Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 22*x^20 + 130*x^18 - 380*x^16 - 7403*x^14 - 30874*x^12 - 41065*x^10 + 53058*x^8 + 199575*x^6 + 151126*x^4 + 156*x^2 - 49)
gp: K = bnfinit(y^22 + 22*y^20 + 130*y^18 - 380*y^16 - 7403*y^14 - 30874*y^12 - 41065*y^10 + 53058*y^8 + 199575*y^6 + 151126*y^4 + 156*y^2 - 49, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 22*x^20 + 130*x^18 - 380*x^16 - 7403*x^14 - 30874*x^12 - 41065*x^10 + 53058*x^8 + 199575*x^6 + 151126*x^4 + 156*x^2 - 49);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 22*x^20 + 130*x^18 - 380*x^16 - 7403*x^14 - 30874*x^12 - 41065*x^10 + 53058*x^8 + 199575*x^6 + 151126*x^4 + 156*x^2 - 49)
\( x^{22} + 22 x^{20} + 130 x^{18} - 380 x^{16} - 7403 x^{14} - 30874 x^{12} - 41065 x^{10} + 53058 x^{8} + \cdots - 49 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[6, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4129233136056857981979443884256982828952059904\)
\(\medspace = 2^{22}\cdot 74843^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(118.43\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(74843\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{12}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{13}+\frac{1}{3}a^{11}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{17}+\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{3}a^{18}+\frac{1}{3}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{19}+\frac{1}{3}a^{7}+\frac{1}{3}a^{3}$, $\frac{1}{79\!\cdots\!53}a^{20}+\frac{12\!\cdots\!32}{79\!\cdots\!53}a^{18}+\frac{18\!\cdots\!34}{79\!\cdots\!53}a^{16}+\frac{55\!\cdots\!42}{79\!\cdots\!53}a^{14}-\frac{38\!\cdots\!81}{79\!\cdots\!53}a^{12}-\frac{12\!\cdots\!24}{79\!\cdots\!53}a^{10}-\frac{82\!\cdots\!73}{26\!\cdots\!51}a^{8}-\frac{85\!\cdots\!54}{26\!\cdots\!51}a^{6}-\frac{30\!\cdots\!05}{79\!\cdots\!53}a^{4}+\frac{19\!\cdots\!95}{79\!\cdots\!53}a^{2}+\frac{11\!\cdots\!28}{26\!\cdots\!51}$, $\frac{1}{55\!\cdots\!71}a^{21}-\frac{67\!\cdots\!21}{55\!\cdots\!71}a^{19}+\frac{18\!\cdots\!34}{55\!\cdots\!71}a^{17}+\frac{32\!\cdots\!93}{55\!\cdots\!71}a^{15}+\frac{17\!\cdots\!27}{55\!\cdots\!71}a^{13}+\frac{17\!\cdots\!33}{55\!\cdots\!71}a^{11}-\frac{13\!\cdots\!23}{55\!\cdots\!71}a^{9}-\frac{79\!\cdots\!64}{55\!\cdots\!71}a^{7}-\frac{21\!\cdots\!62}{55\!\cdots\!71}a^{5}-\frac{25\!\cdots\!52}{18\!\cdots\!57}a^{3}+\frac{22\!\cdots\!41}{55\!\cdots\!71}a$
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: | $13$ | 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: |
$\frac{314619007211583}{26\!\cdots\!51}a^{20}+\frac{17\!\cdots\!06}{79\!\cdots\!53}a^{18}+\frac{60\!\cdots\!11}{79\!\cdots\!53}a^{16}-\frac{19\!\cdots\!78}{26\!\cdots\!51}a^{14}-\frac{16\!\cdots\!39}{26\!\cdots\!51}a^{12}-\frac{38\!\cdots\!18}{26\!\cdots\!51}a^{10}+\frac{89\!\cdots\!45}{26\!\cdots\!51}a^{8}+\frac{41\!\cdots\!59}{79\!\cdots\!53}a^{6}+\frac{38\!\cdots\!54}{79\!\cdots\!53}a^{4}-\frac{90\!\cdots\!78}{79\!\cdots\!53}a^{2}-\frac{41\!\cdots\!46}{79\!\cdots\!53}$, $\frac{424762463121020}{79\!\cdots\!53}a^{20}+\frac{77\!\cdots\!59}{79\!\cdots\!53}a^{18}+\frac{25\!\cdots\!70}{79\!\cdots\!53}a^{16}-\frac{26\!\cdots\!27}{79\!\cdots\!53}a^{14}-\frac{21\!\cdots\!33}{79\!\cdots\!53}a^{12}-\frac{48\!\cdots\!28}{79\!\cdots\!53}a^{10}+\frac{18\!\cdots\!37}{79\!\cdots\!53}a^{8}+\frac{18\!\cdots\!55}{79\!\cdots\!53}a^{6}+\frac{15\!\cdots\!63}{79\!\cdots\!53}a^{4}-\frac{99\!\cdots\!81}{26\!\cdots\!51}a^{2}+\frac{13\!\cdots\!49}{79\!\cdots\!53}$, $\frac{16\!\cdots\!46}{55\!\cdots\!71}a^{21}+\frac{32\!\cdots\!05}{55\!\cdots\!71}a^{19}+\frac{13\!\cdots\!42}{55\!\cdots\!71}a^{17}-\frac{95\!\cdots\!39}{55\!\cdots\!71}a^{15}-\frac{96\!\cdots\!26}{55\!\cdots\!71}a^{13}-\frac{26\!\cdots\!66}{55\!\cdots\!71}a^{11}-\frac{26\!\cdots\!41}{18\!\cdots\!57}a^{9}+\frac{27\!\cdots\!11}{18\!\cdots\!57}a^{7}+\frac{11\!\cdots\!49}{55\!\cdots\!71}a^{5}+\frac{35\!\cdots\!75}{55\!\cdots\!71}a^{3}-\frac{47\!\cdots\!94}{55\!\cdots\!71}a$, $\frac{19\!\cdots\!62}{18\!\cdots\!57}a^{21}+\frac{11\!\cdots\!88}{55\!\cdots\!71}a^{19}+\frac{19\!\cdots\!65}{18\!\cdots\!57}a^{17}-\frac{27\!\cdots\!38}{55\!\cdots\!71}a^{15}-\frac{37\!\cdots\!57}{55\!\cdots\!71}a^{13}-\frac{13\!\cdots\!78}{55\!\cdots\!71}a^{11}-\frac{13\!\cdots\!46}{55\!\cdots\!71}a^{9}+\frac{93\!\cdots\!73}{18\!\cdots\!57}a^{7}+\frac{76\!\cdots\!55}{55\!\cdots\!71}a^{5}+\frac{51\!\cdots\!98}{55\!\cdots\!71}a^{3}+\frac{17\!\cdots\!73}{55\!\cdots\!71}a$, $\frac{40\!\cdots\!80}{55\!\cdots\!71}a^{21}+\frac{76\!\cdots\!56}{55\!\cdots\!71}a^{19}+\frac{96\!\cdots\!95}{18\!\cdots\!57}a^{17}-\frac{23\!\cdots\!53}{55\!\cdots\!71}a^{15}-\frac{22\!\cdots\!14}{55\!\cdots\!71}a^{13}-\frac{57\!\cdots\!60}{55\!\cdots\!71}a^{11}-\frac{18\!\cdots\!20}{18\!\cdots\!57}a^{9}+\frac{61\!\cdots\!03}{18\!\cdots\!57}a^{7}+\frac{76\!\cdots\!39}{18\!\cdots\!57}a^{5}+\frac{52\!\cdots\!26}{55\!\cdots\!71}a^{3}+\frac{98\!\cdots\!80}{55\!\cdots\!71}a$, $\frac{719413123523887}{79\!\cdots\!53}a^{20}+\frac{13\!\cdots\!88}{79\!\cdots\!53}a^{18}+\frac{58\!\cdots\!49}{79\!\cdots\!53}a^{16}-\frac{39\!\cdots\!24}{79\!\cdots\!53}a^{14}-\frac{41\!\cdots\!89}{79\!\cdots\!53}a^{12}-\frac{12\!\cdots\!33}{79\!\cdots\!53}a^{10}-\frac{21\!\cdots\!36}{26\!\cdots\!51}a^{8}+\frac{11\!\cdots\!50}{26\!\cdots\!51}a^{6}+\frac{60\!\cdots\!54}{79\!\cdots\!53}a^{4}+\frac{28\!\cdots\!74}{79\!\cdots\!53}a^{2}-\frac{16\!\cdots\!39}{26\!\cdots\!51}$, $\frac{687283532387287}{18\!\cdots\!57}a^{21}+\frac{15\!\cdots\!24}{18\!\cdots\!57}a^{19}+\frac{94\!\cdots\!85}{18\!\cdots\!57}a^{17}-\frac{23\!\cdots\!48}{18\!\cdots\!57}a^{15}-\frac{52\!\cdots\!87}{18\!\cdots\!57}a^{13}-\frac{22\!\cdots\!42}{18\!\cdots\!57}a^{11}-\frac{32\!\cdots\!59}{18\!\cdots\!57}a^{9}+\frac{35\!\cdots\!09}{18\!\cdots\!57}a^{7}+\frac{15\!\cdots\!82}{18\!\cdots\!57}a^{5}+\frac{12\!\cdots\!00}{18\!\cdots\!57}a^{3}+\frac{92\!\cdots\!34}{18\!\cdots\!57}a$, $\frac{57827341716106}{79\!\cdots\!53}a^{20}+\frac{904692942282182}{79\!\cdots\!53}a^{18}+\frac{733407458825458}{79\!\cdots\!53}a^{16}-\frac{15\!\cdots\!73}{26\!\cdots\!51}a^{14}-\frac{67\!\cdots\!20}{26\!\cdots\!51}a^{12}+\frac{47\!\cdots\!26}{26\!\cdots\!51}a^{10}+\frac{21\!\cdots\!74}{79\!\cdots\!53}a^{8}+\frac{21\!\cdots\!02}{79\!\cdots\!53}a^{6}-\frac{45\!\cdots\!94}{79\!\cdots\!53}a^{4}-\frac{66\!\cdots\!92}{79\!\cdots\!53}a^{2}-\frac{39\!\cdots\!14}{79\!\cdots\!53}$, $\frac{23635741180322}{79\!\cdots\!53}a^{20}+\frac{692129422187914}{79\!\cdots\!53}a^{18}+\frac{17\!\cdots\!37}{26\!\cdots\!51}a^{16}-\frac{43\!\cdots\!04}{26\!\cdots\!51}a^{14}-\frac{10\!\cdots\!82}{26\!\cdots\!51}a^{12}-\frac{32\!\cdots\!96}{26\!\cdots\!51}a^{10}+\frac{28\!\cdots\!53}{79\!\cdots\!53}a^{8}+\frac{47\!\cdots\!14}{79\!\cdots\!53}a^{6}+\frac{20\!\cdots\!52}{79\!\cdots\!53}a^{4}-\frac{49\!\cdots\!30}{79\!\cdots\!53}a^{2}-\frac{53\!\cdots\!41}{26\!\cdots\!51}$, $\frac{21\!\cdots\!81}{18\!\cdots\!57}a^{21}+\frac{46\!\cdots\!89}{18\!\cdots\!57}a^{19}+\frac{25\!\cdots\!72}{18\!\cdots\!57}a^{17}-\frac{27\!\cdots\!75}{55\!\cdots\!71}a^{15}-\frac{45\!\cdots\!36}{55\!\cdots\!71}a^{13}-\frac{17\!\cdots\!53}{55\!\cdots\!71}a^{11}-\frac{21\!\cdots\!39}{55\!\cdots\!71}a^{9}+\frac{33\!\cdots\!82}{55\!\cdots\!71}a^{7}+\frac{10\!\cdots\!32}{55\!\cdots\!71}a^{5}+\frac{79\!\cdots\!41}{55\!\cdots\!71}a^{3}+\frac{66\!\cdots\!31}{55\!\cdots\!71}a$, $\frac{91\!\cdots\!50}{55\!\cdots\!71}a^{21}+\frac{98\!\cdots\!84}{79\!\cdots\!53}a^{20}+\frac{66\!\cdots\!87}{18\!\cdots\!57}a^{19}+\frac{20\!\cdots\!06}{79\!\cdots\!53}a^{18}+\frac{39\!\cdots\!60}{18\!\cdots\!57}a^{17}+\frac{31\!\cdots\!33}{26\!\cdots\!51}a^{16}-\frac{35\!\cdots\!05}{55\!\cdots\!71}a^{15}-\frac{61\!\cdots\!87}{79\!\cdots\!53}a^{14}-\frac{67\!\cdots\!65}{55\!\cdots\!71}a^{13}-\frac{71\!\cdots\!24}{79\!\cdots\!53}a^{12}-\frac{27\!\cdots\!29}{55\!\cdots\!71}a^{11}-\frac{19\!\cdots\!34}{79\!\cdots\!53}a^{10}-\frac{34\!\cdots\!51}{55\!\cdots\!71}a^{9}+\frac{76\!\cdots\!91}{26\!\cdots\!51}a^{8}+\frac{56\!\cdots\!25}{55\!\cdots\!71}a^{7}+\frac{74\!\cdots\!65}{26\!\cdots\!51}a^{6}+\frac{19\!\cdots\!14}{55\!\cdots\!71}a^{5}+\frac{14\!\cdots\!95}{26\!\cdots\!51}a^{4}+\frac{14\!\cdots\!43}{55\!\cdots\!71}a^{3}+\frac{27\!\cdots\!40}{79\!\cdots\!53}a^{2}+\frac{27\!\cdots\!97}{55\!\cdots\!71}a+\frac{50\!\cdots\!76}{79\!\cdots\!53}$, $\frac{34\!\cdots\!03}{18\!\cdots\!57}a^{21}+\frac{64\!\cdots\!53}{26\!\cdots\!51}a^{20}+\frac{24\!\cdots\!71}{55\!\cdots\!71}a^{19}+\frac{46\!\cdots\!81}{79\!\cdots\!53}a^{18}+\frac{58\!\cdots\!25}{18\!\cdots\!57}a^{17}+\frac{33\!\cdots\!78}{79\!\cdots\!53}a^{16}-\frac{25\!\cdots\!91}{18\!\cdots\!57}a^{15}-\frac{14\!\cdots\!20}{79\!\cdots\!53}a^{14}-\frac{25\!\cdots\!97}{18\!\cdots\!57}a^{13}-\frac{14\!\cdots\!35}{79\!\cdots\!53}a^{12}-\frac{15\!\cdots\!10}{18\!\cdots\!57}a^{11}-\frac{86\!\cdots\!32}{79\!\cdots\!53}a^{10}-\frac{40\!\cdots\!56}{18\!\cdots\!57}a^{9}-\frac{23\!\cdots\!63}{79\!\cdots\!53}a^{8}-\frac{16\!\cdots\!07}{55\!\cdots\!71}a^{7}-\frac{31\!\cdots\!39}{79\!\cdots\!53}a^{6}-\frac{28\!\cdots\!84}{18\!\cdots\!57}a^{5}-\frac{55\!\cdots\!28}{26\!\cdots\!51}a^{4}-\frac{73\!\cdots\!43}{55\!\cdots\!71}a^{3}-\frac{46\!\cdots\!46}{26\!\cdots\!51}a^{2}+\frac{93\!\cdots\!11}{18\!\cdots\!57}a+\frac{53\!\cdots\!06}{79\!\cdots\!53}$, $\frac{17\!\cdots\!43}{79\!\cdots\!53}a^{21}-\frac{15\!\cdots\!57}{26\!\cdots\!51}a^{20}+\frac{50\!\cdots\!05}{79\!\cdots\!53}a^{19}-\frac{13\!\cdots\!75}{79\!\cdots\!53}a^{18}+\frac{56\!\cdots\!81}{79\!\cdots\!53}a^{17}-\frac{14\!\cdots\!89}{79\!\cdots\!53}a^{16}+\frac{32\!\cdots\!81}{79\!\cdots\!53}a^{15}-\frac{83\!\cdots\!21}{79\!\cdots\!53}a^{14}+\frac{89\!\cdots\!11}{79\!\cdots\!53}a^{13}-\frac{23\!\cdots\!18}{79\!\cdots\!53}a^{12}+\frac{72\!\cdots\!01}{79\!\cdots\!53}a^{11}-\frac{19\!\cdots\!11}{79\!\cdots\!53}a^{10}-\frac{21\!\cdots\!18}{79\!\cdots\!53}a^{9}+\frac{57\!\cdots\!12}{79\!\cdots\!53}a^{8}-\frac{56\!\cdots\!98}{79\!\cdots\!53}a^{7}+\frac{49\!\cdots\!60}{26\!\cdots\!51}a^{6}-\frac{38\!\cdots\!95}{79\!\cdots\!53}a^{5}+\frac{33\!\cdots\!80}{26\!\cdots\!51}a^{4}-\frac{15\!\cdots\!64}{26\!\cdots\!51}a^{3}+\frac{10\!\cdots\!42}{79\!\cdots\!53}a^{2}+\frac{12\!\cdots\!93}{79\!\cdots\!53}a-\frac{32\!\cdots\!46}{79\!\cdots\!53}$
| 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: | \( 1292783794150000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 1292783794150000 \cdot 1}{2\cdot\sqrt{4129233136056857981979443884256982828952059904}}\cr\approx \mathstrut & 1.56379608887292 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 22*x^20 + 130*x^18 - 380*x^16 - 7403*x^14 - 30874*x^12 - 41065*x^10 + 53058*x^8 + 199575*x^6 + 151126*x^4 + 156*x^2 - 49)
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^22 + 22*x^20 + 130*x^18 - 380*x^16 - 7403*x^14 - 30874*x^12 - 41065*x^10 + 53058*x^8 + 199575*x^6 + 151126*x^4 + 156*x^2 - 49, 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^22 + 22*x^20 + 130*x^18 - 380*x^16 - 7403*x^14 - 30874*x^12 - 41065*x^10 + 53058*x^8 + 199575*x^6 + 151126*x^4 + 156*x^2 - 49);
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^22 + 22*x^20 + 130*x^18 - 380*x^16 - 7403*x^14 - 30874*x^12 - 41065*x^10 + 53058*x^8 + 199575*x^6 + 151126*x^4 + 156*x^2 - 49);
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
$C_2^{10}.\PSL(2,11)$ (as 22T39):
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 non-solvable group of order 675840 |
| The 56 conjugacy class representatives for $C_2^{10}.\PSL(2,11)$ |
| Character table for $C_2^{10}.\PSL(2,11)$ |
Intermediate fields
| 11.11.31376518243389673201.1 |
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
| Degree 22 sibling: | data not computed |
| Minimal sibling: | 22.14.4129233136056857981979443884256982828952059904.8 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.5.0.1}{5} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $2$ | $11$ | $22$ | |||
|
\(74843\)
| $\Q_{74843}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{74843}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |