*x^3 + 814*x^2 + 1246*x - 499);
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 - 2*x^15 - 15*x^14 + 30*x^13 + 79*x^12 - 162*x^11 - 272*x^10 + 555*x^9 + 561*x^8 - 597*x^7 - 1582*x^6 + 2064*x^5 + 769*x^4 - 3099*x^3 + 814*x^2 + 1246*x - 499);
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
| \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.4.320486703125.1, 8.4.381078125.1, 8.8.442050625.1 |
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 8 siblings: | 8.4.381078125.1, 8.4.320486703125.1 |
| Degree 16 sibling: | 16.0.86380562306022715087890625.5 |
| Minimal sibling: | 8.4.381078125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.8.6.2 | $x^{8} + 10 x^{4} - 25$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.2 | $x^{8} + 10 x^{4} - 25$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(29\)
| 29.8.6.2 | $x^{8} - 1914 x^{4} - 2069701$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 29.8.4.1 | $x^{8} + 2784 x^{7} + 2906616 x^{6} + 1348864734 x^{5} + 234834277018 x^{4} + 41857830864 x^{3} + 492109772617 x^{2} + 3561769809750 x + 616658760166$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |