Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 19*x^14 + 35*x^13 + 90*x^12 - 265*x^11 - 5*x^10 + 1444*x^9 + 1264*x^8 - 3145*x^7 - 7994*x^6 + 493*x^5 + 2574*x^4 - 1223*x^3 + 431*x^2 + 143*x + 121)
gp: K = bnfinit(y^16 - 2*y^15 - 19*y^14 + 35*y^13 + 90*y^12 - 265*y^11 - 5*y^10 + 1444*y^9 + 1264*y^8 - 3145*y^7 - 7994*y^6 + 493*y^5 + 2574*y^4 - 1223*y^3 + 431*y^2 + 143*y + 121, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 19*x^14 + 35*x^13 + 90*x^12 - 265*x^11 - 5*x^10 + 1444*x^9 + 1264*x^8 - 3145*x^7 - 7994*x^6 + 493*x^5 + 2574*x^4 - 1223*x^3 + 431*x^2 + 143*x + 121);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 19*x^14 + 35*x^13 + 90*x^12 - 265*x^11 - 5*x^10 + 1444*x^9 + 1264*x^8 - 3145*x^7 - 7994*x^6 + 493*x^5 + 2574*x^4 - 1223*x^3 + 431*x^2 + 143*x + 121)
\( x^{16} - 2 x^{15} - 19 x^{14} + 35 x^{13} + 90 x^{12} - 265 x^{11} - 5 x^{10} + 1444 x^{9} + 1264 x^{8} + \cdots + 121 \)
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: |
\(8583027023095873875496489\)
\(\medspace = 13^{10}\cdot 53^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(36.17\) | 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: | $13^{3/4}53^{1/2}\approx 49.84199864366084$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{39}a^{12}+\frac{4}{39}a^{11}-\frac{2}{39}a^{10}+\frac{2}{39}a^{9}+\frac{5}{13}a^{7}-\frac{1}{13}a^{6}+\frac{2}{13}a^{5}-\frac{4}{39}a^{4}-\frac{10}{39}a^{3}+\frac{14}{39}a^{2}-\frac{11}{39}a-\frac{3}{13}$, $\frac{1}{39}a^{13}-\frac{5}{39}a^{11}-\frac{1}{13}a^{10}+\frac{5}{39}a^{9}+\frac{2}{39}a^{8}+\frac{5}{13}a^{7}+\frac{6}{13}a^{6}+\frac{11}{39}a^{5}+\frac{2}{13}a^{4}+\frac{2}{39}a^{3}-\frac{5}{13}a^{2}-\frac{17}{39}a+\frac{10}{39}$, $\frac{1}{1521}a^{14}+\frac{14}{1521}a^{13}+\frac{11}{1521}a^{12}+\frac{10}{507}a^{11}-\frac{43}{1521}a^{10}-\frac{14}{117}a^{9}-\frac{152}{1521}a^{8}+\frac{3}{13}a^{7}+\frac{371}{1521}a^{6}+\frac{100}{1521}a^{5}+\frac{724}{1521}a^{4}-\frac{244}{507}a^{3}-\frac{536}{1521}a^{2}-\frac{625}{1521}a-\frac{394}{1521}$, $\frac{1}{33\!\cdots\!17}a^{15}+\frac{13\!\cdots\!13}{47\!\cdots\!31}a^{14}+\frac{22\!\cdots\!43}{33\!\cdots\!17}a^{13}-\frac{35\!\cdots\!40}{11\!\cdots\!39}a^{12}+\frac{93\!\cdots\!72}{47\!\cdots\!31}a^{11}-\frac{24\!\cdots\!68}{33\!\cdots\!17}a^{10}-\frac{25\!\cdots\!38}{33\!\cdots\!17}a^{9}+\frac{16\!\cdots\!91}{37\!\cdots\!13}a^{8}+\frac{30\!\cdots\!22}{33\!\cdots\!17}a^{7}+\frac{58\!\cdots\!89}{33\!\cdots\!17}a^{6}+\frac{24\!\cdots\!92}{33\!\cdots\!17}a^{5}-\frac{40\!\cdots\!54}{11\!\cdots\!39}a^{4}+\frac{12\!\cdots\!52}{30\!\cdots\!47}a^{3}-\frac{13\!\cdots\!28}{33\!\cdots\!17}a^{2}-\frac{43\!\cdots\!93}{33\!\cdots\!17}a+\frac{16\!\cdots\!94}{33\!\cdots\!83}$
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: |
$\frac{76\!\cdots\!06}{25\!\cdots\!09}a^{15}-\frac{21\!\cdots\!21}{36\!\cdots\!87}a^{14}-\frac{42\!\cdots\!05}{85\!\cdots\!03}a^{13}+\frac{23\!\cdots\!38}{25\!\cdots\!09}a^{12}+\frac{50\!\cdots\!03}{36\!\cdots\!87}a^{11}-\frac{14\!\cdots\!15}{25\!\cdots\!09}a^{10}+\frac{17\!\cdots\!98}{28\!\cdots\!01}a^{9}+\frac{64\!\cdots\!06}{25\!\cdots\!09}a^{8}+\frac{10\!\cdots\!45}{25\!\cdots\!09}a^{7}-\frac{30\!\cdots\!22}{25\!\cdots\!09}a^{6}-\frac{12\!\cdots\!93}{85\!\cdots\!03}a^{5}-\frac{40\!\cdots\!74}{25\!\cdots\!09}a^{4}-\frac{11\!\cdots\!65}{23\!\cdots\!19}a^{3}-\frac{98\!\cdots\!31}{25\!\cdots\!09}a^{2}+\frac{60\!\cdots\!92}{28\!\cdots\!01}a-\frac{52\!\cdots\!59}{23\!\cdots\!19}$, $\frac{26\!\cdots\!85}{25\!\cdots\!09}a^{15}-\frac{23\!\cdots\!54}{12\!\cdots\!29}a^{14}-\frac{49\!\cdots\!82}{25\!\cdots\!09}a^{13}+\frac{86\!\cdots\!57}{25\!\cdots\!09}a^{12}+\frac{34\!\cdots\!23}{36\!\cdots\!87}a^{11}-\frac{22\!\cdots\!64}{85\!\cdots\!03}a^{10}-\frac{93\!\cdots\!00}{25\!\cdots\!09}a^{9}+\frac{37\!\cdots\!02}{25\!\cdots\!09}a^{8}+\frac{37\!\cdots\!13}{25\!\cdots\!09}a^{7}-\frac{26\!\cdots\!45}{85\!\cdots\!03}a^{6}-\frac{21\!\cdots\!59}{25\!\cdots\!09}a^{5}-\frac{98\!\cdots\!13}{25\!\cdots\!09}a^{4}+\frac{81\!\cdots\!01}{23\!\cdots\!19}a^{3}-\frac{62\!\cdots\!22}{85\!\cdots\!03}a^{2}-\frac{12\!\cdots\!37}{25\!\cdots\!09}a+\frac{67\!\cdots\!53}{23\!\cdots\!19}$, $\frac{11\!\cdots\!97}{85\!\cdots\!03}a^{15}-\frac{93\!\cdots\!83}{36\!\cdots\!87}a^{14}-\frac{62\!\cdots\!97}{25\!\cdots\!09}a^{13}+\frac{11\!\cdots\!95}{25\!\cdots\!09}a^{12}+\frac{13\!\cdots\!42}{12\!\cdots\!29}a^{11}-\frac{80\!\cdots\!07}{25\!\cdots\!09}a^{10}+\frac{60\!\cdots\!82}{25\!\cdots\!09}a^{9}+\frac{43\!\cdots\!08}{25\!\cdots\!09}a^{8}+\frac{15\!\cdots\!86}{85\!\cdots\!03}a^{7}-\frac{80\!\cdots\!57}{25\!\cdots\!09}a^{6}-\frac{25\!\cdots\!38}{25\!\cdots\!09}a^{5}-\frac{50\!\cdots\!87}{25\!\cdots\!09}a^{4}-\frac{10\!\cdots\!88}{77\!\cdots\!73}a^{3}-\frac{28\!\cdots\!97}{25\!\cdots\!09}a^{2}-\frac{11\!\cdots\!09}{25\!\cdots\!09}a+\frac{15\!\cdots\!94}{23\!\cdots\!19}$, $\frac{26\!\cdots\!85}{25\!\cdots\!09}a^{15}-\frac{23\!\cdots\!54}{12\!\cdots\!29}a^{14}-\frac{49\!\cdots\!82}{25\!\cdots\!09}a^{13}+\frac{86\!\cdots\!57}{25\!\cdots\!09}a^{12}+\frac{34\!\cdots\!23}{36\!\cdots\!87}a^{11}-\frac{22\!\cdots\!64}{85\!\cdots\!03}a^{10}-\frac{93\!\cdots\!00}{25\!\cdots\!09}a^{9}+\frac{37\!\cdots\!02}{25\!\cdots\!09}a^{8}+\frac{37\!\cdots\!13}{25\!\cdots\!09}a^{7}-\frac{26\!\cdots\!45}{85\!\cdots\!03}a^{6}-\frac{21\!\cdots\!59}{25\!\cdots\!09}a^{5}-\frac{98\!\cdots\!13}{25\!\cdots\!09}a^{4}+\frac{81\!\cdots\!01}{23\!\cdots\!19}a^{3}-\frac{62\!\cdots\!22}{85\!\cdots\!03}a^{2}+\frac{13\!\cdots\!72}{25\!\cdots\!09}a+\frac{67\!\cdots\!53}{23\!\cdots\!19}$, $\frac{25\!\cdots\!41}{10\!\cdots\!49}a^{15}-\frac{21\!\cdots\!96}{43\!\cdots\!21}a^{14}-\frac{14\!\cdots\!18}{30\!\cdots\!47}a^{13}+\frac{25\!\cdots\!31}{30\!\cdots\!47}a^{12}+\frac{33\!\cdots\!66}{14\!\cdots\!07}a^{11}-\frac{19\!\cdots\!56}{30\!\cdots\!47}a^{10}-\frac{31\!\cdots\!13}{30\!\cdots\!47}a^{9}+\frac{11\!\cdots\!15}{30\!\cdots\!47}a^{8}+\frac{35\!\cdots\!57}{10\!\cdots\!49}a^{7}-\frac{25\!\cdots\!69}{30\!\cdots\!47}a^{6}-\frac{65\!\cdots\!17}{30\!\cdots\!47}a^{5}+\frac{17\!\cdots\!84}{30\!\cdots\!47}a^{4}+\frac{12\!\cdots\!27}{10\!\cdots\!49}a^{3}+\frac{95\!\cdots\!72}{30\!\cdots\!47}a^{2}+\frac{33\!\cdots\!60}{30\!\cdots\!47}a-\frac{74\!\cdots\!85}{30\!\cdots\!47}$, $\frac{32\!\cdots\!14}{30\!\cdots\!47}a^{15}-\frac{36\!\cdots\!57}{14\!\cdots\!07}a^{14}-\frac{60\!\cdots\!56}{30\!\cdots\!47}a^{13}+\frac{12\!\cdots\!90}{30\!\cdots\!47}a^{12}+\frac{37\!\cdots\!20}{43\!\cdots\!21}a^{11}-\frac{97\!\cdots\!17}{33\!\cdots\!83}a^{10}+\frac{13\!\cdots\!83}{30\!\cdots\!47}a^{9}+\frac{43\!\cdots\!30}{30\!\cdots\!47}a^{8}+\frac{36\!\cdots\!52}{30\!\cdots\!47}a^{7}-\frac{37\!\cdots\!66}{10\!\cdots\!49}a^{6}-\frac{27\!\cdots\!53}{30\!\cdots\!47}a^{5}+\frac{34\!\cdots\!41}{30\!\cdots\!47}a^{4}+\frac{14\!\cdots\!87}{30\!\cdots\!47}a^{3}+\frac{20\!\cdots\!06}{33\!\cdots\!83}a^{2}+\frac{50\!\cdots\!37}{30\!\cdots\!47}a+\frac{83\!\cdots\!15}{30\!\cdots\!47}$, $\frac{532277916166}{137908156737351}a^{15}-\frac{138100900973}{19701165248193}a^{14}-\frac{10389105361790}{137908156737351}a^{13}+\frac{16994675919037}{137908156737351}a^{12}+\frac{7511722425209}{19701165248193}a^{11}-\frac{135482267010647}{137908156737351}a^{10}-\frac{32918304843349}{137908156737351}a^{9}+\frac{261956467208509}{45969385579117}a^{8}+\frac{810187276102241}{137908156737351}a^{7}-\frac{16\!\cdots\!94}{137908156737351}a^{6}-\frac{46\!\cdots\!80}{137908156737351}a^{5}-\frac{223270989284104}{137908156737351}a^{4}+\frac{172858860701351}{12537105157941}a^{3}-\frac{396931097457337}{137908156737351}a^{2}-\frac{260575811725193}{137908156737351}a-\frac{2883747659529}{4179035052647}$, $\frac{55\!\cdots\!22}{85\!\cdots\!03}a^{15}-\frac{19\!\cdots\!31}{12\!\cdots\!29}a^{14}-\frac{10\!\cdots\!32}{85\!\cdots\!03}a^{13}+\frac{24\!\cdots\!25}{85\!\cdots\!03}a^{12}+\frac{74\!\cdots\!39}{12\!\cdots\!29}a^{11}-\frac{60\!\cdots\!59}{28\!\cdots\!01}a^{10}-\frac{51\!\cdots\!60}{85\!\cdots\!03}a^{9}+\frac{93\!\cdots\!74}{85\!\cdots\!03}a^{8}+\frac{42\!\cdots\!18}{85\!\cdots\!03}a^{7}-\frac{29\!\cdots\!92}{85\!\cdots\!03}a^{6}-\frac{47\!\cdots\!59}{85\!\cdots\!03}a^{5}+\frac{35\!\cdots\!35}{85\!\cdots\!03}a^{4}+\frac{62\!\cdots\!96}{77\!\cdots\!73}a^{3}+\frac{79\!\cdots\!44}{28\!\cdots\!01}a^{2}-\frac{27\!\cdots\!74}{85\!\cdots\!03}a+\frac{10\!\cdots\!47}{77\!\cdots\!73}$, $\frac{19\!\cdots\!63}{30\!\cdots\!47}a^{15}+\frac{746797016093762}{43\!\cdots\!21}a^{14}-\frac{46\!\cdots\!71}{30\!\cdots\!47}a^{13}-\frac{12\!\cdots\!53}{33\!\cdots\!83}a^{12}+\frac{45\!\cdots\!82}{43\!\cdots\!21}a^{11}-\frac{15\!\cdots\!39}{30\!\cdots\!47}a^{10}-\frac{99\!\cdots\!97}{30\!\cdots\!47}a^{9}+\frac{88\!\cdots\!09}{10\!\cdots\!49}a^{8}+\frac{82\!\cdots\!51}{30\!\cdots\!47}a^{7}+\frac{65\!\cdots\!04}{30\!\cdots\!47}a^{6}-\frac{28\!\cdots\!68}{30\!\cdots\!47}a^{5}-\frac{38\!\cdots\!97}{33\!\cdots\!83}a^{4}-\frac{25\!\cdots\!70}{30\!\cdots\!47}a^{3}+\frac{11\!\cdots\!67}{30\!\cdots\!47}a^{2}+\frac{11\!\cdots\!88}{30\!\cdots\!47}a+\frac{46\!\cdots\!21}{10\!\cdots\!49}$, $\frac{15\!\cdots\!71}{36\!\cdots\!87}a^{15}-\frac{33\!\cdots\!55}{36\!\cdots\!87}a^{14}-\frac{40\!\cdots\!52}{40\!\cdots\!43}a^{13}+\frac{10\!\cdots\!26}{36\!\cdots\!87}a^{12}+\frac{12\!\cdots\!60}{28\!\cdots\!99}a^{11}-\frac{90\!\cdots\!57}{36\!\cdots\!87}a^{10}+\frac{26\!\cdots\!22}{12\!\cdots\!29}a^{9}+\frac{29\!\cdots\!06}{36\!\cdots\!87}a^{8}-\frac{22\!\cdots\!96}{36\!\cdots\!87}a^{7}-\frac{66\!\cdots\!02}{36\!\cdots\!87}a^{6}-\frac{99\!\cdots\!97}{40\!\cdots\!43}a^{5}+\frac{27\!\cdots\!65}{36\!\cdots\!87}a^{4}-\frac{12\!\cdots\!02}{33\!\cdots\!17}a^{3}-\frac{68\!\cdots\!46}{36\!\cdots\!87}a^{2}+\frac{35\!\cdots\!91}{12\!\cdots\!29}a-\frac{26\!\cdots\!33}{33\!\cdots\!17}$, $\frac{17\!\cdots\!56}{37\!\cdots\!13}a^{15}-\frac{31\!\cdots\!06}{47\!\cdots\!31}a^{14}+\frac{11\!\cdots\!24}{33\!\cdots\!17}a^{13}+\frac{40\!\cdots\!39}{33\!\cdots\!17}a^{12}-\frac{28\!\cdots\!23}{15\!\cdots\!77}a^{11}-\frac{19\!\cdots\!27}{33\!\cdots\!17}a^{10}+\frac{53\!\cdots\!57}{33\!\cdots\!17}a^{9}+\frac{13\!\cdots\!86}{33\!\cdots\!17}a^{8}-\frac{27\!\cdots\!77}{37\!\cdots\!13}a^{7}-\frac{22\!\cdots\!41}{33\!\cdots\!17}a^{6}+\frac{51\!\cdots\!95}{33\!\cdots\!17}a^{5}+\frac{13\!\cdots\!27}{33\!\cdots\!17}a^{4}-\frac{11\!\cdots\!95}{10\!\cdots\!49}a^{3}-\frac{48\!\cdots\!09}{33\!\cdots\!17}a^{2}+\frac{34\!\cdots\!00}{33\!\cdots\!17}a-\frac{15\!\cdots\!55}{30\!\cdots\!47}$
| 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: | \( 6129504.74347 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 6129504.74347 \cdot 1}{2\cdot\sqrt{8583027023095873875496489}}\cr\approx \mathstrut & 0.417382838857 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 19*x^14 + 35*x^13 + 90*x^12 - 265*x^11 - 5*x^10 + 1444*x^9 + 1264*x^8 - 3145*x^7 - 7994*x^6 + 493*x^5 + 2574*x^4 - 1223*x^3 + 431*x^2 + 143*x + 121)
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 - 2*x^15 - 19*x^14 + 35*x^13 + 90*x^12 - 265*x^11 - 5*x^10 + 1444*x^9 + 1264*x^8 - 3145*x^7 - 7994*x^6 + 493*x^5 + 2574*x^4 - 1223*x^3 + 431*x^2 + 143*x + 121, 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 - 2*x^15 - 19*x^14 + 35*x^13 + 90*x^12 - 265*x^11 - 5*x^10 + 1444*x^9 + 1264*x^8 - 3145*x^7 - 7994*x^6 + 493*x^5 + 2574*x^4 - 1223*x^3 + 431*x^2 + 143*x + 121);
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 - 2*x^15 - 19*x^14 + 35*x^13 + 90*x^12 - 265*x^11 - 5*x^10 + 1444*x^9 + 1264*x^8 - 3145*x^7 - 7994*x^6 + 493*x^5 + 2574*x^4 - 1223*x^3 + 431*x^2 + 143*x + 121);
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
| \(\Q(\sqrt{53}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{689}) \), 4.4.36517.1 x2, 4.4.8957.1 x2, \(\Q(\sqrt{13}, \sqrt{53})\), 8.4.17335386757.1, 8.4.2929680361933.2, 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: | 8.4.2929680361933.2, 8.4.17335386757.1 |
| Degree 16 sibling: | 16.0.516387172268851080441049.2 |
| Minimal sibling: | 8.4.17335386757.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{8}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.4.3.1 | $x^{4} + 52$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 13.4.3.1 | $x^{4} + 52$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 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 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(53\)
| 53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |