Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 116*x^14 + 177*x^13 + 7745*x^12 - 10421*x^11 - 424462*x^10 - 657538*x^9 + 8943832*x^8 + 23224335*x^7 - 106230562*x^6 - 311941107*x^5 + 640011076*x^4 + 1607965375*x^3 - 1656703698*x^2 - 2425324435*x + 1779444071)
gp: K = bnfinit(y^16 - 2*y^15 - 116*y^14 + 177*y^13 + 7745*y^12 - 10421*y^11 - 424462*y^10 - 657538*y^9 + 8943832*y^8 + 23224335*y^7 - 106230562*y^6 - 311941107*y^5 + 640011076*y^4 + 1607965375*y^3 - 1656703698*y^2 - 2425324435*y + 1779444071, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 116*x^14 + 177*x^13 + 7745*x^12 - 10421*x^11 - 424462*x^10 - 657538*x^9 + 8943832*x^8 + 23224335*x^7 - 106230562*x^6 - 311941107*x^5 + 640011076*x^4 + 1607965375*x^3 - 1656703698*x^2 - 2425324435*x + 1779444071);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 116*x^14 + 177*x^13 + 7745*x^12 - 10421*x^11 - 424462*x^10 - 657538*x^9 + 8943832*x^8 + 23224335*x^7 - 106230562*x^6 - 311941107*x^5 + 640011076*x^4 + 1607965375*x^3 - 1656703698*x^2 - 2425324435*x + 1779444071)
\( x^{16} - 2 x^{15} - 116 x^{14} + 177 x^{13} + 7745 x^{12} - 10421 x^{11} - 424462 x^{10} + \cdots + 1779444071 \)
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: |
\(4880564680360454664521443587092657321\)
\(\medspace = 11^{12}\cdot 41^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(196.35\) | 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: | $11^{3/4}41^{15/16}\approx 196.3493100489201$ | ||
| Ramified primes: |
\(11\), \(41\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{41}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{49\!\cdots\!29}a^{15}-\frac{12\!\cdots\!56}{49\!\cdots\!29}a^{14}-\frac{22\!\cdots\!66}{49\!\cdots\!29}a^{13}-\frac{58\!\cdots\!50}{49\!\cdots\!29}a^{12}-\frac{24\!\cdots\!54}{49\!\cdots\!29}a^{11}-\frac{22\!\cdots\!32}{49\!\cdots\!29}a^{10}-\frac{66\!\cdots\!90}{49\!\cdots\!29}a^{9}+\frac{14\!\cdots\!87}{49\!\cdots\!29}a^{8}+\frac{16\!\cdots\!49}{49\!\cdots\!29}a^{7}-\frac{17\!\cdots\!17}{49\!\cdots\!29}a^{6}-\frac{12\!\cdots\!28}{49\!\cdots\!29}a^{5}+\frac{72\!\cdots\!13}{49\!\cdots\!29}a^{4}-\frac{14\!\cdots\!83}{49\!\cdots\!29}a^{3}-\frac{52\!\cdots\!12}{49\!\cdots\!29}a^{2}-\frac{19\!\cdots\!53}{49\!\cdots\!29}a+\frac{24\!\cdots\!35}{22\!\cdots\!63}$
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: |
$\frac{14\!\cdots\!85}{78\!\cdots\!51}a^{15}-\frac{56\!\cdots\!02}{78\!\cdots\!51}a^{14}-\frac{15\!\cdots\!34}{78\!\cdots\!51}a^{13}+\frac{55\!\cdots\!76}{78\!\cdots\!51}a^{12}+\frac{99\!\cdots\!51}{78\!\cdots\!51}a^{11}-\frac{34\!\cdots\!87}{78\!\cdots\!51}a^{10}-\frac{53\!\cdots\!02}{78\!\cdots\!51}a^{9}+\frac{10\!\cdots\!53}{78\!\cdots\!51}a^{8}+\frac{12\!\cdots\!18}{78\!\cdots\!51}a^{7}+\frac{84\!\cdots\!06}{78\!\cdots\!51}a^{6}-\frac{16\!\cdots\!26}{78\!\cdots\!51}a^{5}-\frac{11\!\cdots\!36}{78\!\cdots\!51}a^{4}+\frac{11\!\cdots\!89}{78\!\cdots\!51}a^{3}+\frac{15\!\cdots\!86}{78\!\cdots\!51}a^{2}-\frac{25\!\cdots\!50}{78\!\cdots\!51}a+\frac{11\!\cdots\!89}{36\!\cdots\!97}$, $\frac{40\!\cdots\!12}{78\!\cdots\!51}a^{15}-\frac{16\!\cdots\!78}{78\!\cdots\!51}a^{14}-\frac{43\!\cdots\!09}{78\!\cdots\!51}a^{13}+\frac{15\!\cdots\!56}{78\!\cdots\!51}a^{12}+\frac{28\!\cdots\!50}{78\!\cdots\!51}a^{11}-\frac{97\!\cdots\!45}{78\!\cdots\!51}a^{10}-\frac{15\!\cdots\!80}{78\!\cdots\!51}a^{9}+\frac{27\!\cdots\!10}{78\!\cdots\!51}a^{8}+\frac{34\!\cdots\!60}{78\!\cdots\!51}a^{7}+\frac{23\!\cdots\!87}{78\!\cdots\!51}a^{6}-\frac{47\!\cdots\!83}{78\!\cdots\!51}a^{5}-\frac{32\!\cdots\!28}{78\!\cdots\!51}a^{4}+\frac{31\!\cdots\!94}{78\!\cdots\!51}a^{3}+\frac{42\!\cdots\!99}{78\!\cdots\!51}a^{2}-\frac{74\!\cdots\!69}{78\!\cdots\!51}a-\frac{33\!\cdots\!46}{36\!\cdots\!97}$, $\frac{23\!\cdots\!25}{78\!\cdots\!51}a^{15}-\frac{70\!\cdots\!53}{78\!\cdots\!51}a^{14}-\frac{27\!\cdots\!47}{78\!\cdots\!51}a^{13}+\frac{81\!\cdots\!87}{78\!\cdots\!51}a^{12}+\frac{17\!\cdots\!08}{78\!\cdots\!51}a^{11}-\frac{55\!\cdots\!51}{78\!\cdots\!51}a^{10}-\frac{94\!\cdots\!48}{78\!\cdots\!51}a^{9}+\frac{13\!\cdots\!74}{78\!\cdots\!51}a^{8}+\frac{20\!\cdots\!14}{78\!\cdots\!51}a^{7}+\frac{35\!\cdots\!21}{78\!\cdots\!51}a^{6}-\frac{28\!\cdots\!37}{78\!\cdots\!51}a^{5}-\frac{69\!\cdots\!58}{78\!\cdots\!51}a^{4}+\frac{19\!\cdots\!53}{78\!\cdots\!51}a^{3}-\frac{75\!\cdots\!67}{78\!\cdots\!51}a^{2}-\frac{42\!\cdots\!96}{78\!\cdots\!51}a+\frac{85\!\cdots\!56}{36\!\cdots\!97}$, $\frac{26\!\cdots\!08}{10\!\cdots\!19}a^{15}+\frac{20\!\cdots\!20}{10\!\cdots\!19}a^{14}-\frac{15\!\cdots\!27}{10\!\cdots\!19}a^{13}-\frac{21\!\cdots\!56}{10\!\cdots\!19}a^{12}+\frac{39\!\cdots\!93}{10\!\cdots\!19}a^{11}+\frac{11\!\cdots\!33}{10\!\cdots\!19}a^{10}-\frac{11\!\cdots\!90}{10\!\cdots\!19}a^{9}-\frac{91\!\cdots\!51}{10\!\cdots\!19}a^{8}-\frac{60\!\cdots\!75}{10\!\cdots\!19}a^{7}-\frac{14\!\cdots\!28}{10\!\cdots\!19}a^{6}+\frac{19\!\cdots\!99}{10\!\cdots\!19}a^{5}+\frac{16\!\cdots\!24}{10\!\cdots\!19}a^{4}+\frac{20\!\cdots\!47}{10\!\cdots\!19}a^{3}-\frac{21\!\cdots\!55}{10\!\cdots\!19}a^{2}-\frac{47\!\cdots\!02}{10\!\cdots\!19}a-\frac{62\!\cdots\!45}{46\!\cdots\!93}$, $\frac{28\!\cdots\!78}{10\!\cdots\!19}a^{15}-\frac{22\!\cdots\!84}{10\!\cdots\!19}a^{14}-\frac{28\!\cdots\!09}{10\!\cdots\!19}a^{13}+\frac{24\!\cdots\!62}{10\!\cdots\!19}a^{12}+\frac{17\!\cdots\!20}{10\!\cdots\!19}a^{11}-\frac{15\!\cdots\!11}{10\!\cdots\!19}a^{10}-\frac{91\!\cdots\!88}{10\!\cdots\!19}a^{9}+\frac{50\!\cdots\!39}{10\!\cdots\!19}a^{8}+\frac{29\!\cdots\!65}{10\!\cdots\!19}a^{7}-\frac{84\!\cdots\!12}{10\!\cdots\!19}a^{6}-\frac{50\!\cdots\!72}{10\!\cdots\!19}a^{5}+\frac{94\!\cdots\!83}{10\!\cdots\!19}a^{4}+\frac{41\!\cdots\!03}{10\!\cdots\!19}a^{3}-\frac{57\!\cdots\!41}{10\!\cdots\!19}a^{2}-\frac{10\!\cdots\!05}{10\!\cdots\!19}a+\frac{33\!\cdots\!08}{46\!\cdots\!93}$, $\frac{20\!\cdots\!94}{10\!\cdots\!19}a^{15}-\frac{22\!\cdots\!27}{10\!\cdots\!19}a^{14}-\frac{17\!\cdots\!42}{10\!\cdots\!19}a^{13}+\frac{23\!\cdots\!39}{10\!\cdots\!19}a^{12}+\frac{10\!\cdots\!56}{10\!\cdots\!19}a^{11}-\frac{14\!\cdots\!95}{10\!\cdots\!19}a^{10}-\frac{53\!\cdots\!90}{10\!\cdots\!19}a^{9}+\frac{54\!\cdots\!23}{10\!\cdots\!19}a^{8}+\frac{22\!\cdots\!64}{10\!\cdots\!19}a^{7}-\frac{97\!\cdots\!51}{10\!\cdots\!19}a^{6}-\frac{42\!\cdots\!30}{10\!\cdots\!19}a^{5}+\frac{11\!\cdots\!08}{10\!\cdots\!19}a^{4}+\frac{38\!\cdots\!36}{10\!\cdots\!19}a^{3}-\frac{64\!\cdots\!42}{10\!\cdots\!19}a^{2}-\frac{10\!\cdots\!25}{10\!\cdots\!19}a+\frac{39\!\cdots\!72}{46\!\cdots\!93}$, $\frac{10\!\cdots\!48}{17\!\cdots\!41}a^{15}-\frac{62\!\cdots\!00}{17\!\cdots\!41}a^{14}-\frac{11\!\cdots\!48}{17\!\cdots\!41}a^{13}+\frac{62\!\cdots\!85}{17\!\cdots\!41}a^{12}+\frac{76\!\cdots\!12}{17\!\cdots\!41}a^{11}-\frac{38\!\cdots\!13}{17\!\cdots\!41}a^{10}-\frac{41\!\cdots\!72}{17\!\cdots\!41}a^{9}+\frac{76\!\cdots\!88}{17\!\cdots\!41}a^{8}+\frac{12\!\cdots\!81}{17\!\cdots\!41}a^{7}+\frac{35\!\cdots\!02}{17\!\cdots\!41}a^{6}-\frac{15\!\cdots\!96}{17\!\cdots\!41}a^{5}-\frac{56\!\cdots\!52}{17\!\cdots\!41}a^{4}+\frac{10\!\cdots\!87}{17\!\cdots\!41}a^{3}-\frac{32\!\cdots\!31}{17\!\cdots\!41}a^{2}-\frac{26\!\cdots\!73}{17\!\cdots\!41}a+\frac{43\!\cdots\!02}{46\!\cdots\!93}$, $\frac{21\!\cdots\!15}{49\!\cdots\!29}a^{15}+\frac{17\!\cdots\!24}{49\!\cdots\!29}a^{14}-\frac{94\!\cdots\!45}{49\!\cdots\!29}a^{13}-\frac{60\!\cdots\!87}{49\!\cdots\!29}a^{12}+\frac{11\!\cdots\!08}{49\!\cdots\!29}a^{11}+\frac{93\!\cdots\!79}{49\!\cdots\!29}a^{10}-\frac{72\!\cdots\!36}{49\!\cdots\!29}a^{9}-\frac{23\!\cdots\!63}{49\!\cdots\!29}a^{8}-\frac{27\!\cdots\!28}{49\!\cdots\!29}a^{7}+\frac{31\!\cdots\!17}{49\!\cdots\!29}a^{6}+\frac{59\!\cdots\!45}{49\!\cdots\!29}a^{5}-\frac{19\!\cdots\!07}{49\!\cdots\!29}a^{4}-\frac{36\!\cdots\!41}{49\!\cdots\!29}a^{3}+\frac{46\!\cdots\!48}{49\!\cdots\!29}a^{2}+\frac{57\!\cdots\!71}{49\!\cdots\!29}a-\frac{20\!\cdots\!34}{22\!\cdots\!63}$, $\frac{58\!\cdots\!57}{49\!\cdots\!29}a^{15}-\frac{77\!\cdots\!83}{49\!\cdots\!29}a^{14}-\frac{68\!\cdots\!71}{49\!\cdots\!29}a^{13}+\frac{56\!\cdots\!28}{49\!\cdots\!29}a^{12}+\frac{46\!\cdots\!22}{49\!\cdots\!29}a^{11}-\frac{29\!\cdots\!59}{49\!\cdots\!29}a^{10}-\frac{25\!\cdots\!04}{49\!\cdots\!29}a^{9}-\frac{55\!\cdots\!10}{49\!\cdots\!29}a^{8}+\frac{48\!\cdots\!78}{49\!\cdots\!29}a^{7}+\frac{17\!\cdots\!42}{49\!\cdots\!29}a^{6}-\frac{51\!\cdots\!05}{49\!\cdots\!29}a^{5}-\frac{21\!\cdots\!16}{49\!\cdots\!29}a^{4}+\frac{23\!\cdots\!86}{49\!\cdots\!29}a^{3}+\frac{11\!\cdots\!05}{49\!\cdots\!29}a^{2}-\frac{24\!\cdots\!29}{49\!\cdots\!29}a-\frac{73\!\cdots\!67}{22\!\cdots\!63}$, $\frac{63\!\cdots\!64}{49\!\cdots\!29}a^{15}-\frac{34\!\cdots\!83}{49\!\cdots\!29}a^{14}-\frac{74\!\cdots\!59}{49\!\cdots\!29}a^{13}+\frac{43\!\cdots\!41}{49\!\cdots\!29}a^{12}+\frac{49\!\cdots\!85}{49\!\cdots\!29}a^{11}+\frac{55\!\cdots\!74}{49\!\cdots\!29}a^{10}-\frac{26\!\cdots\!43}{49\!\cdots\!29}a^{9}-\frac{81\!\cdots\!77}{49\!\cdots\!29}a^{8}+\frac{45\!\cdots\!65}{49\!\cdots\!29}a^{7}+\frac{21\!\cdots\!41}{49\!\cdots\!29}a^{6}-\frac{36\!\cdots\!68}{49\!\cdots\!29}a^{5}-\frac{25\!\cdots\!29}{49\!\cdots\!29}a^{4}+\frac{40\!\cdots\!00}{49\!\cdots\!29}a^{3}+\frac{10\!\cdots\!66}{49\!\cdots\!29}a^{2}+\frac{52\!\cdots\!64}{49\!\cdots\!29}a-\frac{35\!\cdots\!74}{22\!\cdots\!63}$, $\frac{96\!\cdots\!23}{49\!\cdots\!29}a^{15}-\frac{36\!\cdots\!39}{49\!\cdots\!29}a^{14}-\frac{10\!\cdots\!03}{49\!\cdots\!29}a^{13}+\frac{35\!\cdots\!59}{49\!\cdots\!29}a^{12}+\frac{68\!\cdots\!55}{49\!\cdots\!29}a^{11}-\frac{22\!\cdots\!72}{49\!\cdots\!29}a^{10}-\frac{37\!\cdots\!09}{49\!\cdots\!29}a^{9}+\frac{19\!\cdots\!46}{49\!\cdots\!29}a^{8}+\frac{85\!\cdots\!31}{49\!\cdots\!29}a^{7}+\frac{72\!\cdots\!88}{49\!\cdots\!29}a^{6}-\frac{11\!\cdots\!31}{49\!\cdots\!29}a^{5}-\frac{97\!\cdots\!85}{49\!\cdots\!29}a^{4}+\frac{78\!\cdots\!73}{49\!\cdots\!29}a^{3}+\frac{15\!\cdots\!07}{49\!\cdots\!29}a^{2}-\frac{18\!\cdots\!81}{49\!\cdots\!29}a+\frac{44\!\cdots\!39}{22\!\cdots\!63}$
| 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: | \( 2773762487010 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 2773762487010 \cdot 1}{2\cdot\sqrt{4880564680360454664521443587092657321}}\cr\approx \mathstrut & 0.250474591564688 \end{aligned}\] (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 - 116*x^14 + 177*x^13 + 7745*x^12 - 10421*x^11 - 424462*x^10 - 657538*x^9 + 8943832*x^8 + 23224335*x^7 - 106230562*x^6 - 311941107*x^5 + 640011076*x^4 + 1607965375*x^3 - 1656703698*x^2 - 2425324435*x + 1779444071)
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 - 116*x^14 + 177*x^13 + 7745*x^12 - 10421*x^11 - 424462*x^10 - 657538*x^9 + 8943832*x^8 + 23224335*x^7 - 106230562*x^6 - 311941107*x^5 + 640011076*x^4 + 1607965375*x^3 - 1656703698*x^2 - 2425324435*x + 1779444071, 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 - 116*x^14 + 177*x^13 + 7745*x^12 - 10421*x^11 - 424462*x^10 - 657538*x^9 + 8943832*x^8 + 23224335*x^7 - 106230562*x^6 - 311941107*x^5 + 640011076*x^4 + 1607965375*x^3 - 1656703698*x^2 - 2425324435*x + 1779444071);
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 - 116*x^14 + 177*x^13 + 7745*x^12 - 10421*x^11 - 424462*x^10 - 657538*x^9 + 8943832*x^8 + 23224335*x^7 - 106230562*x^6 - 311941107*x^5 + 640011076*x^4 + 1607965375*x^3 - 1656703698*x^2 - 2425324435*x + 1779444071);
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
$\OD_{32}$ (as 16T22):
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 $C_{16} : C_2$ |
| Character table for $C_{16} : C_2$ |
Intermediate fields
| \(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.2851397323891721.1 |
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 |
| 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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | $16$ | R | $16$ | $16$ | $16$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | $16$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.1.0.1}{1} }^{16}$ | R | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | $16$ | $16$ | ${\href{/padicField/59.1.0.1}{1} }^{16}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.16.12.3 | $x^{16} - 330 x^{12} + 92444 x^{8} - 692120 x^{4} + 117128$ | $4$ | $4$ | $12$ | $C_{16} : C_2$ | $[\ ]_{4}^{8}$ |
|
\(41\)
| 41.16.15.1 | $x^{16} + 164$ | $16$ | $1$ | $15$ | $C_{16} : C_2$ | $[\ ]_{16}^{2}$ |