Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 55*x^20 + 1177*x^18 - 12639*x^16 + 73898*x^14 - 237402*x^12 + 390258*x^10 - 246906*x^8 - 33011*x^6 + 40249*x^4 - 451*x^2 + 1)
gp: K = bnfinit(y^22 - 55*y^20 + 1177*y^18 - 12639*y^16 + 73898*y^14 - 237402*y^12 + 390258*y^10 - 246906*y^8 - 33011*y^6 + 40249*y^4 - 451*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 55*x^20 + 1177*x^18 - 12639*x^16 + 73898*x^14 - 237402*x^12 + 390258*x^10 - 246906*x^8 - 33011*x^6 + 40249*x^4 - 451*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 55*x^20 + 1177*x^18 - 12639*x^16 + 73898*x^14 - 237402*x^12 + 390258*x^10 - 246906*x^8 - 33011*x^6 + 40249*x^4 - 451*x^2 + 1)
\( x^{22} - 55 x^{20} + 1177 x^{18} - 12639 x^{16} + 73898 x^{14} - 237402 x^{12} + 390258 x^{10} + \cdots + 1 \)
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: | $[20, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-6320614583435223575185822515817648777854976\)
\(\medspace = -\,2^{38}\cdot 7^{10}\cdot 11^{22}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(88.20\) | 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: |
\(2\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{44}a^{12}+\frac{3}{22}a^{10}-\frac{7}{44}a^{8}-\frac{1}{22}a^{6}-\frac{7}{44}a^{4}-\frac{4}{11}a^{2}+\frac{1}{44}$, $\frac{1}{44}a^{13}-\frac{5}{44}a^{11}-\frac{1}{4}a^{10}+\frac{1}{11}a^{9}-\frac{1}{4}a^{8}-\frac{1}{22}a^{7}-\frac{7}{44}a^{5}-\frac{5}{44}a^{3}+\frac{1}{4}a^{2}-\frac{5}{22}a+\frac{1}{4}$, $\frac{1}{44}a^{14}+\frac{1}{44}a^{10}-\frac{1}{11}a^{8}+\frac{5}{44}a^{6}-\frac{9}{22}a^{4}+\frac{9}{44}a^{2}-\frac{3}{22}$, $\frac{1}{44}a^{15}+\frac{1}{44}a^{11}-\frac{1}{11}a^{9}+\frac{5}{44}a^{7}-\frac{9}{22}a^{5}+\frac{9}{44}a^{3}-\frac{3}{22}a$, $\frac{1}{44}a^{16}-\frac{5}{22}a^{10}-\frac{5}{22}a^{8}+\frac{3}{22}a^{6}-\frac{3}{22}a^{4}-\frac{3}{11}a^{2}-\frac{1}{44}$, $\frac{1}{88}a^{17}-\frac{1}{88}a^{16}-\frac{5}{44}a^{11}+\frac{5}{44}a^{10}+\frac{3}{22}a^{9}-\frac{3}{22}a^{8}-\frac{2}{11}a^{7}+\frac{2}{11}a^{6}+\frac{2}{11}a^{5}-\frac{2}{11}a^{4}+\frac{5}{44}a^{3}-\frac{5}{44}a^{2}+\frac{43}{88}a-\frac{43}{88}$, $\frac{1}{88}a^{18}-\frac{1}{88}a^{16}-\frac{3}{44}a^{10}-\frac{5}{44}a^{8}+\frac{3}{22}a^{6}+\frac{3}{22}a^{4}-\frac{39}{88}a^{2}-\frac{3}{8}$, $\frac{1}{88}a^{19}-\frac{1}{88}a^{16}+\frac{3}{44}a^{11}-\frac{3}{22}a^{10}-\frac{5}{22}a^{9}+\frac{5}{44}a^{8}-\frac{1}{22}a^{7}+\frac{2}{11}a^{6}+\frac{7}{22}a^{5}-\frac{2}{11}a^{4}+\frac{37}{88}a^{3}+\frac{3}{22}a^{2}+\frac{4}{11}a+\frac{23}{88}$, $\frac{1}{10\!\cdots\!28}a^{20}-\frac{3013257342723}{10\!\cdots\!28}a^{18}+\frac{5489254673333}{506853095290564}a^{16}+\frac{3006629573045}{506853095290564}a^{14}+\frac{5209250624075}{506853095290564}a^{12}+\frac{28372247731547}{253426547645282}a^{10}-\frac{57162816319371}{253426547645282}a^{8}+\frac{16695858270279}{506853095290564}a^{6}-\frac{302880706394047}{10\!\cdots\!28}a^{4}+\frac{429433860712951}{10\!\cdots\!28}a^{2}-\frac{239307430839203}{506853095290564}$, $\frac{1}{10\!\cdots\!28}a^{21}-\frac{3013257342723}{10\!\cdots\!28}a^{19}-\frac{540879182665}{10\!\cdots\!28}a^{17}-\frac{1}{88}a^{16}+\frac{3006629573045}{506853095290564}a^{15}+\frac{5209250624075}{506853095290564}a^{13}-\frac{3092958928223}{126713273822641}a^{11}-\frac{3}{22}a^{10}-\frac{56728689992087}{506853095290564}a^{9}+\frac{5}{44}a^{8}+\frac{108850966504927}{506853095290564}a^{7}+\frac{2}{11}a^{6}-\frac{487190922863343}{10\!\cdots\!28}a^{5}-\frac{2}{11}a^{4}-\frac{446039667516205}{10\!\cdots\!28}a^{3}+\frac{3}{22}a^{2}-\frac{213668925503793}{10\!\cdots\!28}a+\frac{23}{88}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: | $20$ | 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: |
$\frac{10957034021973}{126713273822641}a^{21}-\frac{48\!\cdots\!17}{10\!\cdots\!28}a^{19}+\frac{10\!\cdots\!77}{10\!\cdots\!28}a^{17}-\frac{27\!\cdots\!83}{253426547645282}a^{15}+\frac{32\!\cdots\!77}{506853095290564}a^{13}-\frac{10\!\cdots\!85}{506853095290564}a^{11}+\frac{42\!\cdots\!18}{126713273822641}a^{9}-\frac{26\!\cdots\!74}{126713273822641}a^{7}-\frac{15\!\cdots\!75}{506853095290564}a^{5}+\frac{31\!\cdots\!73}{92155108234648}a^{3}-\frac{64\!\cdots\!25}{10\!\cdots\!28}a$, $a$, $\frac{12201156313673}{10\!\cdots\!28}a^{20}-\frac{670759747029771}{10\!\cdots\!28}a^{18}+\frac{17\!\cdots\!23}{126713273822641}a^{16}-\frac{76\!\cdots\!31}{506853095290564}a^{14}+\frac{44\!\cdots\!05}{506853095290564}a^{12}-\frac{35\!\cdots\!84}{126713273822641}a^{10}+\frac{11\!\cdots\!29}{253426547645282}a^{8}-\frac{14\!\cdots\!63}{506853095290564}a^{6}-\frac{43\!\cdots\!99}{10\!\cdots\!28}a^{4}+\frac{47\!\cdots\!39}{10\!\cdots\!28}a^{2}-\frac{380538134352361}{126713273822641}$, $\frac{655178940311487}{10\!\cdots\!28}a^{21}-\frac{36\!\cdots\!53}{10\!\cdots\!28}a^{19}+\frac{19\!\cdots\!47}{253426547645282}a^{17}-\frac{20\!\cdots\!81}{253426547645282}a^{15}+\frac{24\!\cdots\!65}{506853095290564}a^{13}-\frac{77\!\cdots\!35}{506853095290564}a^{11}+\frac{31\!\cdots\!55}{126713273822641}a^{9}-\frac{40\!\cdots\!81}{253426547645282}a^{7}-\frac{22\!\cdots\!69}{10\!\cdots\!28}a^{5}+\frac{26\!\cdots\!75}{10\!\cdots\!28}a^{3}-\frac{37\!\cdots\!13}{253426547645282}a$, $\frac{23011077324471}{10\!\cdots\!28}a^{21}-\frac{158275921054231}{126713273822641}a^{19}+\frac{27\!\cdots\!67}{10\!\cdots\!28}a^{17}-\frac{36\!\cdots\!03}{126713273822641}a^{15}+\frac{21\!\cdots\!84}{126713273822641}a^{13}-\frac{13\!\cdots\!41}{253426547645282}a^{11}+\frac{22\!\cdots\!85}{253426547645282}a^{9}-\frac{14\!\cdots\!21}{253426547645282}a^{7}-\frac{71\!\cdots\!55}{10\!\cdots\!28}a^{5}+\frac{11\!\cdots\!77}{126713273822641}a^{3}-\frac{17\!\cdots\!75}{10\!\cdots\!28}a$, $\frac{19031274677633}{506853095290564}a^{21}-\frac{20\!\cdots\!75}{10\!\cdots\!28}a^{19}+\frac{44\!\cdots\!87}{10\!\cdots\!28}a^{17}-\frac{23\!\cdots\!61}{506853095290564}a^{15}+\frac{13\!\cdots\!27}{506853095290564}a^{13}-\frac{22\!\cdots\!99}{253426547645282}a^{11}+\frac{36\!\cdots\!65}{253426547645282}a^{9}-\frac{45\!\cdots\!67}{506853095290564}a^{7}-\frac{18\!\cdots\!91}{126713273822641}a^{5}+\frac{15\!\cdots\!71}{10\!\cdots\!28}a^{3}+\frac{10\!\cdots\!61}{10\!\cdots\!28}a$, $\frac{145506972600265}{506853095290564}a^{21}-\frac{16\!\cdots\!89}{10\!\cdots\!28}a^{19}+\frac{34\!\cdots\!67}{10\!\cdots\!28}a^{17}-\frac{18\!\cdots\!41}{506853095290564}a^{15}+\frac{10\!\cdots\!97}{506853095290564}a^{13}-\frac{86\!\cdots\!39}{126713273822641}a^{11}+\frac{28\!\cdots\!91}{253426547645282}a^{9}-\frac{35\!\cdots\!45}{506853095290564}a^{7}-\frac{12\!\cdots\!76}{126713273822641}a^{5}+\frac{11\!\cdots\!53}{10\!\cdots\!28}a^{3}-\frac{78\!\cdots\!75}{10\!\cdots\!28}a$, $\frac{2572978654690}{126713273822641}a^{20}-\frac{283023529729063}{253426547645282}a^{18}+\frac{60\!\cdots\!95}{253426547645282}a^{16}-\frac{13\!\cdots\!31}{506853095290564}a^{14}+\frac{76\!\cdots\!43}{506853095290564}a^{12}-\frac{24\!\cdots\!27}{506853095290564}a^{10}+\frac{40\!\cdots\!15}{506853095290564}a^{8}-\frac{25\!\cdots\!33}{506853095290564}a^{6}-\frac{35\!\cdots\!35}{506853095290564}a^{4}+\frac{41\!\cdots\!33}{506853095290564}a^{2}-\frac{24\!\cdots\!21}{506853095290564}$, $\frac{232812699012743}{506853095290564}a^{21}-\frac{25119866886791}{10\!\cdots\!28}a^{20}-\frac{12\!\cdots\!61}{506853095290564}a^{19}+\frac{13\!\cdots\!53}{10\!\cdots\!28}a^{18}+\frac{68\!\cdots\!49}{126713273822641}a^{17}-\frac{14\!\cdots\!39}{506853095290564}a^{16}-\frac{29\!\cdots\!05}{506853095290564}a^{15}+\frac{79\!\cdots\!01}{253426547645282}a^{14}+\frac{15\!\cdots\!87}{46077554117324}a^{13}-\frac{92\!\cdots\!55}{506853095290564}a^{12}-\frac{13\!\cdots\!45}{126713273822641}a^{11}+\frac{74\!\cdots\!45}{126713273822641}a^{10}+\frac{45\!\cdots\!25}{253426547645282}a^{9}-\frac{48\!\cdots\!01}{506853095290564}a^{8}-\frac{57\!\cdots\!41}{506853095290564}a^{7}+\frac{15\!\cdots\!01}{253426547645282}a^{6}-\frac{39\!\cdots\!25}{253426547645282}a^{5}+\frac{86\!\cdots\!89}{10\!\cdots\!28}a^{4}+\frac{93\!\cdots\!27}{506853095290564}a^{3}-\frac{91\!\cdots\!35}{92155108234648}a^{2}-\frac{18\!\cdots\!87}{126713273822641}a+\frac{710926289287206}{126713273822641}$, $\frac{232812699012743}{506853095290564}a^{21}+\frac{25119866886791}{10\!\cdots\!28}a^{20}-\frac{12\!\cdots\!61}{506853095290564}a^{19}-\frac{13\!\cdots\!53}{10\!\cdots\!28}a^{18}+\frac{68\!\cdots\!49}{126713273822641}a^{17}+\frac{14\!\cdots\!39}{506853095290564}a^{16}-\frac{29\!\cdots\!05}{506853095290564}a^{15}-\frac{79\!\cdots\!01}{253426547645282}a^{14}+\frac{15\!\cdots\!87}{46077554117324}a^{13}+\frac{92\!\cdots\!55}{506853095290564}a^{12}-\frac{13\!\cdots\!45}{126713273822641}a^{11}-\frac{74\!\cdots\!45}{126713273822641}a^{10}+\frac{45\!\cdots\!25}{253426547645282}a^{9}+\frac{48\!\cdots\!01}{506853095290564}a^{8}-\frac{57\!\cdots\!41}{506853095290564}a^{7}-\frac{15\!\cdots\!01}{253426547645282}a^{6}-\frac{39\!\cdots\!25}{253426547645282}a^{5}-\frac{86\!\cdots\!89}{10\!\cdots\!28}a^{4}+\frac{93\!\cdots\!27}{506853095290564}a^{3}+\frac{91\!\cdots\!35}{92155108234648}a^{2}-\frac{18\!\cdots\!87}{126713273822641}a-\frac{710926289287206}{126713273822641}$, $\frac{40325569986249}{10\!\cdots\!28}a^{20}-\frac{277212205681424}{126713273822641}a^{18}+\frac{47\!\cdots\!91}{10\!\cdots\!28}a^{16}-\frac{25\!\cdots\!25}{506853095290564}a^{14}+\frac{14\!\cdots\!21}{506853095290564}a^{12}-\frac{47\!\cdots\!75}{506853095290564}a^{10}+\frac{78\!\cdots\!11}{506853095290564}a^{8}-\frac{49\!\cdots\!79}{506853095290564}a^{6}-\frac{14\!\cdots\!79}{10\!\cdots\!28}a^{4}+\frac{80\!\cdots\!59}{506853095290564}a^{2}-\frac{49\!\cdots\!65}{10\!\cdots\!28}$, $\frac{2280181480355}{253426547645282}a^{21}-\frac{12930926725747}{506853095290564}a^{20}-\frac{250911171734067}{506853095290564}a^{19}+\frac{355544712359827}{253426547645282}a^{18}+\frac{13\!\cdots\!23}{126713273822641}a^{17}-\frac{76\!\cdots\!75}{253426547645282}a^{16}-\frac{57\!\cdots\!59}{506853095290564}a^{15}+\frac{81\!\cdots\!67}{253426547645282}a^{14}+\frac{84\!\cdots\!35}{126713273822641}a^{13}-\frac{47\!\cdots\!15}{253426547645282}a^{12}-\frac{24\!\cdots\!97}{11519388529331}a^{11}+\frac{30\!\cdots\!61}{506853095290564}a^{10}+\frac{17\!\cdots\!95}{506853095290564}a^{9}-\frac{50\!\cdots\!09}{506853095290564}a^{8}-\frac{11\!\cdots\!39}{506853095290564}a^{7}+\frac{15\!\cdots\!19}{253426547645282}a^{6}-\frac{39\!\cdots\!03}{126713273822641}a^{5}+\frac{44\!\cdots\!09}{506853095290564}a^{4}+\frac{19\!\cdots\!73}{506853095290564}a^{3}-\frac{50\!\cdots\!31}{506853095290564}a^{2}+\frac{20\!\cdots\!87}{506853095290564}a+\frac{41\!\cdots\!11}{506853095290564}$, $\frac{612915714972013}{10\!\cdots\!28}a^{21}+\frac{4372188254061}{92155108234648}a^{20}-\frac{33\!\cdots\!35}{10\!\cdots\!28}a^{19}-\frac{26\!\cdots\!17}{10\!\cdots\!28}a^{18}+\frac{72\!\cdots\!01}{10\!\cdots\!28}a^{17}+\frac{56\!\cdots\!57}{10\!\cdots\!28}a^{16}-\frac{38\!\cdots\!61}{506853095290564}a^{15}-\frac{30\!\cdots\!85}{506853095290564}a^{14}+\frac{20\!\cdots\!27}{46077554117324}a^{13}+\frac{88\!\cdots\!63}{253426547645282}a^{12}-\frac{18\!\cdots\!78}{126713273822641}a^{11}-\frac{28\!\cdots\!17}{253426547645282}a^{10}+\frac{11\!\cdots\!13}{506853095290564}a^{9}+\frac{23\!\cdots\!74}{126713273822641}a^{8}-\frac{75\!\cdots\!65}{506853095290564}a^{7}-\frac{58\!\cdots\!63}{506853095290564}a^{6}-\frac{20\!\cdots\!55}{10\!\cdots\!28}a^{5}-\frac{16\!\cdots\!47}{10\!\cdots\!28}a^{4}+\frac{24\!\cdots\!59}{10\!\cdots\!28}a^{3}+\frac{19\!\cdots\!37}{10\!\cdots\!28}a^{2}-\frac{15\!\cdots\!39}{10\!\cdots\!28}a-\frac{11\!\cdots\!57}{10\!\cdots\!28}$, $\frac{209425340583135}{506853095290564}a^{21}-\frac{46556315022747}{10\!\cdots\!28}a^{20}-\frac{28\!\cdots\!79}{126713273822641}a^{19}+\frac{25\!\cdots\!83}{10\!\cdots\!28}a^{18}+\frac{49\!\cdots\!29}{10\!\cdots\!28}a^{17}-\frac{54\!\cdots\!07}{10\!\cdots\!28}a^{16}-\frac{26\!\cdots\!67}{506853095290564}a^{15}+\frac{29\!\cdots\!37}{506853095290564}a^{14}+\frac{38\!\cdots\!00}{126713273822641}a^{13}-\frac{17\!\cdots\!49}{506853095290564}a^{12}-\frac{49\!\cdots\!25}{506853095290564}a^{11}+\frac{27\!\cdots\!03}{253426547645282}a^{10}+\frac{81\!\cdots\!99}{506853095290564}a^{9}-\frac{90\!\cdots\!27}{506853095290564}a^{8}-\frac{51\!\cdots\!81}{506853095290564}a^{7}+\frac{56\!\cdots\!97}{506853095290564}a^{6}-\frac{73\!\cdots\!67}{506853095290564}a^{5}+\frac{16\!\cdots\!45}{10\!\cdots\!28}a^{4}+\frac{83\!\cdots\!25}{506853095290564}a^{3}-\frac{18\!\cdots\!43}{10\!\cdots\!28}a^{2}-\frac{50\!\cdots\!63}{10\!\cdots\!28}a+\frac{57\!\cdots\!17}{10\!\cdots\!28}$, $\frac{241928933840469}{10\!\cdots\!28}a^{21}+\frac{10773047594207}{10\!\cdots\!28}a^{20}-\frac{13\!\cdots\!19}{10\!\cdots\!28}a^{19}-\frac{592315366134193}{10\!\cdots\!28}a^{18}+\frac{28\!\cdots\!37}{10\!\cdots\!28}a^{17}+\frac{12\!\cdots\!95}{10\!\cdots\!28}a^{16}-\frac{38\!\cdots\!37}{126713273822641}a^{15}-\frac{15\!\cdots\!83}{11519388529331}a^{14}+\frac{89\!\cdots\!25}{506853095290564}a^{13}+\frac{19\!\cdots\!33}{253426547645282}a^{12}-\frac{71\!\cdots\!60}{126713273822641}a^{11}-\frac{63\!\cdots\!59}{253426547645282}a^{10}+\frac{11\!\cdots\!72}{126713273822641}a^{9}+\frac{20\!\cdots\!37}{506853095290564}a^{8}-\frac{74\!\cdots\!86}{126713273822641}a^{7}-\frac{65\!\cdots\!49}{253426547645282}a^{6}-\frac{81\!\cdots\!91}{10\!\cdots\!28}a^{5}-\frac{36\!\cdots\!59}{10\!\cdots\!28}a^{4}+\frac{97\!\cdots\!67}{10\!\cdots\!28}a^{3}+\frac{42\!\cdots\!13}{10\!\cdots\!28}a^{2}-\frac{86\!\cdots\!33}{10\!\cdots\!28}a-\frac{402791681900907}{92155108234648}$, $\frac{768039756194495}{506853095290564}a^{21}+\frac{22318793269613}{253426547645282}a^{20}-\frac{84\!\cdots\!11}{10\!\cdots\!28}a^{19}-\frac{49\!\cdots\!71}{10\!\cdots\!28}a^{18}+\frac{18\!\cdots\!67}{10\!\cdots\!28}a^{17}+\frac{95\!\cdots\!77}{92155108234648}a^{16}-\frac{97\!\cdots\!67}{506853095290564}a^{15}-\frac{28\!\cdots\!25}{253426547645282}a^{14}+\frac{14\!\cdots\!00}{126713273822641}a^{13}+\frac{74\!\cdots\!45}{11519388529331}a^{12}-\frac{18\!\cdots\!19}{506853095290564}a^{11}-\frac{52\!\cdots\!09}{253426547645282}a^{10}+\frac{74\!\cdots\!61}{126713273822641}a^{9}+\frac{43\!\cdots\!74}{126713273822641}a^{8}-\frac{18\!\cdots\!61}{506853095290564}a^{7}-\frac{54\!\cdots\!35}{253426547645282}a^{6}-\frac{26\!\cdots\!19}{506853095290564}a^{5}-\frac{75\!\cdots\!67}{253426547645282}a^{4}+\frac{61\!\cdots\!33}{10\!\cdots\!28}a^{3}+\frac{35\!\cdots\!35}{10\!\cdots\!28}a^{2}-\frac{44\!\cdots\!59}{10\!\cdots\!28}a-\frac{22\!\cdots\!49}{92155108234648}$, $\frac{15605568520123}{46077554117324}a^{21}-\frac{17644086664713}{10\!\cdots\!28}a^{20}-\frac{47\!\cdots\!83}{253426547645282}a^{19}+\frac{485277839367727}{506853095290564}a^{18}+\frac{10\!\cdots\!13}{253426547645282}a^{17}-\frac{20\!\cdots\!25}{10\!\cdots\!28}a^{16}-\frac{21\!\cdots\!05}{506853095290564}a^{15}+\frac{27\!\cdots\!00}{126713273822641}a^{14}+\frac{63\!\cdots\!15}{253426547645282}a^{13}-\frac{32\!\cdots\!91}{253426547645282}a^{12}-\frac{40\!\cdots\!99}{506853095290564}a^{11}+\frac{52\!\cdots\!40}{126713273822641}a^{10}+\frac{33\!\cdots\!45}{253426547645282}a^{9}-\frac{78\!\cdots\!50}{11519388529331}a^{8}-\frac{42\!\cdots\!55}{506853095290564}a^{7}+\frac{54\!\cdots\!07}{126713273822641}a^{6}-\frac{58\!\cdots\!35}{506853095290564}a^{5}+\frac{58\!\cdots\!57}{10\!\cdots\!28}a^{4}+\frac{62\!\cdots\!15}{46077554117324}a^{3}-\frac{35\!\cdots\!07}{506853095290564}a^{2}-\frac{13\!\cdots\!23}{126713273822641}a+\frac{67\!\cdots\!21}{10\!\cdots\!28}$, $\frac{216607261056765}{10\!\cdots\!28}a^{21}-\frac{4787434373559}{10\!\cdots\!28}a^{20}-\frac{59\!\cdots\!61}{506853095290564}a^{19}+\frac{263030142213345}{10\!\cdots\!28}a^{18}+\frac{25\!\cdots\!35}{10\!\cdots\!28}a^{17}-\frac{14\!\cdots\!85}{253426547645282}a^{16}-\frac{13\!\cdots\!17}{506853095290564}a^{15}+\frac{30\!\cdots\!75}{506853095290564}a^{14}+\frac{80\!\cdots\!23}{506853095290564}a^{13}-\frac{17\!\cdots\!43}{506853095290564}a^{12}-\frac{12\!\cdots\!15}{253426547645282}a^{11}+\frac{55\!\cdots\!97}{506853095290564}a^{10}+\frac{21\!\cdots\!19}{253426547645282}a^{9}-\frac{90\!\cdots\!75}{506853095290564}a^{8}-\frac{26\!\cdots\!35}{506853095290564}a^{7}+\frac{55\!\cdots\!95}{506853095290564}a^{6}-\frac{72\!\cdots\!95}{10\!\cdots\!28}a^{5}+\frac{18\!\cdots\!69}{10\!\cdots\!28}a^{4}+\frac{43\!\cdots\!03}{506853095290564}a^{3}-\frac{17\!\cdots\!23}{10\!\cdots\!28}a^{2}-\frac{75\!\cdots\!63}{10\!\cdots\!28}a+\frac{802071847048671}{506853095290564}$, $\frac{19091890238203}{126713273822641}a^{21}+\frac{11201982493187}{10\!\cdots\!28}a^{20}-\frac{10\!\cdots\!51}{126713273822641}a^{19}-\frac{615874299732083}{10\!\cdots\!28}a^{18}+\frac{17\!\cdots\!59}{10\!\cdots\!28}a^{17}+\frac{11\!\cdots\!25}{92155108234648}a^{16}-\frac{96\!\cdots\!81}{506853095290564}a^{15}-\frac{17\!\cdots\!78}{126713273822641}a^{14}+\frac{56\!\cdots\!03}{506853095290564}a^{13}+\frac{41\!\cdots\!73}{506853095290564}a^{12}-\frac{18\!\cdots\!97}{506853095290564}a^{11}-\frac{13\!\cdots\!27}{506853095290564}a^{10}+\frac{14\!\cdots\!11}{253426547645282}a^{9}+\frac{21\!\cdots\!17}{506853095290564}a^{8}-\frac{18\!\cdots\!31}{506853095290564}a^{7}-\frac{68\!\cdots\!71}{253426547645282}a^{6}-\frac{25\!\cdots\!67}{506853095290564}a^{5}-\frac{37\!\cdots\!77}{10\!\cdots\!28}a^{4}+\frac{30\!\cdots\!89}{506853095290564}a^{3}+\frac{44\!\cdots\!73}{10\!\cdots\!28}a^{2}-\frac{55\!\cdots\!43}{10\!\cdots\!28}a-\frac{31\!\cdots\!37}{10\!\cdots\!28}$, $\frac{5897070983657}{126713273822641}a^{21}-\frac{42289375167343}{10\!\cdots\!28}a^{20}-\frac{25\!\cdots\!51}{10\!\cdots\!28}a^{19}+\frac{23\!\cdots\!91}{10\!\cdots\!28}a^{18}+\frac{69\!\cdots\!39}{126713273822641}a^{17}-\frac{49\!\cdots\!31}{10\!\cdots\!28}a^{16}-\frac{13\!\cdots\!31}{23038777058662}a^{15}+\frac{26\!\cdots\!47}{506853095290564}a^{14}+\frac{86\!\cdots\!79}{253426547645282}a^{13}-\frac{78\!\cdots\!85}{253426547645282}a^{12}-\frac{27\!\cdots\!61}{253426547645282}a^{11}+\frac{50\!\cdots\!31}{506853095290564}a^{10}+\frac{90\!\cdots\!51}{506853095290564}a^{9}-\frac{82\!\cdots\!67}{506853095290564}a^{8}-\frac{27\!\cdots\!89}{253426547645282}a^{7}+\frac{51\!\cdots\!81}{506853095290564}a^{6}-\frac{48\!\cdots\!47}{253426547645282}a^{5}+\frac{14\!\cdots\!55}{10\!\cdots\!28}a^{4}+\frac{18\!\cdots\!75}{10\!\cdots\!28}a^{3}-\frac{16\!\cdots\!25}{10\!\cdots\!28}a^{2}+\frac{21\!\cdots\!45}{506853095290564}a+\frac{742031909324855}{92155108234648}$
| 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: | \( 655923805276000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{1}\cdot 655923805276000 \cdot 1}{2\cdot\sqrt{6320614583435223575185822515817648777854976}}\cr\approx \mathstrut & 0.859455755920438 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 55*x^20 + 1177*x^18 - 12639*x^16 + 73898*x^14 - 237402*x^12 + 390258*x^10 - 246906*x^8 - 33011*x^6 + 40249*x^4 - 451*x^2 + 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^22 - 55*x^20 + 1177*x^18 - 12639*x^16 + 73898*x^14 - 237402*x^12 + 390258*x^10 - 246906*x^8 - 33011*x^6 + 40249*x^4 - 451*x^2 + 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^22 - 55*x^20 + 1177*x^18 - 12639*x^16 + 73898*x^14 - 237402*x^12 + 390258*x^10 - 246906*x^8 - 33011*x^6 + 40249*x^4 - 451*x^2 + 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^22 - 55*x^20 + 1177*x^18 - 12639*x^16 + 73898*x^14 - 237402*x^12 + 390258*x^10 - 246906*x^8 - 33011*x^6 + 40249*x^4 - 451*x^2 + 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
$C_2\times C_2^{10}.F_{11}$ (as 22T37):
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 225280 |
| The 88 conjugacy class representatives for $C_2\times C_2^{10}.F_{11}$ |
| Character table for $C_2\times C_2^{10}.F_{11}$ |
Intermediate fields
| 11.11.4910318845910094848.1 |
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 |
| Degree 44 siblings: | data not computed |
| 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 | R | ${\href{/padicField/3.10.0.1}{10} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $22$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | $20{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $22$ | $22$ | $1$ | $38$ | |||
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.10.5.1 | $x^{10} + 2401 x^{2} - 67228$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 7.10.5.1 | $x^{10} + 2401 x^{2} - 67228$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(11\)
| Deg $22$ | $11$ | $2$ | $22$ |