Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 16*x^14 + 33*x^13 - 16*x^12 - 171*x^11 + 698*x^10 - 1208*x^9 + 1089*x^8 + 1529*x^7 - 8176*x^6 + 18614*x^5 - 29307*x^4 + 31602*x^3 - 22983*x^2 + 8312*x - 1087)
gp: K = bnfinit(y^16 - y^15 - 16*y^14 + 33*y^13 - 16*y^12 - 171*y^11 + 698*y^10 - 1208*y^9 + 1089*y^8 + 1529*y^7 - 8176*y^6 + 18614*y^5 - 29307*y^4 + 31602*y^3 - 22983*y^2 + 8312*y - 1087, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 16*x^14 + 33*x^13 - 16*x^12 - 171*x^11 + 698*x^10 - 1208*x^9 + 1089*x^8 + 1529*x^7 - 8176*x^6 + 18614*x^5 - 29307*x^4 + 31602*x^3 - 22983*x^2 + 8312*x - 1087);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 - 16*x^14 + 33*x^13 - 16*x^12 - 171*x^11 + 698*x^10 - 1208*x^9 + 1089*x^8 + 1529*x^7 - 8176*x^6 + 18614*x^5 - 29307*x^4 + 31602*x^3 - 22983*x^2 + 8312*x - 1087)
\( x^{16} - x^{15} - 16 x^{14} + 33 x^{13} - 16 x^{12} - 171 x^{11} + 698 x^{10} - 1208 x^{9} + 1089 x^{8} + \cdots - 1087 \)
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: |
\(9786054790924809742924193\)
\(\medspace = 17^{15}\cdot 43^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(36.47\) | 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: | $17^{15/16}43^{1/2}\approx 93.38563830116195$ | ||
| Ramified primes: |
\(17\), \(43\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{6767}a^{14}+\frac{2573}{6767}a^{13}+\frac{1933}{6767}a^{12}+\frac{2484}{6767}a^{11}+\frac{2130}{6767}a^{10}+\frac{3057}{6767}a^{9}+\frac{515}{6767}a^{8}-\frac{2610}{6767}a^{7}+\frac{1580}{6767}a^{6}-\frac{2845}{6767}a^{5}-\frac{2985}{6767}a^{4}-\frac{913}{6767}a^{3}+\frac{2761}{6767}a^{2}+\frac{2055}{6767}a-\frac{1135}{6767}$, $\frac{1}{14\!\cdots\!97}a^{15}-\frac{45\!\cdots\!26}{14\!\cdots\!97}a^{14}-\frac{17\!\cdots\!45}{14\!\cdots\!97}a^{13}+\frac{29\!\cdots\!81}{14\!\cdots\!97}a^{12}-\frac{62\!\cdots\!19}{14\!\cdots\!97}a^{11}-\frac{62\!\cdots\!75}{14\!\cdots\!97}a^{10}+\frac{22\!\cdots\!86}{14\!\cdots\!97}a^{9}-\frac{39\!\cdots\!99}{14\!\cdots\!97}a^{8}+\frac{19\!\cdots\!98}{14\!\cdots\!97}a^{7}+\frac{67\!\cdots\!30}{14\!\cdots\!97}a^{6}+\frac{57\!\cdots\!83}{14\!\cdots\!97}a^{5}-\frac{27\!\cdots\!65}{14\!\cdots\!97}a^{4}+\frac{16\!\cdots\!22}{14\!\cdots\!97}a^{3}+\frac{47\!\cdots\!68}{14\!\cdots\!97}a^{2}+\frac{54\!\cdots\!97}{14\!\cdots\!97}a-\frac{16\!\cdots\!23}{14\!\cdots\!97}$
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: |
$\frac{61\!\cdots\!42}{14\!\cdots\!97}a^{15}-\frac{56\!\cdots\!57}{14\!\cdots\!97}a^{14}-\frac{99\!\cdots\!12}{14\!\cdots\!97}a^{13}+\frac{19\!\cdots\!08}{14\!\cdots\!97}a^{12}-\frac{70\!\cdots\!81}{14\!\cdots\!97}a^{11}-\frac{10\!\cdots\!34}{14\!\cdots\!97}a^{10}+\frac{41\!\cdots\!42}{14\!\cdots\!97}a^{9}-\frac{69\!\cdots\!05}{14\!\cdots\!97}a^{8}+\frac{55\!\cdots\!42}{14\!\cdots\!97}a^{7}+\frac{10\!\cdots\!94}{14\!\cdots\!97}a^{6}-\frac{49\!\cdots\!40}{14\!\cdots\!97}a^{5}+\frac{10\!\cdots\!17}{14\!\cdots\!97}a^{4}-\frac{16\!\cdots\!54}{14\!\cdots\!97}a^{3}+\frac{16\!\cdots\!53}{14\!\cdots\!97}a^{2}-\frac{10\!\cdots\!32}{14\!\cdots\!97}a+\frac{22\!\cdots\!76}{14\!\cdots\!97}$, $\frac{93\!\cdots\!46}{14\!\cdots\!97}a^{15}-\frac{51\!\cdots\!15}{14\!\cdots\!97}a^{14}-\frac{15\!\cdots\!47}{14\!\cdots\!97}a^{13}+\frac{24\!\cdots\!73}{14\!\cdots\!97}a^{12}-\frac{69\!\cdots\!71}{14\!\cdots\!97}a^{11}-\frac{16\!\cdots\!59}{14\!\cdots\!97}a^{10}+\frac{58\!\cdots\!93}{14\!\cdots\!97}a^{9}-\frac{90\!\cdots\!49}{14\!\cdots\!97}a^{8}+\frac{66\!\cdots\!02}{14\!\cdots\!97}a^{7}+\frac{16\!\cdots\!29}{14\!\cdots\!97}a^{6}-\frac{69\!\cdots\!62}{14\!\cdots\!97}a^{5}+\frac{14\!\cdots\!30}{14\!\cdots\!97}a^{4}-\frac{21\!\cdots\!39}{14\!\cdots\!97}a^{3}+\frac{21\!\cdots\!88}{14\!\cdots\!97}a^{2}-\frac{13\!\cdots\!28}{14\!\cdots\!97}a+\frac{27\!\cdots\!13}{14\!\cdots\!97}$, $\frac{50\!\cdots\!02}{14\!\cdots\!97}a^{15}-\frac{31\!\cdots\!94}{14\!\cdots\!97}a^{14}-\frac{83\!\cdots\!25}{14\!\cdots\!97}a^{13}+\frac{13\!\cdots\!18}{14\!\cdots\!97}a^{12}-\frac{28\!\cdots\!45}{14\!\cdots\!97}a^{11}-\frac{86\!\cdots\!77}{14\!\cdots\!97}a^{10}+\frac{32\!\cdots\!85}{14\!\cdots\!97}a^{9}-\frac{48\!\cdots\!20}{14\!\cdots\!97}a^{8}+\frac{37\!\cdots\!57}{14\!\cdots\!97}a^{7}+\frac{91\!\cdots\!36}{14\!\cdots\!97}a^{6}-\frac{38\!\cdots\!22}{14\!\cdots\!97}a^{5}+\frac{79\!\cdots\!42}{14\!\cdots\!97}a^{4}-\frac{11\!\cdots\!41}{14\!\cdots\!97}a^{3}+\frac{11\!\cdots\!77}{14\!\cdots\!97}a^{2}-\frac{72\!\cdots\!74}{14\!\cdots\!97}a+\frac{13\!\cdots\!16}{14\!\cdots\!97}$, $\frac{91\!\cdots\!53}{14\!\cdots\!97}a^{15}-\frac{46\!\cdots\!45}{14\!\cdots\!97}a^{14}-\frac{14\!\cdots\!68}{14\!\cdots\!97}a^{13}+\frac{22\!\cdots\!95}{14\!\cdots\!97}a^{12}-\frac{31\!\cdots\!92}{14\!\cdots\!97}a^{11}-\frac{15\!\cdots\!57}{14\!\cdots\!97}a^{10}+\frac{55\!\cdots\!85}{14\!\cdots\!97}a^{9}-\frac{83\!\cdots\!58}{14\!\cdots\!97}a^{8}+\frac{60\!\cdots\!83}{14\!\cdots\!97}a^{7}+\frac{16\!\cdots\!43}{14\!\cdots\!97}a^{6}-\frac{66\!\cdots\!04}{14\!\cdots\!97}a^{5}+\frac{13\!\cdots\!95}{14\!\cdots\!97}a^{4}-\frac{20\!\cdots\!69}{14\!\cdots\!97}a^{3}+\frac{19\!\cdots\!89}{14\!\cdots\!97}a^{2}-\frac{12\!\cdots\!23}{14\!\cdots\!97}a+\frac{25\!\cdots\!52}{14\!\cdots\!97}$, $\frac{51\!\cdots\!28}{14\!\cdots\!97}a^{15}-\frac{11\!\cdots\!67}{14\!\cdots\!97}a^{14}-\frac{82\!\cdots\!46}{14\!\cdots\!97}a^{13}+\frac{10\!\cdots\!71}{14\!\cdots\!97}a^{12}-\frac{78\!\cdots\!81}{14\!\cdots\!97}a^{11}-\frac{84\!\cdots\!32}{14\!\cdots\!97}a^{10}+\frac{29\!\cdots\!51}{14\!\cdots\!97}a^{9}-\frac{40\!\cdots\!53}{14\!\cdots\!97}a^{8}+\frac{28\!\cdots\!57}{14\!\cdots\!97}a^{7}+\frac{91\!\cdots\!34}{14\!\cdots\!97}a^{6}-\frac{34\!\cdots\!42}{14\!\cdots\!97}a^{5}+\frac{69\!\cdots\!75}{14\!\cdots\!97}a^{4}-\frac{10\!\cdots\!28}{14\!\cdots\!97}a^{3}+\frac{94\!\cdots\!64}{14\!\cdots\!97}a^{2}-\frac{57\!\cdots\!15}{14\!\cdots\!97}a+\frac{13\!\cdots\!66}{14\!\cdots\!97}$, $\frac{16\!\cdots\!78}{14\!\cdots\!97}a^{15}+\frac{82\!\cdots\!53}{14\!\cdots\!97}a^{14}-\frac{17\!\cdots\!37}{14\!\cdots\!97}a^{13}+\frac{20\!\cdots\!78}{14\!\cdots\!97}a^{12}-\frac{14\!\cdots\!17}{14\!\cdots\!97}a^{11}-\frac{23\!\cdots\!73}{14\!\cdots\!97}a^{10}+\frac{10\!\cdots\!31}{14\!\cdots\!97}a^{9}-\frac{22\!\cdots\!94}{14\!\cdots\!97}a^{8}+\frac{30\!\cdots\!45}{14\!\cdots\!97}a^{7}+\frac{21\!\cdots\!61}{14\!\cdots\!97}a^{6}-\frac{14\!\cdots\!09}{14\!\cdots\!97}a^{5}+\frac{33\!\cdots\!12}{14\!\cdots\!97}a^{4}-\frac{50\!\cdots\!64}{14\!\cdots\!97}a^{3}+\frac{54\!\cdots\!48}{14\!\cdots\!97}a^{2}-\frac{36\!\cdots\!38}{14\!\cdots\!97}a+\frac{20\!\cdots\!79}{14\!\cdots\!97}$, $\frac{30\!\cdots\!85}{14\!\cdots\!97}a^{15}-\frac{97\!\cdots\!15}{14\!\cdots\!97}a^{14}-\frac{50\!\cdots\!35}{14\!\cdots\!97}a^{13}+\frac{63\!\cdots\!34}{14\!\cdots\!97}a^{12}+\frac{17\!\cdots\!25}{14\!\cdots\!97}a^{11}-\frac{49\!\cdots\!68}{14\!\cdots\!97}a^{10}+\frac{17\!\cdots\!77}{14\!\cdots\!97}a^{9}-\frac{22\!\cdots\!20}{14\!\cdots\!97}a^{8}+\frac{13\!\cdots\!41}{14\!\cdots\!97}a^{7}+\frac{56\!\cdots\!51}{14\!\cdots\!97}a^{6}-\frac{20\!\cdots\!44}{14\!\cdots\!97}a^{5}+\frac{39\!\cdots\!45}{14\!\cdots\!97}a^{4}-\frac{56\!\cdots\!28}{14\!\cdots\!97}a^{3}+\frac{49\!\cdots\!49}{14\!\cdots\!97}a^{2}-\frac{28\!\cdots\!50}{14\!\cdots\!97}a+\frac{49\!\cdots\!23}{14\!\cdots\!97}$, $\frac{61\!\cdots\!72}{14\!\cdots\!97}a^{15}+\frac{85\!\cdots\!38}{14\!\cdots\!97}a^{14}-\frac{84\!\cdots\!07}{14\!\cdots\!97}a^{13}-\frac{88\!\cdots\!94}{14\!\cdots\!97}a^{12}-\frac{20\!\cdots\!21}{14\!\cdots\!97}a^{11}-\frac{11\!\cdots\!13}{14\!\cdots\!97}a^{10}+\frac{16\!\cdots\!39}{14\!\cdots\!97}a^{9}-\frac{21\!\cdots\!68}{14\!\cdots\!97}a^{8}-\frac{10\!\cdots\!68}{22\!\cdots\!91}a^{7}+\frac{11\!\cdots\!04}{14\!\cdots\!97}a^{6}-\frac{23\!\cdots\!11}{14\!\cdots\!97}a^{5}+\frac{44\!\cdots\!22}{14\!\cdots\!97}a^{4}-\frac{42\!\cdots\!75}{14\!\cdots\!97}a^{3}+\frac{31\!\cdots\!20}{14\!\cdots\!97}a^{2}+\frac{15\!\cdots\!16}{14\!\cdots\!97}a-\frac{26\!\cdots\!99}{14\!\cdots\!97}$, $\frac{65\!\cdots\!70}{14\!\cdots\!97}a^{15}+\frac{43\!\cdots\!19}{14\!\cdots\!97}a^{14}-\frac{14\!\cdots\!85}{22\!\cdots\!91}a^{13}+\frac{53\!\cdots\!36}{14\!\cdots\!97}a^{12}-\frac{14\!\cdots\!57}{14\!\cdots\!97}a^{11}-\frac{11\!\cdots\!71}{14\!\cdots\!97}a^{10}+\frac{26\!\cdots\!11}{14\!\cdots\!97}a^{9}-\frac{34\!\cdots\!30}{14\!\cdots\!97}a^{8}+\frac{12\!\cdots\!31}{14\!\cdots\!97}a^{7}+\frac{12\!\cdots\!98}{14\!\cdots\!97}a^{6}-\frac{32\!\cdots\!29}{14\!\cdots\!97}a^{5}+\frac{65\!\cdots\!83}{14\!\cdots\!97}a^{4}-\frac{78\!\cdots\!53}{14\!\cdots\!97}a^{3}+\frac{68\!\cdots\!58}{14\!\cdots\!97}a^{2}-\frac{26\!\cdots\!15}{14\!\cdots\!97}a+\frac{30\!\cdots\!91}{14\!\cdots\!97}$, $\frac{12\!\cdots\!45}{14\!\cdots\!97}a^{15}-\frac{18\!\cdots\!53}{14\!\cdots\!97}a^{14}-\frac{21\!\cdots\!64}{14\!\cdots\!97}a^{13}+\frac{49\!\cdots\!56}{14\!\cdots\!97}a^{12}-\frac{11\!\cdots\!77}{14\!\cdots\!97}a^{11}-\frac{22\!\cdots\!65}{14\!\cdots\!97}a^{10}+\frac{99\!\cdots\!80}{14\!\cdots\!97}a^{9}-\frac{16\!\cdots\!89}{14\!\cdots\!97}a^{8}+\frac{13\!\cdots\!20}{14\!\cdots\!97}a^{7}+\frac{22\!\cdots\!17}{14\!\cdots\!97}a^{6}-\frac{11\!\cdots\!91}{14\!\cdots\!97}a^{5}+\frac{25\!\cdots\!22}{14\!\cdots\!97}a^{4}-\frac{39\!\cdots\!78}{14\!\cdots\!97}a^{3}+\frac{39\!\cdots\!14}{14\!\cdots\!97}a^{2}-\frac{27\!\cdots\!82}{14\!\cdots\!97}a+\frac{62\!\cdots\!71}{14\!\cdots\!97}$, $\frac{20\!\cdots\!55}{14\!\cdots\!97}a^{15}-\frac{13\!\cdots\!20}{14\!\cdots\!97}a^{14}-\frac{33\!\cdots\!62}{14\!\cdots\!97}a^{13}+\frac{55\!\cdots\!08}{14\!\cdots\!97}a^{12}-\frac{13\!\cdots\!91}{14\!\cdots\!97}a^{11}-\frac{35\!\cdots\!16}{14\!\cdots\!97}a^{10}+\frac{13\!\cdots\!95}{14\!\cdots\!97}a^{9}-\frac{20\!\cdots\!85}{14\!\cdots\!97}a^{8}+\frac{15\!\cdots\!39}{14\!\cdots\!97}a^{7}+\frac{36\!\cdots\!55}{14\!\cdots\!97}a^{6}-\frac{15\!\cdots\!58}{14\!\cdots\!97}a^{5}+\frac{32\!\cdots\!43}{14\!\cdots\!97}a^{4}-\frac{48\!\cdots\!19}{14\!\cdots\!97}a^{3}+\frac{47\!\cdots\!85}{14\!\cdots\!97}a^{2}-\frac{30\!\cdots\!78}{14\!\cdots\!97}a+\frac{68\!\cdots\!43}{14\!\cdots\!97}$
| 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: | \( 2917719.3712 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 2917719.3712 \cdot 1}{2\cdot\sqrt{9786054790924809742924193}}\cr\approx \mathstrut & 0.18606690783 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 16*x^14 + 33*x^13 - 16*x^12 - 171*x^11 + 698*x^10 - 1208*x^9 + 1089*x^8 + 1529*x^7 - 8176*x^6 + 18614*x^5 - 29307*x^4 + 31602*x^3 - 22983*x^2 + 8312*x - 1087)
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 - x^15 - 16*x^14 + 33*x^13 - 16*x^12 - 171*x^11 + 698*x^10 - 1208*x^9 + 1089*x^8 + 1529*x^7 - 8176*x^6 + 18614*x^5 - 29307*x^4 + 31602*x^3 - 22983*x^2 + 8312*x - 1087, 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 - x^15 - 16*x^14 + 33*x^13 - 16*x^12 - 171*x^11 + 698*x^10 - 1208*x^9 + 1089*x^8 + 1529*x^7 - 8176*x^6 + 18614*x^5 - 29307*x^4 + 31602*x^3 - 22983*x^2 + 8312*x - 1087);
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 - x^15 - 16*x^14 + 33*x^13 - 16*x^12 - 171*x^11 + 698*x^10 - 1208*x^9 + 1089*x^8 + 1529*x^7 - 8176*x^6 + 18614*x^5 - 29307*x^4 + 31602*x^3 - 22983*x^2 + 8312*x - 1087);
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
| \(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\) |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | R | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.16.15.1 | $x^{16} + 272$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |
|
\(43\)
| 43.8.4.2 | $x^{8} + 9245 x^{4} - 3339294 x^{2} + 10256403$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ |
| 43.8.0.1 | $x^{8} + x^{4} + 39 x^{3} + 20 x^{2} + 24 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |