Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 4*x^14 + 54*x^13 - 119*x^12 - 138*x^11 + 532*x^10 + 463*x^9 - 2582*x^8 + 196*x^7 + 8673*x^6 - 14128*x^5 - 2278*x^4 + 22735*x^3 - 14330*x^2 + 2077*x - 67)
gp: K = bnfinit(y^16 - 6*y^15 + 4*y^14 + 54*y^13 - 119*y^12 - 138*y^11 + 532*y^10 + 463*y^9 - 2582*y^8 + 196*y^7 + 8673*y^6 - 14128*y^5 - 2278*y^4 + 22735*y^3 - 14330*y^2 + 2077*y - 67, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 4*x^14 + 54*x^13 - 119*x^12 - 138*x^11 + 532*x^10 + 463*x^9 - 2582*x^8 + 196*x^7 + 8673*x^6 - 14128*x^5 - 2278*x^4 + 22735*x^3 - 14330*x^2 + 2077*x - 67);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 4*x^14 + 54*x^13 - 119*x^12 - 138*x^11 + 532*x^10 + 463*x^9 - 2582*x^8 + 196*x^7 + 8673*x^6 - 14128*x^5 - 2278*x^4 + 22735*x^3 - 14330*x^2 + 2077*x - 67)
\( x^{16} - 6 x^{15} + 4 x^{14} + 54 x^{13} - 119 x^{12} - 138 x^{11} + 532 x^{10} + 463 x^{9} - 2582 x^{8} + \cdots - 67 \)
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: |
\(3393001956715501807791409\)
\(\medspace = 17^{14}\cdot 67^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.13\) | 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: | $17^{7/8}67^{1/2}\approx 97.65178978680869$ | ||
| 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: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{67\!\cdots\!54}a^{15}+\frac{55\!\cdots\!47}{33\!\cdots\!27}a^{14}+\frac{12\!\cdots\!53}{67\!\cdots\!54}a^{13}-\frac{25\!\cdots\!97}{67\!\cdots\!54}a^{12}-\frac{10\!\cdots\!24}{33\!\cdots\!27}a^{11}+\frac{11\!\cdots\!33}{67\!\cdots\!54}a^{10}+\frac{29\!\cdots\!19}{67\!\cdots\!54}a^{9}-\frac{15\!\cdots\!29}{33\!\cdots\!27}a^{8}+\frac{18\!\cdots\!17}{67\!\cdots\!54}a^{7}-\frac{30\!\cdots\!15}{67\!\cdots\!54}a^{6}+\frac{10\!\cdots\!79}{33\!\cdots\!27}a^{5}+\frac{32\!\cdots\!71}{67\!\cdots\!54}a^{4}-\frac{20\!\cdots\!39}{67\!\cdots\!54}a^{3}-\frac{14\!\cdots\!75}{33\!\cdots\!27}a^{2}+\frac{20\!\cdots\!51}{67\!\cdots\!54}a-\frac{76\!\cdots\!31}{33\!\cdots\!27}$
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
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: |
$\frac{12\!\cdots\!75}{67\!\cdots\!54}a^{15}-\frac{74\!\cdots\!47}{67\!\cdots\!54}a^{14}+\frac{47\!\cdots\!73}{67\!\cdots\!54}a^{13}+\frac{33\!\cdots\!18}{33\!\cdots\!27}a^{12}-\frac{14\!\cdots\!97}{67\!\cdots\!54}a^{11}-\frac{17\!\cdots\!73}{67\!\cdots\!54}a^{10}+\frac{33\!\cdots\!58}{33\!\cdots\!27}a^{9}+\frac{58\!\cdots\!85}{67\!\cdots\!54}a^{8}-\frac{32\!\cdots\!67}{67\!\cdots\!54}a^{7}+\frac{95\!\cdots\!28}{33\!\cdots\!27}a^{6}+\frac{10\!\cdots\!85}{67\!\cdots\!54}a^{5}-\frac{17\!\cdots\!87}{67\!\cdots\!54}a^{4}-\frac{18\!\cdots\!07}{33\!\cdots\!27}a^{3}+\frac{28\!\cdots\!61}{67\!\cdots\!54}a^{2}-\frac{16\!\cdots\!21}{67\!\cdots\!54}a+\frac{14\!\cdots\!97}{67\!\cdots\!54}$, $\frac{17\!\cdots\!09}{67\!\cdots\!54}a^{15}-\frac{52\!\cdots\!64}{33\!\cdots\!27}a^{14}+\frac{62\!\cdots\!55}{67\!\cdots\!54}a^{13}+\frac{48\!\cdots\!15}{33\!\cdots\!27}a^{12}-\frac{10\!\cdots\!53}{33\!\cdots\!27}a^{11}-\frac{26\!\cdots\!77}{67\!\cdots\!54}a^{10}+\frac{46\!\cdots\!62}{33\!\cdots\!27}a^{9}+\frac{45\!\cdots\!82}{33\!\cdots\!27}a^{8}-\frac{45\!\cdots\!87}{67\!\cdots\!54}a^{7}-\frac{36\!\cdots\!70}{33\!\cdots\!27}a^{6}+\frac{78\!\cdots\!75}{33\!\cdots\!27}a^{5}-\frac{23\!\cdots\!23}{67\!\cdots\!54}a^{4}-\frac{33\!\cdots\!66}{33\!\cdots\!27}a^{3}+\frac{20\!\cdots\!64}{33\!\cdots\!27}a^{2}-\frac{21\!\cdots\!25}{67\!\cdots\!54}a+\frac{17\!\cdots\!59}{67\!\cdots\!54}$, $\frac{41\!\cdots\!07}{67\!\cdots\!54}a^{15}-\frac{13\!\cdots\!24}{33\!\cdots\!27}a^{14}+\frac{28\!\cdots\!97}{67\!\cdots\!54}a^{13}+\frac{11\!\cdots\!35}{33\!\cdots\!27}a^{12}-\frac{31\!\cdots\!15}{33\!\cdots\!27}a^{11}-\frac{40\!\cdots\!31}{67\!\cdots\!54}a^{10}+\frac{13\!\cdots\!42}{33\!\cdots\!27}a^{9}+\frac{53\!\cdots\!87}{33\!\cdots\!27}a^{8}-\frac{12\!\cdots\!21}{67\!\cdots\!54}a^{7}+\frac{26\!\cdots\!70}{33\!\cdots\!27}a^{6}+\frac{19\!\cdots\!41}{33\!\cdots\!27}a^{5}-\frac{77\!\cdots\!09}{67\!\cdots\!54}a^{4}+\frac{50\!\cdots\!80}{33\!\cdots\!27}a^{3}+\frac{59\!\cdots\!25}{33\!\cdots\!27}a^{2}-\frac{10\!\cdots\!95}{67\!\cdots\!54}a+\frac{15\!\cdots\!53}{67\!\cdots\!54}$, $\frac{59\!\cdots\!01}{33\!\cdots\!27}a^{15}-\frac{72\!\cdots\!09}{67\!\cdots\!54}a^{14}+\frac{27\!\cdots\!50}{33\!\cdots\!27}a^{13}+\frac{32\!\cdots\!57}{33\!\cdots\!27}a^{12}-\frac{14\!\cdots\!65}{67\!\cdots\!54}a^{11}-\frac{75\!\cdots\!82}{33\!\cdots\!27}a^{10}+\frac{33\!\cdots\!43}{33\!\cdots\!27}a^{9}+\frac{48\!\cdots\!97}{67\!\cdots\!54}a^{8}-\frac{15\!\cdots\!81}{33\!\cdots\!27}a^{7}+\frac{28\!\cdots\!91}{33\!\cdots\!27}a^{6}+\frac{10\!\cdots\!95}{67\!\cdots\!54}a^{5}-\frac{90\!\cdots\!29}{33\!\cdots\!27}a^{4}-\frac{68\!\cdots\!93}{33\!\cdots\!27}a^{3}+\frac{28\!\cdots\!09}{67\!\cdots\!54}a^{2}-\frac{98\!\cdots\!61}{33\!\cdots\!27}a+\frac{74\!\cdots\!38}{33\!\cdots\!27}$, $\frac{60\!\cdots\!73}{67\!\cdots\!54}a^{15}-\frac{83\!\cdots\!69}{33\!\cdots\!27}a^{14}-\frac{78\!\cdots\!27}{67\!\cdots\!54}a^{13}+\frac{16\!\cdots\!61}{33\!\cdots\!27}a^{12}+\frac{15\!\cdots\!34}{33\!\cdots\!27}a^{11}-\frac{24\!\cdots\!09}{67\!\cdots\!54}a^{10}-\frac{12\!\cdots\!85}{33\!\cdots\!27}a^{9}+\frac{51\!\cdots\!44}{33\!\cdots\!27}a^{8}-\frac{33\!\cdots\!05}{67\!\cdots\!54}a^{7}-\frac{19\!\cdots\!63}{33\!\cdots\!27}a^{6}+\frac{19\!\cdots\!95}{33\!\cdots\!27}a^{5}+\frac{67\!\cdots\!71}{67\!\cdots\!54}a^{4}-\frac{11\!\cdots\!14}{33\!\cdots\!27}a^{3}-\frac{11\!\cdots\!24}{33\!\cdots\!27}a^{2}+\frac{30\!\cdots\!01}{67\!\cdots\!54}a-\frac{70\!\cdots\!33}{67\!\cdots\!54}$, $\frac{26\!\cdots\!49}{67\!\cdots\!54}a^{15}-\frac{71\!\cdots\!18}{33\!\cdots\!27}a^{14}+\frac{20\!\cdots\!49}{67\!\cdots\!54}a^{13}+\frac{70\!\cdots\!98}{33\!\cdots\!27}a^{12}-\frac{11\!\cdots\!77}{33\!\cdots\!27}a^{11}-\frac{47\!\cdots\!45}{67\!\cdots\!54}a^{10}+\frac{51\!\cdots\!66}{33\!\cdots\!27}a^{9}+\frac{90\!\cdots\!65}{33\!\cdots\!27}a^{8}-\frac{53\!\cdots\!69}{67\!\cdots\!54}a^{7}-\frac{13\!\cdots\!54}{33\!\cdots\!27}a^{6}+\frac{98\!\cdots\!94}{33\!\cdots\!27}a^{5}-\frac{24\!\cdots\!17}{67\!\cdots\!54}a^{4}-\frac{91\!\cdots\!49}{33\!\cdots\!27}a^{3}+\frac{20\!\cdots\!44}{33\!\cdots\!27}a^{2}-\frac{10\!\cdots\!01}{67\!\cdots\!54}a+\frac{31\!\cdots\!09}{67\!\cdots\!54}$, $\frac{36\!\cdots\!61}{33\!\cdots\!27}a^{15}-\frac{43\!\cdots\!35}{67\!\cdots\!54}a^{14}+\frac{12\!\cdots\!55}{33\!\cdots\!27}a^{13}+\frac{19\!\cdots\!21}{33\!\cdots\!27}a^{12}-\frac{83\!\cdots\!35}{67\!\cdots\!54}a^{11}-\frac{55\!\cdots\!14}{33\!\cdots\!27}a^{10}+\frac{19\!\cdots\!16}{33\!\cdots\!27}a^{9}+\frac{37\!\cdots\!99}{67\!\cdots\!54}a^{8}-\frac{93\!\cdots\!44}{33\!\cdots\!27}a^{7}-\frac{29\!\cdots\!90}{33\!\cdots\!27}a^{6}+\frac{63\!\cdots\!03}{67\!\cdots\!54}a^{5}-\frac{48\!\cdots\!95}{33\!\cdots\!27}a^{4}-\frac{12\!\cdots\!81}{33\!\cdots\!27}a^{3}+\frac{17\!\cdots\!25}{67\!\cdots\!54}a^{2}-\frac{45\!\cdots\!86}{33\!\cdots\!27}a+\frac{11\!\cdots\!14}{33\!\cdots\!27}$, $\frac{27\!\cdots\!82}{33\!\cdots\!27}a^{15}-\frac{15\!\cdots\!95}{33\!\cdots\!27}a^{14}+\frac{21\!\cdots\!81}{33\!\cdots\!27}a^{13}+\frac{15\!\cdots\!10}{33\!\cdots\!27}a^{12}-\frac{23\!\cdots\!68}{33\!\cdots\!27}a^{11}-\frac{52\!\cdots\!39}{33\!\cdots\!27}a^{10}+\frac{11\!\cdots\!87}{33\!\cdots\!27}a^{9}+\frac{19\!\cdots\!49}{33\!\cdots\!27}a^{8}-\frac{59\!\cdots\!04}{33\!\cdots\!27}a^{7}-\frac{30\!\cdots\!25}{33\!\cdots\!27}a^{6}+\frac{22\!\cdots\!69}{33\!\cdots\!27}a^{5}-\frac{25\!\cdots\!87}{33\!\cdots\!27}a^{4}-\frac{21\!\cdots\!77}{33\!\cdots\!27}a^{3}+\frac{49\!\cdots\!70}{33\!\cdots\!27}a^{2}-\frac{98\!\cdots\!34}{33\!\cdots\!27}a+\frac{44\!\cdots\!97}{33\!\cdots\!27}$, $\frac{16\!\cdots\!87}{33\!\cdots\!27}a^{15}-\frac{93\!\cdots\!11}{33\!\cdots\!27}a^{14}+\frac{44\!\cdots\!46}{33\!\cdots\!27}a^{13}+\frac{88\!\cdots\!05}{33\!\cdots\!27}a^{12}-\frac{17\!\cdots\!68}{33\!\cdots\!27}a^{11}-\frac{26\!\cdots\!53}{33\!\cdots\!27}a^{10}+\frac{82\!\cdots\!03}{33\!\cdots\!27}a^{9}+\frac{89\!\cdots\!79}{33\!\cdots\!27}a^{8}-\frac{40\!\cdots\!91}{33\!\cdots\!27}a^{7}-\frac{47\!\cdots\!54}{33\!\cdots\!27}a^{6}+\frac{14\!\cdots\!37}{33\!\cdots\!27}a^{5}-\frac{20\!\cdots\!31}{33\!\cdots\!27}a^{4}-\frac{81\!\cdots\!73}{33\!\cdots\!27}a^{3}+\frac{37\!\cdots\!92}{33\!\cdots\!27}a^{2}-\frac{18\!\cdots\!50}{33\!\cdots\!27}a+\frac{73\!\cdots\!17}{33\!\cdots\!27}$, $\frac{14\!\cdots\!47}{33\!\cdots\!27}a^{15}-\frac{73\!\cdots\!05}{33\!\cdots\!27}a^{14}+\frac{64\!\cdots\!42}{33\!\cdots\!27}a^{13}+\frac{74\!\cdots\!12}{33\!\cdots\!27}a^{12}-\frac{10\!\cdots\!88}{33\!\cdots\!27}a^{11}-\frac{27\!\cdots\!20}{33\!\cdots\!27}a^{10}+\frac{46\!\cdots\!82}{33\!\cdots\!27}a^{9}+\frac{11\!\cdots\!23}{33\!\cdots\!27}a^{8}-\frac{26\!\cdots\!32}{33\!\cdots\!27}a^{7}-\frac{20\!\cdots\!84}{33\!\cdots\!27}a^{6}+\frac{97\!\cdots\!45}{33\!\cdots\!27}a^{5}-\frac{10\!\cdots\!38}{33\!\cdots\!27}a^{4}-\frac{10\!\cdots\!26}{33\!\cdots\!27}a^{3}+\frac{15\!\cdots\!49}{33\!\cdots\!27}a^{2}+\frac{50\!\cdots\!68}{33\!\cdots\!27}a+\frac{51\!\cdots\!32}{33\!\cdots\!27}$, $\frac{10\!\cdots\!98}{33\!\cdots\!27}a^{15}-\frac{12\!\cdots\!83}{67\!\cdots\!54}a^{14}+\frac{47\!\cdots\!59}{33\!\cdots\!27}a^{13}+\frac{56\!\cdots\!52}{33\!\cdots\!27}a^{12}-\frac{26\!\cdots\!81}{67\!\cdots\!54}a^{11}-\frac{13\!\cdots\!13}{33\!\cdots\!27}a^{10}+\frac{58\!\cdots\!88}{33\!\cdots\!27}a^{9}+\frac{90\!\cdots\!75}{67\!\cdots\!54}a^{8}-\frac{28\!\cdots\!89}{33\!\cdots\!27}a^{7}+\frac{38\!\cdots\!64}{33\!\cdots\!27}a^{6}+\frac{18\!\cdots\!75}{67\!\cdots\!54}a^{5}-\frac{15\!\cdots\!59}{33\!\cdots\!27}a^{4}-\frac{21\!\cdots\!85}{33\!\cdots\!27}a^{3}+\frac{51\!\cdots\!13}{67\!\cdots\!54}a^{2}-\frac{15\!\cdots\!55}{33\!\cdots\!27}a+\frac{66\!\cdots\!73}{33\!\cdots\!27}$
| 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: | \( 1961937.73949 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 1961937.73949 \cdot 1}{2\cdot\sqrt{3393001956715501807791409}}\cr\approx \mathstrut & 0.212482226907 \end{aligned}\] (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 + 4*x^14 + 54*x^13 - 119*x^12 - 138*x^11 + 532*x^10 + 463*x^9 - 2582*x^8 + 196*x^7 + 8673*x^6 - 14128*x^5 - 2278*x^4 + 22735*x^3 - 14330*x^2 + 2077*x - 67)
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 + 4*x^14 + 54*x^13 - 119*x^12 - 138*x^11 + 532*x^10 + 463*x^9 - 2582*x^8 + 196*x^7 + 8673*x^6 - 14128*x^5 - 2278*x^4 + 22735*x^3 - 14330*x^2 + 2077*x - 67, 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 + 4*x^14 + 54*x^13 - 119*x^12 - 138*x^11 + 532*x^10 + 463*x^9 - 2582*x^8 + 196*x^7 + 8673*x^6 - 14128*x^5 - 2278*x^4 + 22735*x^3 - 14330*x^2 + 2077*x - 67);
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 + 4*x^14 + 54*x^13 - 119*x^12 - 138*x^11 + 532*x^10 + 463*x^9 - 2582*x^8 + 196*x^7 + 8673*x^6 - 14128*x^5 - 2278*x^4 + 22735*x^3 - 14330*x^2 + 2077*x - 67);
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
| \(\Q(\sqrt{17}) \), 4.2.329171.1, 4.4.4913.1, 4.2.19363.1, 8.4.1842010303097.2, \(\Q(\zeta_{17})^+\), 8.4.108353547241.1 |
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: | 16.0.68372792983010839504523425519489.14 |
| 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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ | |
|
\(67\)
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |