Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 15*x^14 - 9*x^13 + 67*x^12 + 28*x^11 + 318*x^10 - 337*x^9 - 1061*x^8 + 899*x^7 - 457*x^6 + 1394*x^5 - 683*x^4 - 197*x^3 + 80*x^2 - 33*x + 9)
gp: K = bnfinit(y^16 - 3*y^15 - 15*y^14 - 9*y^13 + 67*y^12 + 28*y^11 + 318*y^10 - 337*y^9 - 1061*y^8 + 899*y^7 - 457*y^6 + 1394*y^5 - 683*y^4 - 197*y^3 + 80*y^2 - 33*y + 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 - 15*x^14 - 9*x^13 + 67*x^12 + 28*x^11 + 318*x^10 - 337*x^9 - 1061*x^8 + 899*x^7 - 457*x^6 + 1394*x^5 - 683*x^4 - 197*x^3 + 80*x^2 - 33*x + 9);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 - 15*x^14 - 9*x^13 + 67*x^12 + 28*x^11 + 318*x^10 - 337*x^9 - 1061*x^8 + 899*x^7 - 457*x^6 + 1394*x^5 - 683*x^4 - 197*x^3 + 80*x^2 - 33*x + 9)
\( x^{16} - 3 x^{15} - 15 x^{14} - 9 x^{13} + 67 x^{12} + 28 x^{11} + 318 x^{10} - 337 x^{9} - 1061 x^{8} + \cdots + 9 \)
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: |
\(27466182620559501230159449\)
\(\medspace = 13^{14}\cdot 17^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(38.90\) | 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: | $13^{7/8}17^{1/2}\approx 38.897787010151276$ | ||
| Ramified primes: |
\(13\), \(17\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{13}-\frac{1}{9}a^{11}-\frac{1}{9}a^{9}-\frac{1}{9}a^{8}+\frac{2}{9}a^{7}-\frac{2}{9}a^{6}+\frac{4}{9}a^{5}-\frac{4}{9}a^{4}+\frac{1}{3}a^{3}-\frac{4}{9}a^{2}$, $\frac{1}{11\!\cdots\!33}a^{15}+\frac{32\!\cdots\!07}{10\!\cdots\!03}a^{14}-\frac{24\!\cdots\!59}{11\!\cdots\!33}a^{13}+\frac{16\!\cdots\!17}{11\!\cdots\!33}a^{12}-\frac{73\!\cdots\!35}{11\!\cdots\!33}a^{11}-\frac{21\!\cdots\!39}{11\!\cdots\!33}a^{10}-\frac{29\!\cdots\!92}{11\!\cdots\!33}a^{9}+\frac{28\!\cdots\!48}{19\!\cdots\!53}a^{8}-\frac{13\!\cdots\!69}{35\!\cdots\!01}a^{7}+\frac{31\!\cdots\!71}{11\!\cdots\!33}a^{6}-\frac{54\!\cdots\!22}{13\!\cdots\!37}a^{5}+\frac{65\!\cdots\!40}{11\!\cdots\!33}a^{4}+\frac{29\!\cdots\!17}{20\!\cdots\!57}a^{3}+\frac{26\!\cdots\!23}{11\!\cdots\!33}a^{2}+\frac{32\!\cdots\!83}{39\!\cdots\!11}a-\frac{12\!\cdots\!16}{13\!\cdots\!37}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
$\frac{28\!\cdots\!37}{43\!\cdots\!79}a^{15}-\frac{63\!\cdots\!22}{35\!\cdots\!01}a^{14}-\frac{40\!\cdots\!90}{39\!\cdots\!11}a^{13}-\frac{37\!\cdots\!47}{43\!\cdots\!79}a^{12}+\frac{15\!\cdots\!81}{39\!\cdots\!11}a^{11}+\frac{36\!\cdots\!16}{13\!\cdots\!37}a^{10}+\frac{84\!\cdots\!94}{39\!\cdots\!11}a^{9}-\frac{10\!\cdots\!91}{64\!\cdots\!51}a^{8}-\frac{25\!\cdots\!99}{35\!\cdots\!01}a^{7}+\frac{16\!\cdots\!16}{39\!\cdots\!11}a^{6}-\frac{77\!\cdots\!48}{39\!\cdots\!11}a^{5}+\frac{31\!\cdots\!87}{39\!\cdots\!11}a^{4}-\frac{18\!\cdots\!57}{76\!\cdots\!91}a^{3}-\frac{79\!\cdots\!79}{39\!\cdots\!11}a^{2}+\frac{59\!\cdots\!36}{13\!\cdots\!37}a+\frac{83\!\cdots\!92}{43\!\cdots\!79}$, $\frac{26\!\cdots\!98}{11\!\cdots\!33}a^{15}-\frac{51\!\cdots\!32}{10\!\cdots\!03}a^{14}-\frac{44\!\cdots\!90}{11\!\cdots\!33}a^{13}-\frac{65\!\cdots\!06}{11\!\cdots\!33}a^{12}+\frac{10\!\cdots\!75}{11\!\cdots\!33}a^{11}+\frac{14\!\cdots\!27}{11\!\cdots\!33}a^{10}+\frac{10\!\cdots\!07}{11\!\cdots\!33}a^{9}+\frac{11\!\cdots\!86}{19\!\cdots\!53}a^{8}-\frac{72\!\cdots\!48}{35\!\cdots\!01}a^{7}+\frac{23\!\cdots\!84}{11\!\cdots\!33}a^{6}-\frac{74\!\cdots\!58}{39\!\cdots\!11}a^{5}+\frac{20\!\cdots\!69}{11\!\cdots\!33}a^{4}-\frac{91\!\cdots\!07}{20\!\cdots\!57}a^{3}+\frac{18\!\cdots\!09}{11\!\cdots\!33}a^{2}+\frac{18\!\cdots\!52}{39\!\cdots\!11}a-\frac{34\!\cdots\!62}{13\!\cdots\!37}$, $\frac{55\!\cdots\!39}{11\!\cdots\!33}a^{15}-\frac{16\!\cdots\!98}{10\!\cdots\!03}a^{14}-\frac{89\!\cdots\!85}{11\!\cdots\!33}a^{13}-\frac{95\!\cdots\!90}{11\!\cdots\!33}a^{12}+\frac{56\!\cdots\!89}{11\!\cdots\!33}a^{11}+\frac{29\!\cdots\!39}{11\!\cdots\!33}a^{10}+\frac{11\!\cdots\!62}{11\!\cdots\!33}a^{9}-\frac{51\!\cdots\!97}{19\!\cdots\!53}a^{8}-\frac{86\!\cdots\!24}{11\!\cdots\!67}a^{7}+\frac{62\!\cdots\!57}{11\!\cdots\!33}a^{6}+\frac{29\!\cdots\!66}{39\!\cdots\!11}a^{5}+\frac{79\!\cdots\!59}{11\!\cdots\!33}a^{4}-\frac{21\!\cdots\!37}{20\!\cdots\!57}a^{3}+\frac{13\!\cdots\!05}{11\!\cdots\!33}a^{2}-\frac{34\!\cdots\!33}{39\!\cdots\!11}a-\frac{86\!\cdots\!11}{13\!\cdots\!37}$, $\frac{10\!\cdots\!22}{39\!\cdots\!11}a^{15}-\frac{31\!\cdots\!10}{35\!\cdots\!01}a^{14}-\frac{17\!\cdots\!45}{43\!\cdots\!79}a^{13}-\frac{48\!\cdots\!06}{39\!\cdots\!11}a^{12}+\frac{27\!\cdots\!93}{13\!\cdots\!37}a^{11}+\frac{26\!\cdots\!00}{39\!\cdots\!11}a^{10}+\frac{31\!\cdots\!10}{39\!\cdots\!11}a^{9}-\frac{77\!\cdots\!54}{64\!\cdots\!51}a^{8}-\frac{11\!\cdots\!32}{35\!\cdots\!01}a^{7}+\frac{11\!\cdots\!78}{39\!\cdots\!11}a^{6}-\frac{21\!\cdots\!64}{39\!\cdots\!11}a^{5}+\frac{18\!\cdots\!85}{43\!\cdots\!79}a^{4}-\frac{13\!\cdots\!66}{69\!\cdots\!19}a^{3}-\frac{52\!\cdots\!63}{43\!\cdots\!79}a^{2}-\frac{78\!\cdots\!75}{13\!\cdots\!37}a-\frac{55\!\cdots\!48}{43\!\cdots\!79}$, $\frac{11\!\cdots\!84}{39\!\cdots\!11}a^{15}-\frac{11\!\cdots\!67}{11\!\cdots\!67}a^{14}-\frac{16\!\cdots\!15}{39\!\cdots\!11}a^{13}-\frac{19\!\cdots\!37}{39\!\cdots\!11}a^{12}+\frac{93\!\cdots\!41}{39\!\cdots\!11}a^{11}+\frac{22\!\cdots\!51}{39\!\cdots\!11}a^{10}+\frac{10\!\cdots\!08}{13\!\cdots\!37}a^{9}-\frac{32\!\cdots\!75}{21\!\cdots\!17}a^{8}-\frac{12\!\cdots\!19}{35\!\cdots\!01}a^{7}+\frac{47\!\cdots\!54}{13\!\cdots\!37}a^{6}-\frac{14\!\cdots\!65}{39\!\cdots\!11}a^{5}+\frac{20\!\cdots\!80}{39\!\cdots\!11}a^{4}-\frac{15\!\cdots\!70}{69\!\cdots\!19}a^{3}-\frac{50\!\cdots\!42}{39\!\cdots\!11}a^{2}-\frac{40\!\cdots\!33}{43\!\cdots\!79}a-\frac{86\!\cdots\!36}{43\!\cdots\!79}$, $\frac{21164790259}{5663045130237}a^{15}+\frac{3564137023}{514822284567}a^{14}-\frac{566119371278}{5663045130237}a^{13}-\frac{1834440313489}{5663045130237}a^{12}-\frac{543658015726}{5663045130237}a^{11}+\frac{5652136786022}{5663045130237}a^{10}+\frac{11577580746034}{5663045130237}a^{9}+\frac{30240279273430}{5663045130237}a^{8}-\frac{329793847151}{57202476063}a^{7}-\frac{80874356709907}{5663045130237}a^{6}+\frac{8128484247547}{1887681710079}a^{5}-\frac{41658926721961}{5663045130237}a^{4}+\frac{87975668049044}{5663045130237}a^{3}-\frac{27802968150811}{5663045130237}a^{2}+\frac{2651437946587}{1887681710079}a+\frac{3514699948}{629227236693}$, $\frac{36\!\cdots\!97}{11\!\cdots\!33}a^{15}-\frac{86\!\cdots\!23}{10\!\cdots\!03}a^{14}-\frac{58\!\cdots\!17}{11\!\cdots\!33}a^{13}-\frac{52\!\cdots\!19}{11\!\cdots\!33}a^{12}+\frac{23\!\cdots\!03}{11\!\cdots\!33}a^{11}+\frac{19\!\cdots\!98}{11\!\cdots\!33}a^{10}+\frac{11\!\cdots\!36}{11\!\cdots\!33}a^{9}-\frac{12\!\cdots\!92}{19\!\cdots\!53}a^{8}-\frac{13\!\cdots\!09}{35\!\cdots\!01}a^{7}+\frac{18\!\cdots\!71}{11\!\cdots\!33}a^{6}-\frac{16\!\cdots\!47}{13\!\cdots\!37}a^{5}+\frac{42\!\cdots\!46}{11\!\cdots\!33}a^{4}-\frac{91\!\cdots\!14}{20\!\cdots\!57}a^{3}-\frac{19\!\cdots\!00}{11\!\cdots\!33}a^{2}+\frac{31\!\cdots\!58}{39\!\cdots\!11}a+\frac{97\!\cdots\!62}{13\!\cdots\!37}$, $\frac{38\!\cdots\!15}{39\!\cdots\!11}a^{15}-\frac{10\!\cdots\!42}{35\!\cdots\!01}a^{14}-\frac{56\!\cdots\!22}{39\!\cdots\!11}a^{13}-\frac{29\!\cdots\!94}{39\!\cdots\!11}a^{12}+\frac{25\!\cdots\!22}{39\!\cdots\!11}a^{11}+\frac{85\!\cdots\!70}{39\!\cdots\!11}a^{10}+\frac{12\!\cdots\!99}{39\!\cdots\!11}a^{9}-\frac{23\!\cdots\!97}{64\!\cdots\!51}a^{8}-\frac{11\!\cdots\!28}{11\!\cdots\!67}a^{7}+\frac{37\!\cdots\!74}{39\!\cdots\!11}a^{6}-\frac{26\!\cdots\!94}{43\!\cdots\!79}a^{5}+\frac{60\!\cdots\!05}{39\!\cdots\!11}a^{4}-\frac{51\!\cdots\!20}{69\!\cdots\!19}a^{3}-\frac{47\!\cdots\!23}{39\!\cdots\!11}a^{2}-\frac{49\!\cdots\!69}{13\!\cdots\!37}a-\frac{78\!\cdots\!36}{43\!\cdots\!79}$, $\frac{33\!\cdots\!90}{39\!\cdots\!11}a^{15}-\frac{30\!\cdots\!97}{35\!\cdots\!01}a^{14}-\frac{69\!\cdots\!29}{39\!\cdots\!11}a^{13}-\frac{13\!\cdots\!97}{39\!\cdots\!11}a^{12}+\frac{15\!\cdots\!85}{39\!\cdots\!11}a^{11}+\frac{49\!\cdots\!79}{39\!\cdots\!11}a^{10}+\frac{12\!\cdots\!56}{39\!\cdots\!11}a^{9}+\frac{17\!\cdots\!91}{64\!\cdots\!51}a^{8}-\frac{15\!\cdots\!83}{11\!\cdots\!67}a^{7}-\frac{35\!\cdots\!02}{39\!\cdots\!11}a^{6}+\frac{51\!\cdots\!21}{43\!\cdots\!79}a^{5}-\frac{25\!\cdots\!30}{39\!\cdots\!11}a^{4}+\frac{35\!\cdots\!49}{69\!\cdots\!19}a^{3}-\frac{14\!\cdots\!98}{39\!\cdots\!11}a^{2}+\frac{22\!\cdots\!50}{43\!\cdots\!79}a+\frac{58\!\cdots\!65}{43\!\cdots\!79}$, $\frac{15\!\cdots\!78}{11\!\cdots\!33}a^{15}-\frac{14\!\cdots\!22}{10\!\cdots\!03}a^{14}-\frac{28\!\cdots\!16}{11\!\cdots\!33}a^{13}-\frac{62\!\cdots\!43}{11\!\cdots\!33}a^{12}+\frac{15\!\cdots\!58}{11\!\cdots\!33}a^{11}+\frac{11\!\cdots\!21}{11\!\cdots\!33}a^{10}+\frac{57\!\cdots\!13}{11\!\cdots\!33}a^{9}+\frac{87\!\cdots\!02}{19\!\cdots\!53}a^{8}-\frac{39\!\cdots\!01}{35\!\cdots\!01}a^{7}-\frac{88\!\cdots\!02}{11\!\cdots\!33}a^{6}+\frac{54\!\cdots\!94}{13\!\cdots\!37}a^{5}+\frac{77\!\cdots\!21}{11\!\cdots\!33}a^{4}+\frac{15\!\cdots\!92}{20\!\cdots\!57}a^{3}-\frac{39\!\cdots\!09}{11\!\cdots\!33}a^{2}-\frac{49\!\cdots\!32}{39\!\cdots\!11}a+\frac{15\!\cdots\!83}{13\!\cdots\!37}$, $\frac{10\!\cdots\!04}{11\!\cdots\!33}a^{15}-\frac{24\!\cdots\!22}{10\!\cdots\!03}a^{14}-\frac{16\!\cdots\!31}{11\!\cdots\!33}a^{13}-\frac{15\!\cdots\!71}{11\!\cdots\!33}a^{12}+\frac{66\!\cdots\!56}{11\!\cdots\!33}a^{11}+\frac{55\!\cdots\!82}{11\!\cdots\!33}a^{10}+\frac{34\!\cdots\!87}{11\!\cdots\!33}a^{9}-\frac{37\!\cdots\!34}{19\!\cdots\!53}a^{8}-\frac{37\!\cdots\!72}{35\!\cdots\!01}a^{7}+\frac{51\!\cdots\!37}{11\!\cdots\!33}a^{6}-\frac{36\!\cdots\!51}{39\!\cdots\!11}a^{5}+\frac{13\!\cdots\!80}{11\!\cdots\!33}a^{4}-\frac{31\!\cdots\!13}{20\!\cdots\!57}a^{3}-\frac{46\!\cdots\!25}{11\!\cdots\!33}a^{2}-\frac{12\!\cdots\!26}{39\!\cdots\!11}a+\frac{13\!\cdots\!30}{13\!\cdots\!37}$
| 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: | \( 13224148.4074 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 13224148.4074 \cdot 1}{2\cdot\sqrt{27466182620559501230159449}}\cr\approx \mathstrut & 0.503382375460 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 15*x^14 - 9*x^13 + 67*x^12 + 28*x^11 + 318*x^10 - 337*x^9 - 1061*x^8 + 899*x^7 - 457*x^6 + 1394*x^5 - 683*x^4 - 197*x^3 + 80*x^2 - 33*x + 9)
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 - 3*x^15 - 15*x^14 - 9*x^13 + 67*x^12 + 28*x^11 + 318*x^10 - 337*x^9 - 1061*x^8 + 899*x^7 - 457*x^6 + 1394*x^5 - 683*x^4 - 197*x^3 + 80*x^2 - 33*x + 9, 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 - 3*x^15 - 15*x^14 - 9*x^13 + 67*x^12 + 28*x^11 + 318*x^10 - 337*x^9 - 1061*x^8 + 899*x^7 - 457*x^6 + 1394*x^5 - 683*x^4 - 197*x^3 + 80*x^2 - 33*x + 9);
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 - 3*x^15 - 15*x^14 - 9*x^13 + 67*x^12 + 28*x^11 + 318*x^10 - 337*x^9 - 1061*x^8 + 899*x^7 - 457*x^6 + 1394*x^5 - 683*x^4 - 197*x^3 + 80*x^2 - 33*x + 9);
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
| \(\Q(\sqrt{13}) \), \(\Q(\sqrt{221}) \), \(\Q(\sqrt{17}) \), 4.4.37349.1 x2, 4.4.634933.1 x2, \(\Q(\sqrt{13}, \sqrt{17})\), 8.4.5240818888357.2, 8.4.5240818888357.1, 8.8.403139914489.1 |
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: | 8.4.5240818888357.2, 8.4.5240818888357.1 |
| Degree 16 sibling: | 16.0.95038694188787201488441.2 |
| Minimal sibling: | 8.4.5240818888357.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{8}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | R | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.8.7.1 | $x^{8} + 52$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 13.8.7.1 | $x^{8} + 52$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
|
\(17\)
| 17.8.4.1 | $x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 17.8.4.1 | $x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |