Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 - 23*x^20 + 427*x^19 - 333*x^18 - 8101*x^17 + 19825*x^16 + 54452*x^15 - 199899*x^14 - 268055*x^13 + 1619754*x^12 - 923591*x^11 - 3715861*x^10 + 3187790*x^9 + 19675195*x^8 - 56103742*x^7 + 59561975*x^6 - 21663766*x^5 + 150503099*x^4 - 437293737*x^3 + 963528183*x^2 - 867599298*x + 639704333)
gp: K = bnfinit(y^22 - 9*y^21 - 23*y^20 + 427*y^19 - 333*y^18 - 8101*y^17 + 19825*y^16 + 54452*y^15 - 199899*y^14 - 268055*y^13 + 1619754*y^12 - 923591*y^11 - 3715861*y^10 + 3187790*y^9 + 19675195*y^8 - 56103742*y^7 + 59561975*y^6 - 21663766*y^5 + 150503099*y^4 - 437293737*y^3 + 963528183*y^2 - 867599298*y + 639704333, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 9*x^21 - 23*x^20 + 427*x^19 - 333*x^18 - 8101*x^17 + 19825*x^16 + 54452*x^15 - 199899*x^14 - 268055*x^13 + 1619754*x^12 - 923591*x^11 - 3715861*x^10 + 3187790*x^9 + 19675195*x^8 - 56103742*x^7 + 59561975*x^6 - 21663766*x^5 + 150503099*x^4 - 437293737*x^3 + 963528183*x^2 - 867599298*x + 639704333);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 9*x^21 - 23*x^20 + 427*x^19 - 333*x^18 - 8101*x^17 + 19825*x^16 + 54452*x^15 - 199899*x^14 - 268055*x^13 + 1619754*x^12 - 923591*x^11 - 3715861*x^10 + 3187790*x^9 + 19675195*x^8 - 56103742*x^7 + 59561975*x^6 - 21663766*x^5 + 150503099*x^4 - 437293737*x^3 + 963528183*x^2 - 867599298*x + 639704333)
\( x^{22} - 9 x^{21} - 23 x^{20} + 427 x^{19} - 333 x^{18} - 8101 x^{17} + 19825 x^{16} + 54452 x^{15} + \cdots + 639704333 \)
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: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-1922554115559596594815766124679345986918150636343\)
\(\medspace = -\,7^{11}\cdot 89^{20}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(156.57\) | 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: | $7^{1/2}89^{10/11}\approx 156.57475950951633$ | ||
| Ramified primes: |
\(7\), \(89\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $22$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(623=7\cdot 89\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{623}(512,·)$, $\chi_{623}(1,·)$, $\chi_{623}(134,·)$, $\chi_{623}(449,·)$, $\chi_{623}(8,·)$, $\chi_{623}(461,·)$, $\chi_{623}(78,·)$, $\chi_{623}(601,·)$, $\chi_{623}(153,·)$, $\chi_{623}(538,·)$, $\chi_{623}(90,·)$, $\chi_{623}(223,·)$, $\chi_{623}(97,·)$, $\chi_{623}(484,·)$, $\chi_{623}(358,·)$, $\chi_{623}(167,·)$, $\chi_{623}(64,·)$, $\chi_{623}(477,·)$, $\chi_{623}(372,·)$, $\chi_{623}(566,·)$, $\chi_{623}(573,·)$, $\chi_{623}(447,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{1024}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{37}a^{15}-\frac{8}{37}a^{14}+\frac{13}{37}a^{13}+\frac{8}{37}a^{12}-\frac{5}{37}a^{11}+\frac{3}{37}a^{10}+\frac{9}{37}a^{9}+\frac{9}{37}a^{8}-\frac{6}{37}a^{7}-\frac{18}{37}a^{6}+\frac{17}{37}a^{5}+\frac{3}{37}a^{4}+\frac{12}{37}a^{3}+\frac{4}{37}a^{2}-\frac{4}{37}a-\frac{1}{37}$, $\frac{1}{37}a^{16}-\frac{14}{37}a^{14}+\frac{1}{37}a^{13}-\frac{15}{37}a^{12}-\frac{4}{37}a^{10}+\frac{7}{37}a^{9}-\frac{8}{37}a^{8}+\frac{8}{37}a^{7}-\frac{16}{37}a^{6}-\frac{9}{37}a^{5}-\frac{1}{37}a^{4}-\frac{11}{37}a^{3}-\frac{9}{37}a^{2}+\frac{4}{37}a-\frac{8}{37}$, $\frac{1}{37}a^{17}-\frac{18}{37}a^{13}+\frac{1}{37}a^{12}+\frac{12}{37}a^{10}+\frac{7}{37}a^{9}-\frac{14}{37}a^{8}+\frac{11}{37}a^{7}-\frac{2}{37}a^{6}+\frac{15}{37}a^{5}-\frac{6}{37}a^{4}+\frac{11}{37}a^{3}-\frac{14}{37}a^{2}+\frac{10}{37}a-\frac{14}{37}$, $\frac{1}{37}a^{18}-\frac{18}{37}a^{14}+\frac{1}{37}a^{13}+\frac{12}{37}a^{11}+\frac{7}{37}a^{10}-\frac{14}{37}a^{9}+\frac{11}{37}a^{8}-\frac{2}{37}a^{7}+\frac{15}{37}a^{6}-\frac{6}{37}a^{5}+\frac{11}{37}a^{4}-\frac{14}{37}a^{3}+\frac{10}{37}a^{2}-\frac{14}{37}a$, $\frac{1}{37}a^{19}+\frac{5}{37}a^{14}+\frac{12}{37}a^{13}+\frac{8}{37}a^{12}-\frac{9}{37}a^{11}+\frac{3}{37}a^{10}-\frac{12}{37}a^{9}+\frac{12}{37}a^{8}+\frac{18}{37}a^{7}+\frac{3}{37}a^{6}-\frac{16}{37}a^{5}+\frac{3}{37}a^{4}+\frac{4}{37}a^{3}-\frac{16}{37}a^{2}+\frac{2}{37}a-\frac{18}{37}$, $\frac{1}{677618999}a^{20}+\frac{863887}{677618999}a^{19}+\frac{2420329}{677618999}a^{18}-\frac{467112}{677618999}a^{17}-\frac{3118574}{677618999}a^{16}+\frac{6239916}{677618999}a^{15}-\frac{273271215}{677618999}a^{14}+\frac{174789670}{677618999}a^{13}+\frac{68525117}{677618999}a^{12}-\frac{90058046}{677618999}a^{11}+\frac{326778181}{677618999}a^{10}+\frac{168887638}{677618999}a^{9}-\frac{182055027}{677618999}a^{8}+\frac{298447882}{677618999}a^{7}+\frac{319570531}{677618999}a^{6}-\frac{7597661}{18314027}a^{5}-\frac{103630759}{677618999}a^{4}-\frac{3476195}{18314027}a^{3}-\frac{153955802}{677618999}a^{2}+\frac{258323764}{677618999}a+\frac{241861221}{677618999}$, $\frac{1}{97\!\cdots\!29}a^{21}-\frac{62\!\cdots\!96}{97\!\cdots\!29}a^{20}-\frac{35\!\cdots\!59}{26\!\cdots\!17}a^{19}-\frac{39\!\cdots\!35}{97\!\cdots\!29}a^{18}-\frac{24\!\cdots\!44}{97\!\cdots\!29}a^{17}-\frac{10\!\cdots\!23}{97\!\cdots\!29}a^{16}-\frac{79\!\cdots\!65}{97\!\cdots\!29}a^{15}+\frac{25\!\cdots\!54}{97\!\cdots\!29}a^{14}+\frac{27\!\cdots\!69}{97\!\cdots\!29}a^{13}-\frac{34\!\cdots\!97}{97\!\cdots\!29}a^{12}-\frac{24\!\cdots\!96}{97\!\cdots\!29}a^{11}+\frac{93\!\cdots\!24}{97\!\cdots\!29}a^{10}-\frac{24\!\cdots\!20}{97\!\cdots\!29}a^{9}-\frac{27\!\cdots\!78}{97\!\cdots\!29}a^{8}-\frac{48\!\cdots\!20}{97\!\cdots\!29}a^{7}-\frac{36\!\cdots\!57}{97\!\cdots\!29}a^{6}+\frac{70\!\cdots\!32}{26\!\cdots\!17}a^{5}+\frac{54\!\cdots\!94}{97\!\cdots\!29}a^{4}-\frac{10\!\cdots\!84}{26\!\cdots\!17}a^{3}-\frac{85\!\cdots\!64}{97\!\cdots\!29}a^{2}+\frac{12\!\cdots\!44}{97\!\cdots\!29}a-\frac{18\!\cdots\!02}{97\!\cdots\!29}$
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
$C_{1084931}$, which has order $1084931$ (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: | $10$ | 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: |
$\frac{17\!\cdots\!26}{97\!\cdots\!29}a^{21}-\frac{90\!\cdots\!54}{97\!\cdots\!29}a^{20}-\frac{26\!\cdots\!86}{26\!\cdots\!17}a^{19}+\frac{54\!\cdots\!34}{97\!\cdots\!29}a^{18}+\frac{23\!\cdots\!00}{97\!\cdots\!29}a^{17}-\frac{14\!\cdots\!14}{97\!\cdots\!29}a^{16}-\frac{25\!\cdots\!96}{97\!\cdots\!29}a^{15}+\frac{19\!\cdots\!21}{97\!\cdots\!29}a^{14}+\frac{19\!\cdots\!69}{97\!\cdots\!29}a^{13}-\frac{15\!\cdots\!97}{97\!\cdots\!29}a^{12}-\frac{76\!\cdots\!09}{97\!\cdots\!29}a^{11}+\frac{85\!\cdots\!87}{97\!\cdots\!29}a^{10}+\frac{11\!\cdots\!81}{97\!\cdots\!29}a^{9}-\frac{28\!\cdots\!04}{97\!\cdots\!29}a^{8}+\frac{12\!\cdots\!09}{97\!\cdots\!29}a^{7}+\frac{70\!\cdots\!29}{97\!\cdots\!29}a^{6}-\frac{27\!\cdots\!05}{26\!\cdots\!17}a^{5}-\frac{28\!\cdots\!52}{97\!\cdots\!29}a^{4}+\frac{25\!\cdots\!03}{71\!\cdots\!41}a^{3}+\frac{42\!\cdots\!63}{97\!\cdots\!29}a^{2}-\frac{57\!\cdots\!99}{97\!\cdots\!29}a+\frac{14\!\cdots\!58}{97\!\cdots\!29}$, $\frac{21\!\cdots\!64}{97\!\cdots\!29}a^{21}-\frac{10\!\cdots\!26}{97\!\cdots\!29}a^{20}-\frac{33\!\cdots\!78}{26\!\cdots\!17}a^{19}+\frac{63\!\cdots\!55}{97\!\cdots\!29}a^{18}+\frac{30\!\cdots\!23}{97\!\cdots\!29}a^{17}-\frac{17\!\cdots\!13}{97\!\cdots\!29}a^{16}-\frac{37\!\cdots\!88}{97\!\cdots\!29}a^{15}+\frac{23\!\cdots\!33}{97\!\cdots\!29}a^{14}+\frac{29\!\cdots\!17}{97\!\cdots\!29}a^{13}-\frac{20\!\cdots\!03}{97\!\cdots\!29}a^{12}-\frac{13\!\cdots\!93}{97\!\cdots\!29}a^{11}+\frac{11\!\cdots\!03}{97\!\cdots\!29}a^{10}+\frac{24\!\cdots\!93}{97\!\cdots\!29}a^{9}-\frac{38\!\cdots\!17}{97\!\cdots\!29}a^{8}+\frac{11\!\cdots\!33}{97\!\cdots\!29}a^{7}+\frac{10\!\cdots\!45}{97\!\cdots\!29}a^{6}-\frac{37\!\cdots\!59}{26\!\cdots\!17}a^{5}-\frac{43\!\cdots\!52}{97\!\cdots\!29}a^{4}+\frac{11\!\cdots\!32}{26\!\cdots\!17}a^{3}+\frac{64\!\cdots\!68}{97\!\cdots\!29}a^{2}-\frac{90\!\cdots\!67}{97\!\cdots\!29}a+\frac{19\!\cdots\!09}{97\!\cdots\!29}$, $\frac{47\!\cdots\!84}{97\!\cdots\!29}a^{21}-\frac{13\!\cdots\!16}{97\!\cdots\!29}a^{20}-\frac{94\!\cdots\!64}{26\!\cdots\!17}a^{19}+\frac{11\!\cdots\!46}{97\!\cdots\!29}a^{18}+\frac{10\!\cdots\!00}{97\!\cdots\!29}a^{17}-\frac{36\!\cdots\!45}{97\!\cdots\!29}a^{16}-\frac{16\!\cdots\!82}{97\!\cdots\!29}a^{15}+\frac{62\!\cdots\!60}{97\!\cdots\!29}a^{14}+\frac{13\!\cdots\!30}{97\!\cdots\!29}a^{13}-\frac{57\!\cdots\!43}{97\!\cdots\!29}a^{12}-\frac{79\!\cdots\!14}{97\!\cdots\!29}a^{11}+\frac{36\!\cdots\!65}{97\!\cdots\!29}a^{10}+\frac{17\!\cdots\!08}{97\!\cdots\!29}a^{9}-\frac{12\!\cdots\!53}{97\!\cdots\!29}a^{8}-\frac{99\!\cdots\!54}{97\!\cdots\!29}a^{7}+\frac{44\!\cdots\!97}{97\!\cdots\!29}a^{6}-\frac{14\!\cdots\!42}{26\!\cdots\!17}a^{5}-\frac{17\!\cdots\!76}{97\!\cdots\!29}a^{4}+\frac{27\!\cdots\!80}{26\!\cdots\!17}a^{3}+\frac{29\!\cdots\!12}{97\!\cdots\!29}a^{2}-\frac{43\!\cdots\!61}{97\!\cdots\!29}a+\frac{71\!\cdots\!33}{97\!\cdots\!29}$, $\frac{24\!\cdots\!60}{97\!\cdots\!29}a^{21}-\frac{11\!\cdots\!76}{97\!\cdots\!29}a^{20}-\frac{37\!\cdots\!56}{26\!\cdots\!17}a^{19}+\frac{73\!\cdots\!11}{97\!\cdots\!29}a^{18}+\frac{33\!\cdots\!13}{97\!\cdots\!29}a^{17}-\frac{19\!\cdots\!60}{97\!\cdots\!29}a^{16}-\frac{39\!\cdots\!50}{97\!\cdots\!29}a^{15}+\frac{26\!\cdots\!70}{97\!\cdots\!29}a^{14}+\frac{30\!\cdots\!06}{97\!\cdots\!29}a^{13}-\frac{22\!\cdots\!88}{97\!\cdots\!29}a^{12}-\frac{13\!\cdots\!74}{97\!\cdots\!29}a^{11}+\frac{12\!\cdots\!96}{97\!\cdots\!29}a^{10}+\frac{22\!\cdots\!28}{97\!\cdots\!29}a^{9}-\frac{41\!\cdots\!68}{97\!\cdots\!29}a^{8}+\frac{14\!\cdots\!92}{97\!\cdots\!29}a^{7}+\frac{10\!\cdots\!71}{97\!\cdots\!29}a^{6}-\frac{40\!\cdots\!85}{26\!\cdots\!17}a^{5}-\frac{44\!\cdots\!18}{97\!\cdots\!29}a^{4}+\frac{12\!\cdots\!12}{26\!\cdots\!17}a^{3}+\frac{66\!\cdots\!54}{97\!\cdots\!29}a^{2}-\frac{91\!\cdots\!26}{97\!\cdots\!29}a+\frac{21\!\cdots\!47}{97\!\cdots\!29}$, $\frac{10\!\cdots\!18}{97\!\cdots\!29}a^{21}-\frac{53\!\cdots\!60}{97\!\cdots\!29}a^{20}-\frac{14\!\cdots\!28}{26\!\cdots\!17}a^{19}+\frac{31\!\cdots\!72}{97\!\cdots\!29}a^{18}+\frac{12\!\cdots\!90}{97\!\cdots\!29}a^{17}-\frac{81\!\cdots\!25}{97\!\cdots\!29}a^{16}-\frac{12\!\cdots\!56}{97\!\cdots\!29}a^{15}+\frac{10\!\cdots\!89}{97\!\cdots\!29}a^{14}+\frac{91\!\cdots\!95}{97\!\cdots\!29}a^{13}-\frac{87\!\cdots\!22}{97\!\cdots\!29}a^{12}-\frac{32\!\cdots\!91}{97\!\cdots\!29}a^{11}+\frac{45\!\cdots\!53}{97\!\cdots\!29}a^{10}+\frac{35\!\cdots\!06}{97\!\cdots\!29}a^{9}-\frac{15\!\cdots\!51}{97\!\cdots\!29}a^{8}+\frac{78\!\cdots\!59}{97\!\cdots\!29}a^{7}+\frac{34\!\cdots\!39}{97\!\cdots\!29}a^{6}-\frac{14\!\cdots\!44}{26\!\cdots\!17}a^{5}-\frac{13\!\cdots\!85}{97\!\cdots\!29}a^{4}+\frac{52\!\cdots\!33}{26\!\cdots\!17}a^{3}+\frac{20\!\cdots\!55}{97\!\cdots\!29}a^{2}-\frac{26\!\cdots\!59}{97\!\cdots\!29}a+\frac{74\!\cdots\!10}{97\!\cdots\!29}$, $\frac{13\!\cdots\!74}{97\!\cdots\!29}a^{21}-\frac{96\!\cdots\!34}{97\!\cdots\!29}a^{20}-\frac{12\!\cdots\!34}{26\!\cdots\!17}a^{19}+\frac{50\!\cdots\!06}{97\!\cdots\!29}a^{18}+\frac{45\!\cdots\!98}{97\!\cdots\!29}a^{17}-\frac{10\!\cdots\!00}{97\!\cdots\!29}a^{16}+\frac{76\!\cdots\!74}{97\!\cdots\!29}a^{15}+\frac{10\!\cdots\!89}{97\!\cdots\!29}a^{14}-\frac{10\!\cdots\!29}{97\!\cdots\!29}a^{13}-\frac{73\!\cdots\!36}{97\!\cdots\!29}a^{12}+\frac{11\!\cdots\!01}{97\!\cdots\!29}a^{11}+\frac{20\!\cdots\!32}{97\!\cdots\!29}a^{10}-\frac{35\!\cdots\!61}{97\!\cdots\!29}a^{9}-\frac{59\!\cdots\!19}{97\!\cdots\!29}a^{8}+\frac{22\!\cdots\!91}{97\!\cdots\!29}a^{7}-\frac{25\!\cdots\!98}{97\!\cdots\!29}a^{6}-\frac{67\!\cdots\!03}{26\!\cdots\!17}a^{5}+\frac{15\!\cdots\!48}{97\!\cdots\!29}a^{4}+\frac{58\!\cdots\!81}{26\!\cdots\!17}a^{3}-\frac{25\!\cdots\!00}{97\!\cdots\!29}a^{2}+\frac{44\!\cdots\!94}{97\!\cdots\!29}a+\frac{39\!\cdots\!23}{97\!\cdots\!29}$, $\frac{65\!\cdots\!90}{97\!\cdots\!29}a^{21}-\frac{33\!\cdots\!77}{97\!\cdots\!29}a^{20}-\frac{97\!\cdots\!38}{26\!\cdots\!17}a^{19}+\frac{20\!\cdots\!63}{97\!\cdots\!29}a^{18}+\frac{84\!\cdots\!48}{97\!\cdots\!29}a^{17}-\frac{52\!\cdots\!91}{97\!\cdots\!29}a^{16}-\frac{93\!\cdots\!66}{97\!\cdots\!29}a^{15}+\frac{69\!\cdots\!55}{97\!\cdots\!29}a^{14}+\frac{69\!\cdots\!81}{97\!\cdots\!29}a^{13}-\frac{58\!\cdots\!59}{97\!\cdots\!29}a^{12}-\frac{27\!\cdots\!93}{97\!\cdots\!29}a^{11}+\frac{31\!\cdots\!36}{97\!\cdots\!29}a^{10}+\frac{40\!\cdots\!62}{97\!\cdots\!29}a^{9}-\frac{10\!\cdots\!53}{97\!\cdots\!29}a^{8}+\frac{45\!\cdots\!15}{97\!\cdots\!29}a^{7}+\frac{25\!\cdots\!95}{97\!\cdots\!29}a^{6}-\frac{98\!\cdots\!54}{26\!\cdots\!17}a^{5}-\frac{10\!\cdots\!97}{97\!\cdots\!29}a^{4}+\frac{33\!\cdots\!45}{26\!\cdots\!17}a^{3}+\frac{15\!\cdots\!84}{97\!\cdots\!29}a^{2}-\frac{20\!\cdots\!74}{97\!\cdots\!29}a+\frac{51\!\cdots\!74}{97\!\cdots\!29}$, $\frac{69\!\cdots\!54}{97\!\cdots\!29}a^{21}-\frac{37\!\cdots\!89}{97\!\cdots\!29}a^{20}-\frac{99\!\cdots\!32}{26\!\cdots\!17}a^{19}+\frac{22\!\cdots\!03}{97\!\cdots\!29}a^{18}+\frac{80\!\cdots\!00}{97\!\cdots\!29}a^{17}-\frac{56\!\cdots\!06}{97\!\cdots\!29}a^{16}-\frac{79\!\cdots\!36}{97\!\cdots\!29}a^{15}+\frac{71\!\cdots\!14}{97\!\cdots\!29}a^{14}+\frac{54\!\cdots\!76}{97\!\cdots\!29}a^{13}-\frac{58\!\cdots\!61}{97\!\cdots\!29}a^{12}-\frac{16\!\cdots\!22}{97\!\cdots\!29}a^{11}+\frac{29\!\cdots\!62}{97\!\cdots\!29}a^{10}+\frac{87\!\cdots\!83}{97\!\cdots\!29}a^{9}-\frac{10\!\cdots\!81}{97\!\cdots\!29}a^{8}+\frac{58\!\cdots\!78}{97\!\cdots\!29}a^{7}+\frac{21\!\cdots\!60}{97\!\cdots\!29}a^{6}-\frac{90\!\cdots\!32}{26\!\cdots\!17}a^{5}-\frac{81\!\cdots\!06}{97\!\cdots\!29}a^{4}+\frac{35\!\cdots\!49}{26\!\cdots\!17}a^{3}+\frac{11\!\cdots\!47}{97\!\cdots\!29}a^{2}-\frac{15\!\cdots\!31}{97\!\cdots\!29}a+\frac{47\!\cdots\!51}{97\!\cdots\!29}$, $\frac{16\!\cdots\!26}{97\!\cdots\!29}a^{21}-\frac{78\!\cdots\!51}{97\!\cdots\!29}a^{20}-\frac{25\!\cdots\!18}{26\!\cdots\!17}a^{19}+\frac{48\!\cdots\!52}{97\!\cdots\!29}a^{18}+\frac{23\!\cdots\!91}{97\!\cdots\!29}a^{17}-\frac{13\!\cdots\!49}{97\!\cdots\!29}a^{16}-\frac{29\!\cdots\!22}{97\!\cdots\!29}a^{15}+\frac{18\!\cdots\!92}{97\!\cdots\!29}a^{14}+\frac{22\!\cdots\!52}{97\!\cdots\!29}a^{13}-\frac{15\!\cdots\!67}{97\!\cdots\!29}a^{12}-\frac{10\!\cdots\!76}{97\!\cdots\!29}a^{11}+\frac{86\!\cdots\!49}{97\!\cdots\!29}a^{10}+\frac{18\!\cdots\!20}{97\!\cdots\!29}a^{9}-\frac{29\!\cdots\!33}{97\!\cdots\!29}a^{8}+\frac{87\!\cdots\!06}{97\!\cdots\!29}a^{7}+\frac{79\!\cdots\!76}{97\!\cdots\!29}a^{6}-\frac{28\!\cdots\!48}{26\!\cdots\!17}a^{5}-\frac{33\!\cdots\!07}{97\!\cdots\!29}a^{4}+\frac{87\!\cdots\!07}{26\!\cdots\!17}a^{3}+\frac{49\!\cdots\!81}{97\!\cdots\!29}a^{2}-\frac{69\!\cdots\!41}{97\!\cdots\!29}a+\frac{15\!\cdots\!87}{97\!\cdots\!29}$, $\frac{51\!\cdots\!62}{97\!\cdots\!29}a^{21}-\frac{26\!\cdots\!33}{97\!\cdots\!29}a^{20}-\frac{76\!\cdots\!30}{26\!\cdots\!17}a^{19}+\frac{16\!\cdots\!67}{97\!\cdots\!29}a^{18}+\frac{65\!\cdots\!78}{97\!\cdots\!29}a^{17}-\frac{41\!\cdots\!78}{97\!\cdots\!29}a^{16}-\frac{71\!\cdots\!76}{97\!\cdots\!29}a^{15}+\frac{54\!\cdots\!60}{97\!\cdots\!29}a^{14}+\frac{51\!\cdots\!56}{97\!\cdots\!29}a^{13}-\frac{45\!\cdots\!36}{97\!\cdots\!29}a^{12}-\frac{19\!\cdots\!66}{97\!\cdots\!29}a^{11}+\frac{24\!\cdots\!00}{97\!\cdots\!29}a^{10}+\frac{27\!\cdots\!58}{97\!\cdots\!29}a^{9}-\frac{81\!\cdots\!65}{97\!\cdots\!29}a^{8}+\frac{37\!\cdots\!82}{97\!\cdots\!29}a^{7}+\frac{19\!\cdots\!54}{97\!\cdots\!29}a^{6}-\frac{75\!\cdots\!67}{26\!\cdots\!17}a^{5}-\frac{80\!\cdots\!25}{97\!\cdots\!29}a^{4}+\frac{26\!\cdots\!71}{26\!\cdots\!17}a^{3}+\frac{11\!\cdots\!63}{97\!\cdots\!29}a^{2}-\frac{15\!\cdots\!35}{97\!\cdots\!29}a+\frac{39\!\cdots\!29}{97\!\cdots\!29}$
| 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: | \( 866679281.3791491 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 866679281.3791491 \cdot 1084931}{2\cdot\sqrt{1922554115559596594815766124679345986918150636343}}\cr\approx \mathstrut & 0.204300675637054 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 - 23*x^20 + 427*x^19 - 333*x^18 - 8101*x^17 + 19825*x^16 + 54452*x^15 - 199899*x^14 - 268055*x^13 + 1619754*x^12 - 923591*x^11 - 3715861*x^10 + 3187790*x^9 + 19675195*x^8 - 56103742*x^7 + 59561975*x^6 - 21663766*x^5 + 150503099*x^4 - 437293737*x^3 + 963528183*x^2 - 867599298*x + 639704333)
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 - 9*x^21 - 23*x^20 + 427*x^19 - 333*x^18 - 8101*x^17 + 19825*x^16 + 54452*x^15 - 199899*x^14 - 268055*x^13 + 1619754*x^12 - 923591*x^11 - 3715861*x^10 + 3187790*x^9 + 19675195*x^8 - 56103742*x^7 + 59561975*x^6 - 21663766*x^5 + 150503099*x^4 - 437293737*x^3 + 963528183*x^2 - 867599298*x + 639704333, 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 - 9*x^21 - 23*x^20 + 427*x^19 - 333*x^18 - 8101*x^17 + 19825*x^16 + 54452*x^15 - 199899*x^14 - 268055*x^13 + 1619754*x^12 - 923591*x^11 - 3715861*x^10 + 3187790*x^9 + 19675195*x^8 - 56103742*x^7 + 59561975*x^6 - 21663766*x^5 + 150503099*x^4 - 437293737*x^3 + 963528183*x^2 - 867599298*x + 639704333);
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 - 9*x^21 - 23*x^20 + 427*x^19 - 333*x^18 - 8101*x^17 + 19825*x^16 + 54452*x^15 - 199899*x^14 - 268055*x^13 + 1619754*x^12 - 923591*x^11 - 3715861*x^10 + 3187790*x^9 + 19675195*x^8 - 56103742*x^7 + 59561975*x^6 - 21663766*x^5 + 150503099*x^4 - 437293737*x^3 + 963528183*x^2 - 867599298*x + 639704333);
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
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 cyclic group of order 22 |
| The 22 conjugacy class representatives for $C_{22}$ |
| Character table for $C_{22}$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), 11.11.31181719929966183601.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | $22$ | $22$ | R | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/37.1.0.1}{1} }^{22}$ | $22$ | ${\href{/padicField/43.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.22.11.2 | $x^{22} + 77 x^{20} + 2695 x^{18} + 56595 x^{16} + 792330 x^{14} + 7764836 x^{12} + 8 x^{11} + 54353222 x^{10} - 3080 x^{9} + 271785360 x^{8} + 129360 x^{7} + 951192165 x^{6} - 1267728 x^{5} + 2218656055 x^{4} + 3169320 x^{3} + 3108706756 x^{2} - 1479008 x + 1977091468$ | $2$ | $11$ | $11$ | 22T1 | $[\ ]_{2}^{11}$ |
|
\(89\)
| 89.22.20.1 | $x^{22} + 902 x^{21} + 369853 x^{20} + 91002780 x^{19} + 14930003715 x^{18} + 1715001858174 x^{17} + 140764522403703 x^{16} + 8257274182897872 x^{15} + 339392828633234970 x^{14} + 9318023686889578540 x^{13} + 154241038345248674930 x^{12} + 1182833460736280446298 x^{11} + 462723115035746105068 x^{10} + 83862213182039091470 x^{9} + 9163606381183345740 x^{8} + 668840534327058852 x^{7} + 34357879837310865 x^{6} + 13716942446122674 x^{5} + 729899506792768125 x^{4} + 29911184461831569270 x^{3} + 817207425550012839749 x^{2} + 13396219542127922375920 x + 99818152545378148140642$ | $11$ | $2$ | $20$ | 22T1 | $[\ ]_{11}^{2}$ |