Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 17*x^14 + 124*x^13 - 1476*x^12 + 412*x^11 + 47837*x^10 - 80542*x^9 - 257892*x^8 + 1864157*x^7 - 3412593*x^6 - 16023466*x^5 + 39257376*x^4 + 37514732*x^3 - 115827035*x^2 + 48133287*x + 23651317)
gp: K = bnfinit(y^16 - 3*y^15 - 17*y^14 + 124*y^13 - 1476*y^12 + 412*y^11 + 47837*y^10 - 80542*y^9 - 257892*y^8 + 1864157*y^7 - 3412593*y^6 - 16023466*y^5 + 39257376*y^4 + 37514732*y^3 - 115827035*y^2 + 48133287*y + 23651317, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 - 17*x^14 + 124*x^13 - 1476*x^12 + 412*x^11 + 47837*x^10 - 80542*x^9 - 257892*x^8 + 1864157*x^7 - 3412593*x^6 - 16023466*x^5 + 39257376*x^4 + 37514732*x^3 - 115827035*x^2 + 48133287*x + 23651317);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 - 17*x^14 + 124*x^13 - 1476*x^12 + 412*x^11 + 47837*x^10 - 80542*x^9 - 257892*x^8 + 1864157*x^7 - 3412593*x^6 - 16023466*x^5 + 39257376*x^4 + 37514732*x^3 - 115827035*x^2 + 48133287*x + 23651317)
\( x^{16} - 3 x^{15} - 17 x^{14} + 124 x^{13} - 1476 x^{12} + 412 x^{11} + 47837 x^{10} - 80542 x^{9} + \cdots + 23651317 \)
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: |
\(584820771442796870588072252861681953\)
\(\medspace = 31^{4}\cdot 97^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(171.96\) | 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: | $31^{1/2}97^{15/16}\approx 405.76945138876613$ | ||
| Ramified primes: |
\(31\), \(97\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{97}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{701}a^{14}+\frac{180}{701}a^{13}+\frac{213}{701}a^{12}-\frac{254}{701}a^{11}-\frac{281}{701}a^{10}+\frac{250}{701}a^{9}-\frac{349}{701}a^{8}-\frac{338}{701}a^{7}-\frac{85}{701}a^{6}-\frac{127}{701}a^{5}-\frac{49}{701}a^{4}+\frac{182}{701}a^{3}-\frac{64}{701}a^{2}-\frac{107}{701}a+\frac{197}{701}$, $\frac{1}{43\!\cdots\!99}a^{15}-\frac{97\!\cdots\!75}{43\!\cdots\!99}a^{14}-\frac{47\!\cdots\!76}{43\!\cdots\!99}a^{13}+\frac{86\!\cdots\!48}{43\!\cdots\!99}a^{12}-\frac{29\!\cdots\!84}{43\!\cdots\!99}a^{11}-\frac{11\!\cdots\!12}{43\!\cdots\!99}a^{10}-\frac{42\!\cdots\!42}{43\!\cdots\!99}a^{9}-\frac{18\!\cdots\!66}{43\!\cdots\!99}a^{8}-\frac{16\!\cdots\!58}{43\!\cdots\!99}a^{7}+\frac{76\!\cdots\!21}{43\!\cdots\!99}a^{6}-\frac{19\!\cdots\!89}{43\!\cdots\!99}a^{5}+\frac{16\!\cdots\!57}{43\!\cdots\!99}a^{4}+\frac{85\!\cdots\!31}{43\!\cdots\!99}a^{3}+\frac{67\!\cdots\!31}{43\!\cdots\!99}a^{2}-\frac{11\!\cdots\!80}{43\!\cdots\!99}a-\frac{43\!\cdots\!88}{43\!\cdots\!99}$
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
$C_{2}$, which has order $2$ (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: |
$\frac{63\!\cdots\!60}{73\!\cdots\!97}a^{15}-\frac{26\!\cdots\!21}{73\!\cdots\!97}a^{14}+\frac{48\!\cdots\!45}{73\!\cdots\!97}a^{13}+\frac{31\!\cdots\!29}{73\!\cdots\!97}a^{12}-\frac{10\!\cdots\!44}{73\!\cdots\!97}a^{11}+\frac{27\!\cdots\!05}{73\!\cdots\!97}a^{10}+\frac{11\!\cdots\!67}{10\!\cdots\!97}a^{9}-\frac{37\!\cdots\!23}{73\!\cdots\!97}a^{8}+\frac{31\!\cdots\!26}{73\!\cdots\!97}a^{7}-\frac{57\!\cdots\!48}{73\!\cdots\!97}a^{6}-\frac{21\!\cdots\!30}{73\!\cdots\!97}a^{5}+\frac{46\!\cdots\!22}{73\!\cdots\!97}a^{4}-\frac{13\!\cdots\!50}{73\!\cdots\!97}a^{3}-\frac{14\!\cdots\!10}{73\!\cdots\!97}a^{2}+\frac{63\!\cdots\!89}{73\!\cdots\!97}a-\frac{40\!\cdots\!37}{73\!\cdots\!97}$, $\frac{26\!\cdots\!28}{43\!\cdots\!99}a^{15}-\frac{13\!\cdots\!93}{43\!\cdots\!99}a^{14}-\frac{17\!\cdots\!02}{43\!\cdots\!99}a^{13}+\frac{25\!\cdots\!86}{43\!\cdots\!99}a^{12}-\frac{44\!\cdots\!05}{43\!\cdots\!99}a^{11}+\frac{11\!\cdots\!36}{43\!\cdots\!99}a^{10}+\frac{75\!\cdots\!35}{43\!\cdots\!99}a^{9}-\frac{30\!\cdots\!95}{43\!\cdots\!99}a^{8}+\frac{43\!\cdots\!55}{43\!\cdots\!99}a^{7}+\frac{23\!\cdots\!61}{43\!\cdots\!99}a^{6}-\frac{11\!\cdots\!84}{43\!\cdots\!99}a^{5}-\frac{31\!\cdots\!90}{43\!\cdots\!99}a^{4}+\frac{41\!\cdots\!82}{43\!\cdots\!99}a^{3}-\frac{13\!\cdots\!75}{43\!\cdots\!99}a^{2}-\frac{53\!\cdots\!74}{43\!\cdots\!99}a+\frac{93\!\cdots\!77}{43\!\cdots\!99}$, $\frac{21\!\cdots\!13}{43\!\cdots\!99}a^{15}-\frac{77\!\cdots\!79}{43\!\cdots\!99}a^{14}-\frac{33\!\cdots\!60}{43\!\cdots\!99}a^{13}+\frac{28\!\cdots\!08}{43\!\cdots\!99}a^{12}-\frac{32\!\cdots\!49}{43\!\cdots\!99}a^{11}+\frac{25\!\cdots\!71}{43\!\cdots\!99}a^{10}+\frac{10\!\cdots\!25}{43\!\cdots\!99}a^{9}-\frac{22\!\cdots\!86}{43\!\cdots\!99}a^{8}-\frac{54\!\cdots\!20}{43\!\cdots\!99}a^{7}+\frac{44\!\cdots\!59}{43\!\cdots\!99}a^{6}-\frac{97\!\cdots\!90}{43\!\cdots\!99}a^{5}-\frac{33\!\cdots\!13}{43\!\cdots\!99}a^{4}+\frac{11\!\cdots\!41}{43\!\cdots\!99}a^{3}+\frac{56\!\cdots\!37}{43\!\cdots\!99}a^{2}-\frac{33\!\cdots\!37}{43\!\cdots\!99}a+\frac{14\!\cdots\!62}{43\!\cdots\!99}$, $\frac{33\!\cdots\!29}{43\!\cdots\!99}a^{15}-\frac{13\!\cdots\!80}{43\!\cdots\!99}a^{14}-\frac{32\!\cdots\!31}{43\!\cdots\!99}a^{13}+\frac{37\!\cdots\!99}{43\!\cdots\!99}a^{12}-\frac{51\!\cdots\!26}{43\!\cdots\!99}a^{11}+\frac{72\!\cdots\!79}{43\!\cdots\!99}a^{10}+\frac{13\!\cdots\!39}{43\!\cdots\!99}a^{9}-\frac{32\!\cdots\!44}{43\!\cdots\!99}a^{8}-\frac{28\!\cdots\!50}{43\!\cdots\!99}a^{7}+\frac{49\!\cdots\!12}{43\!\cdots\!99}a^{6}-\frac{12\!\cdots\!12}{43\!\cdots\!99}a^{5}-\frac{32\!\cdots\!92}{43\!\cdots\!99}a^{4}+\frac{10\!\cdots\!22}{43\!\cdots\!99}a^{3}+\frac{38\!\cdots\!93}{43\!\cdots\!99}a^{2}-\frac{24\!\cdots\!13}{43\!\cdots\!99}a+\frac{16\!\cdots\!89}{43\!\cdots\!99}$, $\frac{36\!\cdots\!94}{43\!\cdots\!99}a^{15}-\frac{43\!\cdots\!75}{43\!\cdots\!99}a^{14}+\frac{15\!\cdots\!10}{43\!\cdots\!99}a^{13}+\frac{18\!\cdots\!63}{43\!\cdots\!99}a^{12}-\frac{86\!\cdots\!85}{43\!\cdots\!99}a^{11}+\frac{62\!\cdots\!14}{43\!\cdots\!99}a^{10}-\frac{68\!\cdots\!91}{43\!\cdots\!99}a^{9}-\frac{95\!\cdots\!33}{43\!\cdots\!99}a^{8}+\frac{46\!\cdots\!04}{43\!\cdots\!99}a^{7}-\frac{65\!\cdots\!97}{43\!\cdots\!99}a^{6}-\frac{30\!\cdots\!81}{43\!\cdots\!99}a^{5}+\frac{13\!\cdots\!65}{43\!\cdots\!99}a^{4}-\frac{13\!\cdots\!66}{43\!\cdots\!99}a^{3}-\frac{32\!\cdots\!05}{43\!\cdots\!99}a^{2}+\frac{73\!\cdots\!52}{43\!\cdots\!99}a+\frac{63\!\cdots\!91}{43\!\cdots\!99}$, $\frac{33\!\cdots\!74}{43\!\cdots\!99}a^{15}-\frac{93\!\cdots\!50}{43\!\cdots\!99}a^{14}-\frac{18\!\cdots\!87}{43\!\cdots\!99}a^{13}+\frac{22\!\cdots\!46}{43\!\cdots\!99}a^{12}-\frac{50\!\cdots\!05}{43\!\cdots\!99}a^{11}+\frac{41\!\cdots\!19}{43\!\cdots\!99}a^{10}+\frac{94\!\cdots\!70}{43\!\cdots\!99}a^{9}-\frac{10\!\cdots\!12}{43\!\cdots\!99}a^{8}+\frac{36\!\cdots\!56}{43\!\cdots\!99}a^{7}+\frac{22\!\cdots\!37}{43\!\cdots\!99}a^{6}-\frac{68\!\cdots\!55}{43\!\cdots\!99}a^{5}-\frac{16\!\cdots\!57}{43\!\cdots\!99}a^{4}-\frac{32\!\cdots\!20}{43\!\cdots\!99}a^{3}-\frac{20\!\cdots\!71}{43\!\cdots\!99}a^{2}+\frac{22\!\cdots\!43}{43\!\cdots\!99}a+\frac{12\!\cdots\!72}{43\!\cdots\!99}$, $\frac{28\!\cdots\!75}{43\!\cdots\!99}a^{15}-\frac{17\!\cdots\!94}{43\!\cdots\!99}a^{14}-\frac{54\!\cdots\!20}{43\!\cdots\!99}a^{13}+\frac{56\!\cdots\!98}{43\!\cdots\!99}a^{12}-\frac{47\!\cdots\!34}{43\!\cdots\!99}a^{11}+\frac{11\!\cdots\!50}{43\!\cdots\!99}a^{10}+\frac{18\!\cdots\!02}{43\!\cdots\!99}a^{9}-\frac{64\!\cdots\!26}{43\!\cdots\!99}a^{8}-\frac{13\!\cdots\!48}{43\!\cdots\!99}a^{7}+\frac{88\!\cdots\!69}{43\!\cdots\!99}a^{6}-\frac{23\!\cdots\!38}{43\!\cdots\!99}a^{5}-\frac{60\!\cdots\!77}{43\!\cdots\!99}a^{4}+\frac{32\!\cdots\!69}{43\!\cdots\!99}a^{3}+\frac{11\!\cdots\!82}{43\!\cdots\!99}a^{2}-\frac{96\!\cdots\!41}{43\!\cdots\!99}a+\frac{74\!\cdots\!27}{43\!\cdots\!99}$, $\frac{24\!\cdots\!27}{43\!\cdots\!99}a^{15}-\frac{17\!\cdots\!44}{43\!\cdots\!99}a^{14}+\frac{33\!\cdots\!90}{43\!\cdots\!99}a^{13}+\frac{16\!\cdots\!31}{43\!\cdots\!99}a^{12}-\frac{42\!\cdots\!10}{43\!\cdots\!99}a^{11}+\frac{19\!\cdots\!54}{43\!\cdots\!99}a^{10}+\frac{34\!\cdots\!61}{43\!\cdots\!99}a^{9}-\frac{34\!\cdots\!31}{43\!\cdots\!99}a^{8}+\frac{85\!\cdots\!55}{43\!\cdots\!99}a^{7}+\frac{88\!\cdots\!17}{43\!\cdots\!99}a^{6}-\frac{12\!\cdots\!79}{43\!\cdots\!99}a^{5}+\frac{13\!\cdots\!77}{43\!\cdots\!99}a^{4}+\frac{37\!\cdots\!44}{43\!\cdots\!99}a^{3}-\frac{70\!\cdots\!46}{43\!\cdots\!99}a^{2}+\frac{23\!\cdots\!92}{43\!\cdots\!99}a+\frac{13\!\cdots\!97}{43\!\cdots\!99}$, $\frac{59\!\cdots\!23}{43\!\cdots\!99}a^{15}-\frac{16\!\cdots\!29}{43\!\cdots\!99}a^{14}-\frac{50\!\cdots\!33}{43\!\cdots\!99}a^{13}+\frac{56\!\cdots\!02}{43\!\cdots\!99}a^{12}-\frac{90\!\cdots\!17}{43\!\cdots\!99}a^{11}+\frac{51\!\cdots\!30}{43\!\cdots\!99}a^{10}+\frac{20\!\cdots\!96}{43\!\cdots\!99}a^{9}-\frac{35\!\cdots\!65}{43\!\cdots\!99}a^{8}+\frac{20\!\cdots\!18}{43\!\cdots\!99}a^{7}+\frac{74\!\cdots\!76}{43\!\cdots\!99}a^{6}-\frac{15\!\cdots\!66}{43\!\cdots\!99}a^{5}-\frac{31\!\cdots\!61}{43\!\cdots\!99}a^{4}+\frac{66\!\cdots\!16}{43\!\cdots\!99}a^{3}-\frac{14\!\cdots\!95}{43\!\cdots\!99}a^{2}-\frac{22\!\cdots\!83}{43\!\cdots\!99}a-\frac{33\!\cdots\!69}{43\!\cdots\!99}$, $\frac{13\!\cdots\!20}{43\!\cdots\!99}a^{15}-\frac{89\!\cdots\!45}{43\!\cdots\!99}a^{14}+\frac{11\!\cdots\!61}{43\!\cdots\!99}a^{13}+\frac{12\!\cdots\!96}{43\!\cdots\!99}a^{12}-\frac{24\!\cdots\!11}{43\!\cdots\!99}a^{11}+\frac{98\!\cdots\!50}{43\!\cdots\!99}a^{10}+\frac{26\!\cdots\!09}{43\!\cdots\!99}a^{9}-\frac{21\!\cdots\!81}{43\!\cdots\!99}a^{8}+\frac{48\!\cdots\!88}{43\!\cdots\!99}a^{7}+\frac{76\!\cdots\!28}{43\!\cdots\!99}a^{6}-\frac{79\!\cdots\!21}{43\!\cdots\!99}a^{5}+\frac{93\!\cdots\!48}{43\!\cdots\!99}a^{4}+\frac{23\!\cdots\!97}{43\!\cdots\!99}a^{3}-\frac{53\!\cdots\!38}{43\!\cdots\!99}a^{2}+\frac{31\!\cdots\!98}{43\!\cdots\!99}a-\frac{83\!\cdots\!49}{43\!\cdots\!99}$, $\frac{30\!\cdots\!00}{43\!\cdots\!99}a^{15}-\frac{17\!\cdots\!05}{43\!\cdots\!99}a^{14}-\frac{40\!\cdots\!57}{43\!\cdots\!99}a^{13}+\frac{38\!\cdots\!54}{43\!\cdots\!99}a^{12}-\frac{54\!\cdots\!59}{43\!\cdots\!99}a^{11}+\frac{16\!\cdots\!83}{43\!\cdots\!99}a^{10}+\frac{99\!\cdots\!67}{43\!\cdots\!99}a^{9}-\frac{51\!\cdots\!55}{43\!\cdots\!99}a^{8}+\frac{62\!\cdots\!46}{43\!\cdots\!99}a^{7}+\frac{39\!\cdots\!91}{43\!\cdots\!99}a^{6}-\frac{21\!\cdots\!61}{43\!\cdots\!99}a^{5}+\frac{90\!\cdots\!42}{43\!\cdots\!99}a^{4}+\frac{95\!\cdots\!55}{43\!\cdots\!99}a^{3}-\frac{14\!\cdots\!34}{43\!\cdots\!99}a^{2}+\frac{44\!\cdots\!09}{43\!\cdots\!99}a+\frac{26\!\cdots\!76}{43\!\cdots\!99}$
| 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: | \( 321228696477 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 321228696477 \cdot 2}{2\cdot\sqrt{584820771442796870588072252861681953}}\cr\approx \mathstrut & 0.167595510021600 \end{aligned}\] (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 - 17*x^14 + 124*x^13 - 1476*x^12 + 412*x^11 + 47837*x^10 - 80542*x^9 - 257892*x^8 + 1864157*x^7 - 3412593*x^6 - 16023466*x^5 + 39257376*x^4 + 37514732*x^3 - 115827035*x^2 + 48133287*x + 23651317)
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 - 17*x^14 + 124*x^13 - 1476*x^12 + 412*x^11 + 47837*x^10 - 80542*x^9 - 257892*x^8 + 1864157*x^7 - 3412593*x^6 - 16023466*x^5 + 39257376*x^4 + 37514732*x^3 - 115827035*x^2 + 48133287*x + 23651317, 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 - 17*x^14 + 124*x^13 - 1476*x^12 + 412*x^11 + 47837*x^10 - 80542*x^9 - 257892*x^8 + 1864157*x^7 - 3412593*x^6 - 16023466*x^5 + 39257376*x^4 + 37514732*x^3 - 115827035*x^2 + 48133287*x + 23651317);
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 - 17*x^14 + 124*x^13 - 1476*x^12 + 412*x^11 + 47837*x^10 - 80542*x^9 - 257892*x^8 + 1864157*x^7 - 3412593*x^6 - 16023466*x^5 + 39257376*x^4 + 37514732*x^3 - 115827035*x^2 + 48133287*x + 23651317);
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
| \(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | $16$ | $16$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | R | $16$ | $16$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | $16$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(31\)
| 31.8.4.2 | $x^{8} + 2883 x^{4} - 476656 x^{2} + 2770563$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ |
| 31.8.0.1 | $x^{8} + 25 x^{3} + 12 x^{2} + 24 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(97\)
| 97.16.15.5 | $x^{16} + 582$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |