Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 27*x^14 + 124*x^13 + 150*x^12 - 1002*x^11 - 284*x^10 + 3913*x^9 - 973*x^8 - 6413*x^7 + 4219*x^6 + 2351*x^5 - 4826*x^4 + 1377*x^3 + 1890*x^2 - 184*x + 211)
gp: K = bnfinit(y^16 - 4*y^15 - 27*y^14 + 124*y^13 + 150*y^12 - 1002*y^11 - 284*y^10 + 3913*y^9 - 973*y^8 - 6413*y^7 + 4219*y^6 + 2351*y^5 - 4826*y^4 + 1377*y^3 + 1890*y^2 - 184*y + 211, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 - 27*x^14 + 124*x^13 + 150*x^12 - 1002*x^11 - 284*x^10 + 3913*x^9 - 973*x^8 - 6413*x^7 + 4219*x^6 + 2351*x^5 - 4826*x^4 + 1377*x^3 + 1890*x^2 - 184*x + 211);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 - 27*x^14 + 124*x^13 + 150*x^12 - 1002*x^11 - 284*x^10 + 3913*x^9 - 973*x^8 - 6413*x^7 + 4219*x^6 + 2351*x^5 - 4826*x^4 + 1377*x^3 + 1890*x^2 - 184*x + 211)
\( x^{16} - 4 x^{15} - 27 x^{14} + 124 x^{13} + 150 x^{12} - 1002 x^{11} - 284 x^{10} + 3913 x^{9} + \cdots + 211 \)
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: |
\(142661082295126092995678329\)
\(\medspace = 13^{8}\cdot 53^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(43.12\) | 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: | $13^{1/2}53^{3/4}\approx 70.82369457825733$ | ||
| Ramified primes: |
\(13\), \(53\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{80\!\cdots\!41}a^{15}+\frac{20\!\cdots\!35}{26\!\cdots\!47}a^{14}-\frac{19\!\cdots\!88}{80\!\cdots\!41}a^{13}+\frac{31\!\cdots\!70}{80\!\cdots\!41}a^{12}-\frac{25\!\cdots\!10}{80\!\cdots\!41}a^{11}+\frac{12\!\cdots\!93}{26\!\cdots\!47}a^{10}+\frac{13\!\cdots\!09}{26\!\cdots\!47}a^{9}-\frac{28\!\cdots\!86}{26\!\cdots\!47}a^{8}+\frac{19\!\cdots\!84}{80\!\cdots\!41}a^{7}+\frac{75\!\cdots\!11}{80\!\cdots\!41}a^{6}-\frac{14\!\cdots\!44}{26\!\cdots\!47}a^{5}+\frac{31\!\cdots\!47}{80\!\cdots\!41}a^{4}-\frac{80\!\cdots\!19}{26\!\cdots\!47}a^{3}+\frac{28\!\cdots\!82}{80\!\cdots\!41}a^{2}+\frac{20\!\cdots\!01}{80\!\cdots\!41}a-\frac{30\!\cdots\!60}{80\!\cdots\!41}$
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: |
$\frac{78\!\cdots\!80}{61\!\cdots\!41}a^{15}-\frac{12\!\cdots\!83}{20\!\cdots\!47}a^{14}-\frac{19\!\cdots\!88}{61\!\cdots\!41}a^{13}+\frac{10\!\cdots\!32}{61\!\cdots\!41}a^{12}+\frac{70\!\cdots\!10}{61\!\cdots\!41}a^{11}-\frac{27\!\cdots\!18}{20\!\cdots\!47}a^{10}+\frac{15\!\cdots\!47}{20\!\cdots\!47}a^{9}+\frac{97\!\cdots\!14}{20\!\cdots\!47}a^{8}-\frac{99\!\cdots\!91}{61\!\cdots\!41}a^{7}-\frac{34\!\cdots\!05}{61\!\cdots\!41}a^{6}+\frac{50\!\cdots\!25}{20\!\cdots\!47}a^{5}-\frac{56\!\cdots\!05}{61\!\cdots\!41}a^{4}+\frac{58\!\cdots\!47}{20\!\cdots\!47}a^{3}-\frac{70\!\cdots\!49}{61\!\cdots\!41}a^{2}+\frac{74\!\cdots\!80}{61\!\cdots\!41}a+\frac{15\!\cdots\!22}{61\!\cdots\!41}$, $\frac{70\!\cdots\!93}{26\!\cdots\!47}a^{15}+\frac{16\!\cdots\!73}{26\!\cdots\!47}a^{14}-\frac{71\!\cdots\!55}{26\!\cdots\!47}a^{13}-\frac{39\!\cdots\!07}{26\!\cdots\!47}a^{12}+\frac{21\!\cdots\!48}{26\!\cdots\!47}a^{11}+\frac{13\!\cdots\!85}{26\!\cdots\!47}a^{10}-\frac{16\!\cdots\!74}{26\!\cdots\!47}a^{9}+\frac{25\!\cdots\!85}{26\!\cdots\!47}a^{8}+\frac{64\!\cdots\!16}{26\!\cdots\!47}a^{7}-\frac{36\!\cdots\!84}{26\!\cdots\!47}a^{6}-\frac{10\!\cdots\!39}{26\!\cdots\!47}a^{5}+\frac{74\!\cdots\!02}{26\!\cdots\!47}a^{4}+\frac{48\!\cdots\!25}{26\!\cdots\!47}a^{3}-\frac{69\!\cdots\!52}{26\!\cdots\!47}a^{2}+\frac{12\!\cdots\!74}{26\!\cdots\!47}a+\frac{43\!\cdots\!78}{26\!\cdots\!47}$, $\frac{96\!\cdots\!21}{59\!\cdots\!81}a^{15}-\frac{15\!\cdots\!45}{19\!\cdots\!27}a^{14}-\frac{23\!\cdots\!85}{59\!\cdots\!81}a^{13}+\frac{14\!\cdots\!78}{59\!\cdots\!81}a^{12}+\frac{76\!\cdots\!55}{59\!\cdots\!81}a^{11}-\frac{37\!\cdots\!80}{19\!\cdots\!27}a^{10}+\frac{48\!\cdots\!63}{19\!\cdots\!27}a^{9}+\frac{13\!\cdots\!16}{19\!\cdots\!27}a^{8}-\frac{19\!\cdots\!61}{59\!\cdots\!81}a^{7}-\frac{57\!\cdots\!21}{59\!\cdots\!81}a^{6}+\frac{11\!\cdots\!97}{19\!\cdots\!27}a^{5}-\frac{12\!\cdots\!05}{59\!\cdots\!81}a^{4}-\frac{25\!\cdots\!50}{19\!\cdots\!27}a^{3}+\frac{12\!\cdots\!73}{59\!\cdots\!81}a^{2}-\frac{19\!\cdots\!15}{59\!\cdots\!81}a+\frac{14\!\cdots\!83}{59\!\cdots\!81}$, $\frac{780834010806}{20\!\cdots\!93}a^{15}-\frac{5652267450465}{20\!\cdots\!93}a^{14}-\frac{13037295380436}{20\!\cdots\!93}a^{13}+\frac{168586198467931}{20\!\cdots\!93}a^{12}-\frac{133584733095027}{20\!\cdots\!93}a^{11}-\frac{12\!\cdots\!93}{20\!\cdots\!93}a^{10}+\frac{17\!\cdots\!30}{20\!\cdots\!93}a^{9}+\frac{46\!\cdots\!63}{20\!\cdots\!93}a^{8}-\frac{84\!\cdots\!94}{20\!\cdots\!93}a^{7}-\frac{55\!\cdots\!52}{20\!\cdots\!93}a^{6}+\frac{15\!\cdots\!89}{20\!\cdots\!93}a^{5}-\frac{43\!\cdots\!38}{20\!\cdots\!93}a^{4}-\frac{70\!\cdots\!18}{20\!\cdots\!93}a^{3}+\frac{12\!\cdots\!34}{20\!\cdots\!93}a^{2}-\frac{19\!\cdots\!41}{20\!\cdots\!93}a+\frac{380734477356597}{20\!\cdots\!93}$, $\frac{16\!\cdots\!43}{13\!\cdots\!63}a^{15}-\frac{28\!\cdots\!08}{44\!\cdots\!21}a^{14}-\frac{37\!\cdots\!89}{13\!\cdots\!63}a^{13}+\frac{25\!\cdots\!79}{13\!\cdots\!63}a^{12}+\frac{46\!\cdots\!38}{13\!\cdots\!63}a^{11}-\frac{60\!\cdots\!29}{44\!\cdots\!21}a^{10}+\frac{27\!\cdots\!03}{44\!\cdots\!21}a^{9}+\frac{20\!\cdots\!47}{44\!\cdots\!21}a^{8}-\frac{53\!\cdots\!77}{13\!\cdots\!63}a^{7}-\frac{56\!\cdots\!79}{13\!\cdots\!63}a^{6}+\frac{28\!\cdots\!52}{44\!\cdots\!21}a^{5}-\frac{49\!\cdots\!73}{13\!\cdots\!63}a^{4}-\frac{67\!\cdots\!54}{44\!\cdots\!21}a^{3}+\frac{19\!\cdots\!01}{13\!\cdots\!63}a^{2}-\frac{17\!\cdots\!69}{13\!\cdots\!63}a-\frac{53\!\cdots\!79}{13\!\cdots\!63}$, $\frac{10\!\cdots\!33}{26\!\cdots\!47}a^{15}-\frac{51\!\cdots\!53}{26\!\cdots\!47}a^{14}-\frac{26\!\cdots\!11}{26\!\cdots\!47}a^{13}+\frac{15\!\cdots\!75}{26\!\cdots\!47}a^{12}+\frac{76\!\cdots\!42}{26\!\cdots\!47}a^{11}-\frac{11\!\cdots\!41}{26\!\cdots\!47}a^{10}+\frac{27\!\cdots\!74}{26\!\cdots\!47}a^{9}+\frac{42\!\cdots\!84}{26\!\cdots\!47}a^{8}-\frac{29\!\cdots\!88}{26\!\cdots\!47}a^{7}-\frac{53\!\cdots\!92}{26\!\cdots\!47}a^{6}+\frac{62\!\cdots\!97}{26\!\cdots\!47}a^{5}-\frac{11\!\cdots\!26}{26\!\cdots\!47}a^{4}-\frac{35\!\cdots\!11}{26\!\cdots\!47}a^{3}+\frac{23\!\cdots\!41}{26\!\cdots\!47}a^{2}+\frac{15\!\cdots\!15}{26\!\cdots\!47}a+\frac{48\!\cdots\!16}{26\!\cdots\!47}$, $\frac{45\!\cdots\!57}{26\!\cdots\!47}a^{15}-\frac{90\!\cdots\!02}{26\!\cdots\!47}a^{14}-\frac{14\!\cdots\!78}{26\!\cdots\!47}a^{13}+\frac{29\!\cdots\!29}{26\!\cdots\!47}a^{12}+\frac{15\!\cdots\!42}{26\!\cdots\!47}a^{11}-\frac{24\!\cdots\!26}{26\!\cdots\!47}a^{10}-\frac{78\!\cdots\!75}{26\!\cdots\!47}a^{9}+\frac{90\!\cdots\!87}{26\!\cdots\!47}a^{8}+\frac{19\!\cdots\!73}{26\!\cdots\!47}a^{7}-\frac{14\!\cdots\!23}{26\!\cdots\!47}a^{6}-\frac{17\!\cdots\!91}{26\!\cdots\!47}a^{5}+\frac{10\!\cdots\!83}{26\!\cdots\!47}a^{4}-\frac{64\!\cdots\!19}{26\!\cdots\!47}a^{3}-\frac{11\!\cdots\!45}{26\!\cdots\!47}a^{2}-\frac{22\!\cdots\!26}{26\!\cdots\!47}a-\frac{24\!\cdots\!99}{26\!\cdots\!47}$, $\frac{28\!\cdots\!39}{80\!\cdots\!41}a^{15}-\frac{76\!\cdots\!92}{80\!\cdots\!41}a^{14}-\frac{86\!\cdots\!96}{80\!\cdots\!41}a^{13}+\frac{23\!\cdots\!84}{80\!\cdots\!41}a^{12}+\frac{24\!\cdots\!40}{26\!\cdots\!47}a^{11}-\frac{18\!\cdots\!09}{80\!\cdots\!41}a^{10}-\frac{33\!\cdots\!42}{80\!\cdots\!41}a^{9}+\frac{69\!\cdots\!23}{80\!\cdots\!41}a^{8}+\frac{25\!\cdots\!57}{26\!\cdots\!47}a^{7}-\frac{11\!\cdots\!79}{80\!\cdots\!41}a^{6}-\frac{27\!\cdots\!97}{26\!\cdots\!47}a^{5}+\frac{80\!\cdots\!72}{80\!\cdots\!41}a^{4}+\frac{35\!\cdots\!49}{80\!\cdots\!41}a^{3}-\frac{99\!\cdots\!18}{80\!\cdots\!41}a^{2}+\frac{14\!\cdots\!82}{80\!\cdots\!41}a+\frac{58\!\cdots\!70}{80\!\cdots\!41}$, $\frac{65\!\cdots\!99}{26\!\cdots\!47}a^{15}-\frac{25\!\cdots\!39}{26\!\cdots\!47}a^{14}-\frac{19\!\cdots\!71}{26\!\cdots\!47}a^{13}+\frac{81\!\cdots\!79}{26\!\cdots\!47}a^{12}+\frac{14\!\cdots\!80}{26\!\cdots\!47}a^{11}-\frac{70\!\cdots\!19}{26\!\cdots\!47}a^{10}-\frac{59\!\cdots\!76}{26\!\cdots\!47}a^{9}+\frac{29\!\cdots\!31}{26\!\cdots\!47}a^{8}+\frac{11\!\cdots\!38}{26\!\cdots\!47}a^{7}-\frac{56\!\cdots\!22}{26\!\cdots\!47}a^{6}-\frac{10\!\cdots\!21}{26\!\cdots\!47}a^{5}+\frac{38\!\cdots\!48}{26\!\cdots\!47}a^{4}+\frac{39\!\cdots\!99}{26\!\cdots\!47}a^{3}-\frac{24\!\cdots\!92}{26\!\cdots\!47}a^{2}+\frac{13\!\cdots\!50}{26\!\cdots\!47}a-\frac{94\!\cdots\!33}{26\!\cdots\!47}$, $\frac{10\!\cdots\!75}{26\!\cdots\!47}a^{15}-\frac{27\!\cdots\!60}{26\!\cdots\!47}a^{14}-\frac{29\!\cdots\!50}{26\!\cdots\!47}a^{13}+\frac{86\!\cdots\!46}{26\!\cdots\!47}a^{12}+\frac{22\!\cdots\!49}{26\!\cdots\!47}a^{11}-\frac{66\!\cdots\!90}{26\!\cdots\!47}a^{10}-\frac{86\!\cdots\!85}{26\!\cdots\!47}a^{9}+\frac{24\!\cdots\!98}{26\!\cdots\!47}a^{8}+\frac{10\!\cdots\!53}{26\!\cdots\!47}a^{7}-\frac{36\!\cdots\!57}{26\!\cdots\!47}a^{6}+\frac{90\!\cdots\!33}{26\!\cdots\!47}a^{5}+\frac{14\!\cdots\!97}{26\!\cdots\!47}a^{4}-\frac{22\!\cdots\!17}{26\!\cdots\!47}a^{3}-\frac{38\!\cdots\!03}{26\!\cdots\!47}a^{2}+\frac{20\!\cdots\!54}{26\!\cdots\!47}a-\frac{14\!\cdots\!66}{26\!\cdots\!47}$, $\frac{89\!\cdots\!11}{80\!\cdots\!41}a^{15}-\frac{22\!\cdots\!71}{80\!\cdots\!41}a^{14}-\frac{26\!\cdots\!54}{80\!\cdots\!41}a^{13}+\frac{67\!\cdots\!74}{80\!\cdots\!41}a^{12}+\frac{65\!\cdots\!11}{26\!\cdots\!47}a^{11}-\frac{49\!\cdots\!99}{80\!\cdots\!41}a^{10}-\frac{68\!\cdots\!74}{80\!\cdots\!41}a^{9}+\frac{17\!\cdots\!81}{80\!\cdots\!41}a^{8}+\frac{18\!\cdots\!68}{26\!\cdots\!47}a^{7}-\frac{24\!\cdots\!18}{80\!\cdots\!41}a^{6}+\frac{46\!\cdots\!07}{26\!\cdots\!47}a^{5}+\frac{88\!\cdots\!75}{80\!\cdots\!41}a^{4}-\frac{16\!\cdots\!92}{80\!\cdots\!41}a^{3}+\frac{15\!\cdots\!00}{80\!\cdots\!41}a^{2}+\frac{30\!\cdots\!95}{80\!\cdots\!41}a-\frac{17\!\cdots\!59}{80\!\cdots\!41}$
| 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: | \( 13402915.644 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 13402915.644 \cdot 1}{2\cdot\sqrt{142661082295126092995678329}}\cr\approx \mathstrut & 0.22385972714 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 27*x^14 + 124*x^13 + 150*x^12 - 1002*x^11 - 284*x^10 + 3913*x^9 - 973*x^8 - 6413*x^7 + 4219*x^6 + 2351*x^5 - 4826*x^4 + 1377*x^3 + 1890*x^2 - 184*x + 211)
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 - 4*x^15 - 27*x^14 + 124*x^13 + 150*x^12 - 1002*x^11 - 284*x^10 + 3913*x^9 - 973*x^8 - 6413*x^7 + 4219*x^6 + 2351*x^5 - 4826*x^4 + 1377*x^3 + 1890*x^2 - 184*x + 211, 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 - 4*x^15 - 27*x^14 + 124*x^13 + 150*x^12 - 1002*x^11 - 284*x^10 + 3913*x^9 - 973*x^8 - 6413*x^7 + 4219*x^6 + 2351*x^5 - 4826*x^4 + 1377*x^3 + 1890*x^2 - 184*x + 211);
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 - 4*x^15 - 27*x^14 + 124*x^13 + 150*x^12 - 1002*x^11 - 284*x^10 + 3913*x^9 - 973*x^8 - 6413*x^7 + 4219*x^6 + 2351*x^5 - 4826*x^4 + 1377*x^3 + 1890*x^2 - 184*x + 211);
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
| \(\Q(\sqrt{13}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{689}) \), 4.4.8957.1 x2, 4.4.36517.1 x2, \(\Q(\sqrt{13}, \sqrt{53})\), 8.4.11944081475573.3, 8.4.4252075997.1, 8.8.225360027841.1 |
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: | 8.4.11944081475573.3, 8.4.4252075997.1 |
| Degree 16 sibling: | 16.0.2371212900396504113756570569.4 |
| Minimal sibling: | 8.4.4252075997.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(53\)
| 53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 53.4.3.2 | $x^{4} + 53$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 53.4.3.2 | $x^{4} + 53$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |