Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 6*x^14 - 14*x^13 - 27*x^12 + 16*x^11 - 29*x^10 + 465*x^9 + 1927*x^8 + 2351*x^7 + 3*x^6 - 7778*x^5 - 8065*x^4 + 4528*x^3 + 5600*x^2 + 1216*x + 256)
gp: K = bnfinit(y^16 - y^15 - 6*y^14 - 14*y^13 - 27*y^12 + 16*y^11 - 29*y^10 + 465*y^9 + 1927*y^8 + 2351*y^7 + 3*y^6 - 7778*y^5 - 8065*y^4 + 4528*y^3 + 5600*y^2 + 1216*y + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 6*x^14 - 14*x^13 - 27*x^12 + 16*x^11 - 29*x^10 + 465*x^9 + 1927*x^8 + 2351*x^7 + 3*x^6 - 7778*x^5 - 8065*x^4 + 4528*x^3 + 5600*x^2 + 1216*x + 256);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 - 6*x^14 - 14*x^13 - 27*x^12 + 16*x^11 - 29*x^10 + 465*x^9 + 1927*x^8 + 2351*x^7 + 3*x^6 - 7778*x^5 - 8065*x^4 + 4528*x^3 + 5600*x^2 + 1216*x + 256)
\( x^{16} - x^{15} - 6 x^{14} - 14 x^{13} - 27 x^{12} + 16 x^{11} - 29 x^{10} + 465 x^{9} + 1927 x^{8} + \cdots + 256 \)
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: |
\(2040294908815649000428369\)
\(\medspace = 17^{14}\cdot 59^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(33.06\) | 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: | $17^{7/8}59^{1/2}\approx 91.6365672680861$ | ||
| Ramified primes: |
\(17\), \(59\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{208}a^{14}-\frac{21}{208}a^{13}+\frac{11}{104}a^{12}-\frac{11}{104}a^{11}+\frac{5}{208}a^{10}+\frac{3}{52}a^{9}+\frac{59}{208}a^{8}-\frac{11}{208}a^{7}-\frac{77}{208}a^{6}-\frac{1}{16}a^{5}+\frac{79}{208}a^{4}-\frac{51}{104}a^{3}+\frac{31}{208}a^{2}-\frac{25}{52}a-\frac{5}{13}$, $\frac{1}{26\!\cdots\!12}a^{15}-\frac{27\!\cdots\!61}{26\!\cdots\!12}a^{14}-\frac{32\!\cdots\!91}{13\!\cdots\!56}a^{13}+\frac{11\!\cdots\!05}{13\!\cdots\!56}a^{12}+\frac{54\!\cdots\!45}{26\!\cdots\!12}a^{11}+\frac{10\!\cdots\!61}{16\!\cdots\!32}a^{10}+\frac{20\!\cdots\!27}{26\!\cdots\!12}a^{9}-\frac{58\!\cdots\!45}{39\!\cdots\!36}a^{8}+\frac{44\!\cdots\!39}{26\!\cdots\!12}a^{7}+\frac{11\!\cdots\!03}{30\!\cdots\!08}a^{6}-\frac{98\!\cdots\!61}{26\!\cdots\!12}a^{5}+\frac{55\!\cdots\!71}{13\!\cdots\!56}a^{4}-\frac{52\!\cdots\!97}{26\!\cdots\!12}a^{3}-\frac{62\!\cdots\!53}{16\!\cdots\!32}a^{2}-\frac{26\!\cdots\!96}{41\!\cdots\!83}a+\frac{13\!\cdots\!20}{41\!\cdots\!83}$
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: |
$\frac{17\!\cdots\!81}{26\!\cdots\!12}a^{15}-\frac{29\!\cdots\!77}{26\!\cdots\!12}a^{14}-\frac{44\!\cdots\!31}{13\!\cdots\!56}a^{13}-\frac{68\!\cdots\!27}{10\!\cdots\!12}a^{12}-\frac{35\!\cdots\!11}{26\!\cdots\!12}a^{11}+\frac{71\!\cdots\!77}{33\!\cdots\!64}a^{10}-\frac{91\!\cdots\!09}{26\!\cdots\!12}a^{9}+\frac{13\!\cdots\!11}{39\!\cdots\!36}a^{8}+\frac{28\!\cdots\!11}{26\!\cdots\!12}a^{7}+\frac{23\!\cdots\!91}{30\!\cdots\!08}a^{6}-\frac{17\!\cdots\!21}{26\!\cdots\!12}a^{5}-\frac{64\!\cdots\!49}{13\!\cdots\!56}a^{4}-\frac{45\!\cdots\!25}{26\!\cdots\!12}a^{3}+\frac{17\!\cdots\!43}{33\!\cdots\!64}a^{2}+\frac{16\!\cdots\!69}{16\!\cdots\!32}a-\frac{16\!\cdots\!73}{41\!\cdots\!83}$, $\frac{26\!\cdots\!75}{16\!\cdots\!32}a^{15}+\frac{55\!\cdots\!81}{16\!\cdots\!32}a^{14}-\frac{34\!\cdots\!59}{16\!\cdots\!32}a^{13}-\frac{43\!\cdots\!35}{16\!\cdots\!32}a^{12}-\frac{20\!\cdots\!83}{16\!\cdots\!32}a^{11}-\frac{13\!\cdots\!28}{41\!\cdots\!83}a^{10}-\frac{23\!\cdots\!13}{41\!\cdots\!83}a^{9}+\frac{90\!\cdots\!31}{24\!\cdots\!96}a^{8}+\frac{48\!\cdots\!77}{83\!\cdots\!66}a^{7}+\frac{42\!\cdots\!34}{47\!\cdots\!47}a^{6}+\frac{83\!\cdots\!23}{83\!\cdots\!66}a^{5}-\frac{15\!\cdots\!59}{16\!\cdots\!32}a^{4}-\frac{93\!\cdots\!48}{32\!\cdots\!91}a^{3}+\frac{15\!\cdots\!24}{41\!\cdots\!83}a^{2}+\frac{48\!\cdots\!07}{16\!\cdots\!32}a-\frac{56\!\cdots\!48}{41\!\cdots\!83}$, $\frac{67\!\cdots\!19}{66\!\cdots\!28}a^{15}-\frac{64\!\cdots\!17}{66\!\cdots\!28}a^{14}+\frac{17\!\cdots\!71}{83\!\cdots\!66}a^{13}-\frac{75\!\cdots\!17}{25\!\cdots\!28}a^{12}-\frac{37\!\cdots\!41}{66\!\cdots\!28}a^{11}-\frac{29\!\cdots\!79}{33\!\cdots\!64}a^{10}-\frac{12\!\cdots\!59}{66\!\cdots\!28}a^{9}+\frac{56\!\cdots\!79}{99\!\cdots\!84}a^{8}+\frac{96\!\cdots\!95}{66\!\cdots\!28}a^{7}+\frac{49\!\cdots\!59}{75\!\cdots\!52}a^{6}+\frac{79\!\cdots\!51}{66\!\cdots\!28}a^{5}+\frac{96\!\cdots\!41}{16\!\cdots\!32}a^{4}-\frac{55\!\cdots\!91}{66\!\cdots\!28}a^{3}-\frac{11\!\cdots\!61}{33\!\cdots\!64}a^{2}-\frac{11\!\cdots\!19}{16\!\cdots\!32}a+\frac{70\!\cdots\!41}{41\!\cdots\!83}$, $\frac{15\!\cdots\!75}{26\!\cdots\!12}a^{15}-\frac{43\!\cdots\!71}{26\!\cdots\!12}a^{14}-\frac{24\!\cdots\!33}{13\!\cdots\!56}a^{13}-\frac{62\!\cdots\!05}{13\!\cdots\!56}a^{12}+\frac{47\!\cdots\!19}{26\!\cdots\!12}a^{11}+\frac{96\!\cdots\!03}{33\!\cdots\!64}a^{10}-\frac{55\!\cdots\!19}{26\!\cdots\!12}a^{9}+\frac{12\!\cdots\!33}{39\!\cdots\!36}a^{8}+\frac{16\!\cdots\!57}{26\!\cdots\!12}a^{7}-\frac{58\!\cdots\!83}{30\!\cdots\!08}a^{6}-\frac{61\!\cdots\!95}{26\!\cdots\!12}a^{5}-\frac{64\!\cdots\!27}{13\!\cdots\!56}a^{4}+\frac{16\!\cdots\!97}{26\!\cdots\!12}a^{3}+\frac{26\!\cdots\!89}{33\!\cdots\!64}a^{2}+\frac{63\!\cdots\!63}{41\!\cdots\!83}a-\frac{12\!\cdots\!47}{41\!\cdots\!83}$, $\frac{26\!\cdots\!75}{16\!\cdots\!32}a^{15}+\frac{55\!\cdots\!81}{16\!\cdots\!32}a^{14}-\frac{34\!\cdots\!59}{16\!\cdots\!32}a^{13}-\frac{43\!\cdots\!35}{16\!\cdots\!32}a^{12}-\frac{20\!\cdots\!83}{16\!\cdots\!32}a^{11}-\frac{13\!\cdots\!28}{41\!\cdots\!83}a^{10}-\frac{23\!\cdots\!13}{41\!\cdots\!83}a^{9}+\frac{90\!\cdots\!31}{24\!\cdots\!96}a^{8}+\frac{48\!\cdots\!77}{83\!\cdots\!66}a^{7}+\frac{42\!\cdots\!34}{47\!\cdots\!47}a^{6}+\frac{83\!\cdots\!23}{83\!\cdots\!66}a^{5}-\frac{15\!\cdots\!59}{16\!\cdots\!32}a^{4}-\frac{93\!\cdots\!48}{32\!\cdots\!91}a^{3}+\frac{15\!\cdots\!24}{41\!\cdots\!83}a^{2}+\frac{48\!\cdots\!07}{16\!\cdots\!32}a-\frac{47\!\cdots\!31}{41\!\cdots\!83}$, $\frac{11\!\cdots\!81}{13\!\cdots\!56}a^{15}-\frac{16\!\cdots\!87}{13\!\cdots\!56}a^{14}-\frac{11\!\cdots\!17}{25\!\cdots\!28}a^{13}-\frac{70\!\cdots\!29}{66\!\cdots\!28}a^{12}-\frac{27\!\cdots\!03}{13\!\cdots\!56}a^{11}+\frac{11\!\cdots\!11}{66\!\cdots\!28}a^{10}-\frac{61\!\cdots\!73}{13\!\cdots\!56}a^{9}+\frac{83\!\cdots\!41}{19\!\cdots\!68}a^{8}+\frac{15\!\cdots\!41}{10\!\cdots\!12}a^{7}+\frac{23\!\cdots\!29}{15\!\cdots\!04}a^{6}-\frac{31\!\cdots\!59}{13\!\cdots\!56}a^{5}-\frac{92\!\cdots\!27}{16\!\cdots\!32}a^{4}-\frac{44\!\cdots\!69}{13\!\cdots\!56}a^{3}+\frac{29\!\cdots\!73}{66\!\cdots\!28}a^{2}+\frac{34\!\cdots\!34}{41\!\cdots\!83}a-\frac{10\!\cdots\!72}{41\!\cdots\!83}$, $\frac{31\!\cdots\!11}{20\!\cdots\!24}a^{15}-\frac{62\!\cdots\!51}{26\!\cdots\!12}a^{14}-\frac{10\!\cdots\!15}{13\!\cdots\!56}a^{13}-\frac{23\!\cdots\!09}{13\!\cdots\!56}a^{12}-\frac{90\!\cdots\!17}{26\!\cdots\!12}a^{11}+\frac{25\!\cdots\!65}{66\!\cdots\!28}a^{10}-\frac{21\!\cdots\!55}{26\!\cdots\!12}a^{9}+\frac{29\!\cdots\!93}{39\!\cdots\!36}a^{8}+\frac{67\!\cdots\!77}{26\!\cdots\!12}a^{7}+\frac{70\!\cdots\!49}{30\!\cdots\!08}a^{6}-\frac{13\!\cdots\!03}{20\!\cdots\!24}a^{5}-\frac{13\!\cdots\!65}{13\!\cdots\!56}a^{4}-\frac{13\!\cdots\!63}{26\!\cdots\!12}a^{3}+\frac{63\!\cdots\!59}{66\!\cdots\!28}a^{2}+\frac{30\!\cdots\!05}{16\!\cdots\!32}a+\frac{15\!\cdots\!38}{41\!\cdots\!83}$, $\frac{85\!\cdots\!03}{26\!\cdots\!12}a^{15}-\frac{35\!\cdots\!47}{26\!\cdots\!12}a^{14}+\frac{10\!\cdots\!17}{13\!\cdots\!56}a^{13}-\frac{38\!\cdots\!61}{10\!\cdots\!12}a^{12}+\frac{14\!\cdots\!83}{26\!\cdots\!12}a^{11}+\frac{70\!\cdots\!61}{66\!\cdots\!28}a^{10}-\frac{59\!\cdots\!67}{26\!\cdots\!12}a^{9}+\frac{86\!\cdots\!21}{39\!\cdots\!36}a^{8}+\frac{11\!\cdots\!37}{26\!\cdots\!12}a^{7}-\frac{48\!\cdots\!59}{30\!\cdots\!08}a^{6}-\frac{31\!\cdots\!63}{26\!\cdots\!12}a^{5}-\frac{20\!\cdots\!41}{13\!\cdots\!56}a^{4}+\frac{46\!\cdots\!05}{26\!\cdots\!12}a^{3}+\frac{80\!\cdots\!67}{66\!\cdots\!28}a^{2}-\frac{11\!\cdots\!41}{16\!\cdots\!32}a-\frac{42\!\cdots\!12}{41\!\cdots\!83}$, $\frac{59\!\cdots\!35}{24\!\cdots\!68}a^{15}-\frac{24\!\cdots\!33}{23\!\cdots\!42}a^{14}+\frac{14\!\cdots\!59}{24\!\cdots\!68}a^{13}-\frac{94\!\cdots\!51}{30\!\cdots\!46}a^{12}+\frac{13\!\cdots\!81}{24\!\cdots\!68}a^{11}+\frac{14\!\cdots\!27}{19\!\cdots\!36}a^{10}-\frac{19\!\cdots\!83}{24\!\cdots\!68}a^{9}+\frac{14\!\cdots\!21}{92\!\cdots\!76}a^{8}+\frac{24\!\cdots\!21}{61\!\cdots\!92}a^{7}-\frac{23\!\cdots\!45}{13\!\cdots\!56}a^{6}-\frac{14\!\cdots\!23}{12\!\cdots\!84}a^{5}-\frac{36\!\cdots\!61}{24\!\cdots\!68}a^{4}+\frac{20\!\cdots\!03}{24\!\cdots\!68}a^{3}+\frac{33\!\cdots\!65}{24\!\cdots\!68}a^{2}+\frac{28\!\cdots\!35}{61\!\cdots\!92}a+\frac{14\!\cdots\!68}{15\!\cdots\!73}$, $\frac{83\!\cdots\!07}{26\!\cdots\!12}a^{15}-\frac{65\!\cdots\!63}{26\!\cdots\!12}a^{14}+\frac{42\!\cdots\!87}{13\!\cdots\!56}a^{13}+\frac{35\!\cdots\!83}{13\!\cdots\!56}a^{12}+\frac{19\!\cdots\!71}{26\!\cdots\!12}a^{11}+\frac{36\!\cdots\!65}{83\!\cdots\!66}a^{10}-\frac{29\!\cdots\!15}{26\!\cdots\!12}a^{9}+\frac{14\!\cdots\!41}{39\!\cdots\!36}a^{8}-\frac{17\!\cdots\!03}{26\!\cdots\!12}a^{7}-\frac{51\!\cdots\!59}{30\!\cdots\!08}a^{6}-\frac{24\!\cdots\!55}{26\!\cdots\!12}a^{5}-\frac{10\!\cdots\!83}{10\!\cdots\!12}a^{4}+\frac{32\!\cdots\!65}{26\!\cdots\!12}a^{3}-\frac{21\!\cdots\!39}{41\!\cdots\!83}a^{2}-\frac{60\!\cdots\!01}{16\!\cdots\!32}a+\frac{33\!\cdots\!95}{41\!\cdots\!83}$, $\frac{72\!\cdots\!59}{26\!\cdots\!12}a^{15}+\frac{19\!\cdots\!05}{26\!\cdots\!12}a^{14}-\frac{33\!\cdots\!41}{13\!\cdots\!56}a^{13}-\frac{10\!\cdots\!77}{13\!\cdots\!56}a^{12}-\frac{66\!\cdots\!69}{26\!\cdots\!12}a^{11}-\frac{10\!\cdots\!87}{33\!\cdots\!64}a^{10}-\frac{67\!\cdots\!47}{26\!\cdots\!12}a^{9}+\frac{19\!\cdots\!93}{39\!\cdots\!36}a^{8}+\frac{26\!\cdots\!61}{26\!\cdots\!12}a^{7}+\frac{57\!\cdots\!61}{23\!\cdots\!16}a^{6}+\frac{94\!\cdots\!37}{26\!\cdots\!12}a^{5}+\frac{15\!\cdots\!53}{13\!\cdots\!56}a^{4}-\frac{12\!\cdots\!11}{26\!\cdots\!12}a^{3}-\frac{13\!\cdots\!21}{33\!\cdots\!64}a^{2}-\frac{56\!\cdots\!21}{83\!\cdots\!66}a-\frac{31\!\cdots\!15}{32\!\cdots\!91}$
| 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: | \( 1322848.31436 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 1322848.31436 \cdot 1}{2\cdot\sqrt{2040294908815649000428369}}\cr\approx \mathstrut & 0.184753646306 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 6*x^14 - 14*x^13 - 27*x^12 + 16*x^11 - 29*x^10 + 465*x^9 + 1927*x^8 + 2351*x^7 + 3*x^6 - 7778*x^5 - 8065*x^4 + 4528*x^3 + 5600*x^2 + 1216*x + 256)
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 - 6*x^14 - 14*x^13 - 27*x^12 + 16*x^11 - 29*x^10 + 465*x^9 + 1927*x^8 + 2351*x^7 + 3*x^6 - 7778*x^5 - 8065*x^4 + 4528*x^3 + 5600*x^2 + 1216*x + 256, 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 - 6*x^14 - 14*x^13 - 27*x^12 + 16*x^11 - 29*x^10 + 465*x^9 + 1927*x^8 + 2351*x^7 + 3*x^6 - 7778*x^5 - 8065*x^4 + 4528*x^3 + 5600*x^2 + 1216*x + 256);
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 - 6*x^14 - 14*x^13 - 27*x^12 + 16*x^11 - 29*x^10 + 465*x^9 + 1927*x^8 + 2351*x^7 + 3*x^6 - 7778*x^5 - 8065*x^4 + 4528*x^3 + 5600*x^2 + 1216*x + 256);
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
| \(\Q(\sqrt{17}) \), 4.2.289867.1, 4.4.4913.1, 4.2.17051.1, 8.4.1428388920713.1, \(\Q(\zeta_{17})^+\), 8.4.84022877689.1 |
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: | 16.0.24722989956581301387479701814209.3 |
| 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 | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | R |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.16.14.1 | $x^{16} + 128 x^{15} + 7192 x^{14} + 232064 x^{13} + 4716796 x^{12} + 62185088 x^{11} + 525781480 x^{10} + 2696730752 x^{9} + 7365142088 x^{8} + 8090194432 x^{7} + 4732152320 x^{6} + 1682759680 x^{5} + 456414056 x^{4} + 996830464 x^{3} + 7439529968 x^{2} + 33582546688 x + 66368009604$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |
|
\(59\)
| 59.4.0.1 | $x^{4} + 2 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 59.4.0.1 | $x^{4} + 2 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 59.8.4.1 | $x^{8} + 240 x^{6} + 80 x^{5} + 21130 x^{4} - 9280 x^{3} + 808256 x^{2} - 825840 x + 11417625$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |