Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + x^14 - 40*x^13 - 136*x^12 - 203*x^11 - 694*x^10 + 178*x^9 + 427*x^8 + 2206*x^7 + 2572*x^6 + 2059*x^5 + 1224*x^4 + 116*x^3 - 115*x^2 - 11*x + 1)
gp: K = bnfinit(y^16 - 3*y^15 + y^14 - 40*y^13 - 136*y^12 - 203*y^11 - 694*y^10 + 178*y^9 + 427*y^8 + 2206*y^7 + 2572*y^6 + 2059*y^5 + 1224*y^4 + 116*y^3 - 115*y^2 - 11*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + x^14 - 40*x^13 - 136*x^12 - 203*x^11 - 694*x^10 + 178*x^9 + 427*x^8 + 2206*x^7 + 2572*x^6 + 2059*x^5 + 1224*x^4 + 116*x^3 - 115*x^2 - 11*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + x^14 - 40*x^13 - 136*x^12 - 203*x^11 - 694*x^10 + 178*x^9 + 427*x^8 + 2206*x^7 + 2572*x^6 + 2059*x^5 + 1224*x^4 + 116*x^3 - 115*x^2 - 11*x + 1)
\( x^{16} - 3 x^{15} + x^{14} - 40 x^{13} - 136 x^{12} - 203 x^{11} - 694 x^{10} + 178 x^{9} + 427 x^{8} + \cdots + 1 \)
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: |
\(821630081083204084623649\)
\(\medspace = 17^{14}\cdot 47^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.24\) | 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: | $17^{7/8}47^{1/2}\approx 81.7884043055344$ | ||
| Ramified primes: |
\(17\), \(47\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{83\!\cdots\!98}a^{15}-\frac{10\!\cdots\!49}{83\!\cdots\!98}a^{14}+\frac{76\!\cdots\!53}{41\!\cdots\!49}a^{13}+\frac{44\!\cdots\!58}{41\!\cdots\!49}a^{12}+\frac{11\!\cdots\!43}{83\!\cdots\!98}a^{11}+\frac{11\!\cdots\!21}{41\!\cdots\!49}a^{10}-\frac{19\!\cdots\!57}{41\!\cdots\!49}a^{9}-\frac{44\!\cdots\!09}{83\!\cdots\!98}a^{8}+\frac{15\!\cdots\!42}{41\!\cdots\!49}a^{7}-\frac{94\!\cdots\!10}{41\!\cdots\!49}a^{6}+\frac{23\!\cdots\!35}{83\!\cdots\!98}a^{5}-\frac{15\!\cdots\!43}{41\!\cdots\!49}a^{4}+\frac{13\!\cdots\!65}{41\!\cdots\!49}a^{3}-\frac{10\!\cdots\!45}{83\!\cdots\!98}a^{2}+\frac{18\!\cdots\!67}{41\!\cdots\!49}a+\frac{21\!\cdots\!51}{83\!\cdots\!98}$
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: |
$\frac{55\!\cdots\!81}{83\!\cdots\!98}a^{15}+\frac{32\!\cdots\!15}{83\!\cdots\!98}a^{14}-\frac{16\!\cdots\!97}{83\!\cdots\!98}a^{13}-\frac{85\!\cdots\!91}{83\!\cdots\!98}a^{12}-\frac{27\!\cdots\!33}{83\!\cdots\!98}a^{11}-\frac{67\!\cdots\!27}{83\!\cdots\!98}a^{10}-\frac{11\!\cdots\!63}{83\!\cdots\!98}a^{9}-\frac{29\!\cdots\!27}{83\!\cdots\!98}a^{8}+\frac{25\!\cdots\!83}{83\!\cdots\!98}a^{7}+\frac{19\!\cdots\!19}{83\!\cdots\!98}a^{6}+\frac{11\!\cdots\!95}{83\!\cdots\!98}a^{5}+\frac{85\!\cdots\!51}{83\!\cdots\!98}a^{4}+\frac{77\!\cdots\!25}{83\!\cdots\!98}a^{3}+\frac{32\!\cdots\!39}{83\!\cdots\!98}a^{2}-\frac{47\!\cdots\!81}{83\!\cdots\!98}a-\frac{39\!\cdots\!26}{41\!\cdots\!49}$, $\frac{46\!\cdots\!95}{83\!\cdots\!98}a^{15}-\frac{13\!\cdots\!63}{83\!\cdots\!98}a^{14}+\frac{14\!\cdots\!12}{41\!\cdots\!49}a^{13}-\frac{91\!\cdots\!00}{41\!\cdots\!49}a^{12}-\frac{65\!\cdots\!85}{83\!\cdots\!98}a^{11}-\frac{49\!\cdots\!29}{41\!\cdots\!49}a^{10}-\frac{16\!\cdots\!64}{41\!\cdots\!49}a^{9}+\frac{61\!\cdots\!13}{83\!\cdots\!98}a^{8}+\frac{11\!\cdots\!39}{41\!\cdots\!49}a^{7}+\frac{51\!\cdots\!33}{41\!\cdots\!49}a^{6}+\frac{12\!\cdots\!93}{83\!\cdots\!98}a^{5}+\frac{46\!\cdots\!75}{41\!\cdots\!49}a^{4}+\frac{28\!\cdots\!50}{41\!\cdots\!49}a^{3}-\frac{64\!\cdots\!25}{83\!\cdots\!98}a^{2}-\frac{45\!\cdots\!23}{41\!\cdots\!49}a-\frac{48\!\cdots\!87}{83\!\cdots\!98}$, $\frac{71\!\cdots\!43}{83\!\cdots\!98}a^{15}-\frac{13\!\cdots\!59}{41\!\cdots\!49}a^{14}+\frac{14\!\cdots\!81}{41\!\cdots\!49}a^{13}-\frac{15\!\cdots\!83}{41\!\cdots\!49}a^{12}-\frac{35\!\cdots\!13}{41\!\cdots\!49}a^{11}-\frac{38\!\cdots\!58}{41\!\cdots\!49}a^{10}-\frac{20\!\cdots\!15}{41\!\cdots\!49}a^{9}+\frac{25\!\cdots\!96}{41\!\cdots\!49}a^{8}-\frac{23\!\cdots\!23}{41\!\cdots\!49}a^{7}+\frac{76\!\cdots\!59}{41\!\cdots\!49}a^{6}+\frac{27\!\cdots\!52}{41\!\cdots\!49}a^{5}+\frac{33\!\cdots\!68}{41\!\cdots\!49}a^{4}+\frac{59\!\cdots\!42}{41\!\cdots\!49}a^{3}-\frac{11\!\cdots\!84}{41\!\cdots\!49}a^{2}+\frac{10\!\cdots\!80}{41\!\cdots\!49}a+\frac{13\!\cdots\!17}{83\!\cdots\!98}$, $\frac{45\!\cdots\!31}{83\!\cdots\!98}a^{15}-\frac{91\!\cdots\!95}{41\!\cdots\!49}a^{14}+\frac{10\!\cdots\!92}{41\!\cdots\!49}a^{13}-\frac{96\!\cdots\!45}{41\!\cdots\!49}a^{12}-\frac{21\!\cdots\!27}{41\!\cdots\!49}a^{11}-\frac{17\!\cdots\!35}{41\!\cdots\!49}a^{10}-\frac{11\!\cdots\!61}{41\!\cdots\!49}a^{9}+\frac{19\!\cdots\!44}{41\!\cdots\!49}a^{8}+\frac{43\!\cdots\!24}{41\!\cdots\!49}a^{7}+\frac{44\!\cdots\!47}{41\!\cdots\!49}a^{6}+\frac{78\!\cdots\!50}{41\!\cdots\!49}a^{5}+\frac{38\!\cdots\!41}{41\!\cdots\!49}a^{4}-\frac{10\!\cdots\!63}{41\!\cdots\!49}a^{3}-\frac{16\!\cdots\!42}{41\!\cdots\!49}a^{2}-\frac{21\!\cdots\!57}{41\!\cdots\!49}a+\frac{24\!\cdots\!17}{83\!\cdots\!98}$, $\frac{36\!\cdots\!02}{41\!\cdots\!49}a^{15}-\frac{13\!\cdots\!96}{41\!\cdots\!49}a^{14}+\frac{24\!\cdots\!25}{83\!\cdots\!98}a^{13}-\frac{30\!\cdots\!53}{83\!\cdots\!98}a^{12}-\frac{39\!\cdots\!99}{41\!\cdots\!49}a^{11}-\frac{90\!\cdots\!79}{83\!\cdots\!98}a^{10}-\frac{43\!\cdots\!51}{83\!\cdots\!98}a^{9}+\frac{22\!\cdots\!69}{41\!\cdots\!49}a^{8}+\frac{77\!\cdots\!03}{83\!\cdots\!98}a^{7}+\frac{15\!\cdots\!33}{83\!\cdots\!98}a^{6}+\frac{39\!\cdots\!22}{41\!\cdots\!49}a^{5}+\frac{74\!\cdots\!87}{83\!\cdots\!98}a^{4}+\frac{20\!\cdots\!21}{83\!\cdots\!98}a^{3}-\frac{90\!\cdots\!05}{41\!\cdots\!49}a^{2}-\frac{32\!\cdots\!89}{83\!\cdots\!98}a+\frac{78\!\cdots\!39}{83\!\cdots\!98}$, $\frac{19\!\cdots\!45}{83\!\cdots\!98}a^{15}-\frac{25\!\cdots\!29}{83\!\cdots\!98}a^{14}-\frac{84\!\cdots\!79}{83\!\cdots\!98}a^{13}-\frac{34\!\cdots\!13}{41\!\cdots\!49}a^{12}-\frac{38\!\cdots\!63}{83\!\cdots\!98}a^{11}-\frac{76\!\cdots\!95}{83\!\cdots\!98}a^{10}-\frac{91\!\cdots\!16}{41\!\cdots\!49}a^{9}-\frac{16\!\cdots\!19}{83\!\cdots\!98}a^{8}+\frac{21\!\cdots\!59}{83\!\cdots\!98}a^{7}+\frac{25\!\cdots\!03}{41\!\cdots\!49}a^{6}+\frac{11\!\cdots\!47}{83\!\cdots\!98}a^{5}+\frac{88\!\cdots\!89}{83\!\cdots\!98}a^{4}+\frac{34\!\cdots\!79}{41\!\cdots\!49}a^{3}+\frac{17\!\cdots\!03}{83\!\cdots\!98}a^{2}-\frac{66\!\cdots\!19}{83\!\cdots\!98}a-\frac{55\!\cdots\!61}{83\!\cdots\!98}$, $\frac{16\!\cdots\!25}{83\!\cdots\!98}a^{15}-\frac{27\!\cdots\!09}{41\!\cdots\!49}a^{14}+\frac{33\!\cdots\!81}{83\!\cdots\!98}a^{13}-\frac{32\!\cdots\!11}{41\!\cdots\!49}a^{12}-\frac{96\!\cdots\!51}{41\!\cdots\!49}a^{11}-\frac{22\!\cdots\!31}{83\!\cdots\!98}a^{10}-\frac{47\!\cdots\!95}{41\!\cdots\!49}a^{9}+\frac{40\!\cdots\!27}{41\!\cdots\!49}a^{8}+\frac{77\!\cdots\!25}{83\!\cdots\!98}a^{7}+\frac{15\!\cdots\!51}{41\!\cdots\!49}a^{6}+\frac{12\!\cdots\!96}{41\!\cdots\!49}a^{5}+\frac{91\!\cdots\!89}{83\!\cdots\!98}a^{4}+\frac{42\!\cdots\!35}{41\!\cdots\!49}a^{3}-\frac{46\!\cdots\!39}{41\!\cdots\!49}a^{2}-\frac{43\!\cdots\!87}{83\!\cdots\!98}a+\frac{47\!\cdots\!11}{83\!\cdots\!98}$, $\frac{54\!\cdots\!59}{83\!\cdots\!98}a^{15}-\frac{88\!\cdots\!73}{83\!\cdots\!98}a^{14}-\frac{21\!\cdots\!33}{83\!\cdots\!98}a^{13}-\frac{19\!\cdots\!35}{83\!\cdots\!98}a^{12}-\frac{10\!\cdots\!77}{83\!\cdots\!98}a^{11}-\frac{18\!\cdots\!75}{83\!\cdots\!98}a^{10}-\frac{46\!\cdots\!25}{83\!\cdots\!98}a^{9}-\frac{33\!\cdots\!49}{83\!\cdots\!98}a^{8}+\frac{67\!\cdots\!31}{83\!\cdots\!98}a^{7}+\frac{12\!\cdots\!57}{83\!\cdots\!98}a^{6}+\frac{29\!\cdots\!81}{83\!\cdots\!98}a^{5}+\frac{19\!\cdots\!25}{83\!\cdots\!98}a^{4}+\frac{14\!\cdots\!03}{83\!\cdots\!98}a^{3}+\frac{27\!\cdots\!65}{83\!\cdots\!98}a^{2}-\frac{24\!\cdots\!95}{83\!\cdots\!98}a-\frac{46\!\cdots\!19}{41\!\cdots\!49}$, $\frac{11\!\cdots\!72}{41\!\cdots\!49}a^{15}-\frac{39\!\cdots\!73}{41\!\cdots\!49}a^{14}+\frac{50\!\cdots\!69}{83\!\cdots\!98}a^{13}-\frac{48\!\cdots\!97}{41\!\cdots\!49}a^{12}-\frac{14\!\cdots\!57}{41\!\cdots\!49}a^{11}-\frac{39\!\cdots\!49}{83\!\cdots\!98}a^{10}-\frac{77\!\cdots\!00}{41\!\cdots\!49}a^{9}+\frac{44\!\cdots\!84}{41\!\cdots\!49}a^{8}+\frac{63\!\cdots\!91}{83\!\cdots\!98}a^{7}+\frac{25\!\cdots\!92}{41\!\cdots\!49}a^{6}+\frac{22\!\cdots\!46}{41\!\cdots\!49}a^{5}+\frac{38\!\cdots\!39}{83\!\cdots\!98}a^{4}+\frac{10\!\cdots\!69}{41\!\cdots\!49}a^{3}-\frac{54\!\cdots\!98}{41\!\cdots\!49}a^{2}-\frac{56\!\cdots\!37}{83\!\cdots\!98}a+\frac{43\!\cdots\!38}{41\!\cdots\!49}$, $\frac{44\!\cdots\!25}{41\!\cdots\!49}a^{15}-\frac{28\!\cdots\!81}{83\!\cdots\!98}a^{14}+\frac{20\!\cdots\!85}{83\!\cdots\!98}a^{13}-\frac{37\!\cdots\!39}{83\!\cdots\!98}a^{12}-\frac{10\!\cdots\!71}{83\!\cdots\!98}a^{11}-\frac{16\!\cdots\!17}{83\!\cdots\!98}a^{10}-\frac{61\!\cdots\!19}{83\!\cdots\!98}a^{9}+\frac{26\!\cdots\!83}{83\!\cdots\!98}a^{8}+\frac{63\!\cdots\!73}{83\!\cdots\!98}a^{7}+\frac{20\!\cdots\!19}{83\!\cdots\!98}a^{6}+\frac{17\!\cdots\!49}{83\!\cdots\!98}a^{5}+\frac{21\!\cdots\!07}{83\!\cdots\!98}a^{4}+\frac{12\!\cdots\!69}{83\!\cdots\!98}a^{3}+\frac{47\!\cdots\!65}{83\!\cdots\!98}a^{2}+\frac{16\!\cdots\!55}{83\!\cdots\!98}a-\frac{12\!\cdots\!05}{83\!\cdots\!98}$, $\frac{26\!\cdots\!34}{41\!\cdots\!49}a^{15}-\frac{17\!\cdots\!49}{83\!\cdots\!98}a^{14}+\frac{86\!\cdots\!87}{83\!\cdots\!98}a^{13}-\frac{10\!\cdots\!80}{41\!\cdots\!49}a^{12}-\frac{66\!\cdots\!47}{83\!\cdots\!98}a^{11}-\frac{86\!\cdots\!81}{83\!\cdots\!98}a^{10}-\frac{16\!\cdots\!69}{41\!\cdots\!49}a^{9}+\frac{19\!\cdots\!97}{83\!\cdots\!98}a^{8}+\frac{23\!\cdots\!95}{83\!\cdots\!98}a^{7}+\frac{54\!\cdots\!65}{41\!\cdots\!49}a^{6}+\frac{10\!\cdots\!51}{83\!\cdots\!98}a^{5}+\frac{63\!\cdots\!47}{83\!\cdots\!98}a^{4}+\frac{11\!\cdots\!50}{41\!\cdots\!49}a^{3}-\frac{22\!\cdots\!77}{83\!\cdots\!98}a^{2}-\frac{16\!\cdots\!51}{83\!\cdots\!98}a-\frac{10\!\cdots\!53}{41\!\cdots\!49}$
| 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: | \( 864723.662197 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 864723.662197 \cdot 1}{2\cdot\sqrt{821630081083204084623649}}\cr\approx \mathstrut & 0.190313072466 \end{aligned}\] (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 + x^14 - 40*x^13 - 136*x^12 - 203*x^11 - 694*x^10 + 178*x^9 + 427*x^8 + 2206*x^7 + 2572*x^6 + 2059*x^5 + 1224*x^4 + 116*x^3 - 115*x^2 - 11*x + 1)
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 + x^14 - 40*x^13 - 136*x^12 - 203*x^11 - 694*x^10 + 178*x^9 + 427*x^8 + 2206*x^7 + 2572*x^6 + 2059*x^5 + 1224*x^4 + 116*x^3 - 115*x^2 - 11*x + 1, 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 + x^14 - 40*x^13 - 136*x^12 - 203*x^11 - 694*x^10 + 178*x^9 + 427*x^8 + 2206*x^7 + 2572*x^6 + 2059*x^5 + 1224*x^4 + 116*x^3 - 115*x^2 - 11*x + 1);
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 + x^14 - 40*x^13 - 136*x^12 - 203*x^11 - 694*x^10 + 178*x^9 + 427*x^8 + 2206*x^7 + 2572*x^6 + 2059*x^5 + 1224*x^4 + 116*x^3 - 115*x^2 - 11*x + 1);
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
| \(\Q(\sqrt{17}) \), 4.4.4913.1, 4.2.230911.1, 4.2.13583.1, 8.4.906438128657.1, \(\Q(\zeta_{17})^+\), 8.4.53319889921.1 |
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: | 16.0.4009292695690170390860412175969.16 |
| 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 | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ | |
|
\(47\)
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |