Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 + 18*x^20 - 48*x^19 + 125*x^18 - 249*x^17 + 478*x^16 - 766*x^15 + 1216*x^14 - 1618*x^13 + 2242*x^12 - 2454*x^11 + 2995*x^10 - 2616*x^9 + 2692*x^8 - 1937*x^7 + 1684*x^6 - 1102*x^5 + 757*x^4 - 253*x^3 + 163*x^2 + 2*x + 1)
gp: K = bnfinit(y^22 - 4*y^21 + 18*y^20 - 48*y^19 + 125*y^18 - 249*y^17 + 478*y^16 - 766*y^15 + 1216*y^14 - 1618*y^13 + 2242*y^12 - 2454*y^11 + 2995*y^10 - 2616*y^9 + 2692*y^8 - 1937*y^7 + 1684*y^6 - 1102*y^5 + 757*y^4 - 253*y^3 + 163*y^2 + 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 4*x^21 + 18*x^20 - 48*x^19 + 125*x^18 - 249*x^17 + 478*x^16 - 766*x^15 + 1216*x^14 - 1618*x^13 + 2242*x^12 - 2454*x^11 + 2995*x^10 - 2616*x^9 + 2692*x^8 - 1937*x^7 + 1684*x^6 - 1102*x^5 + 757*x^4 - 253*x^3 + 163*x^2 + 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 4*x^21 + 18*x^20 - 48*x^19 + 125*x^18 - 249*x^17 + 478*x^16 - 766*x^15 + 1216*x^14 - 1618*x^13 + 2242*x^12 - 2454*x^11 + 2995*x^10 - 2616*x^9 + 2692*x^8 - 1937*x^7 + 1684*x^6 - 1102*x^5 + 757*x^4 - 253*x^3 + 163*x^2 + 2*x + 1)
\( x^{22} - 4 x^{21} + 18 x^{20} - 48 x^{19} + 125 x^{18} - 249 x^{17} + 478 x^{16} - 766 x^{15} + 1216 x^{14} + \cdots + 1 \)
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: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-15257581934366008831157764375\)
\(\medspace = -\,5^{4}\cdot 7^{11}\cdot 83^{4}\cdot 127^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.10\) | 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: | $5^{1/2}7^{1/2}83^{2/3}127^{2/3}\approx 2844.1679033165196$ | ||
| Ramified primes: |
\(5\), \(7\), \(83\), \(127\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{73}a^{20}+\frac{6}{73}a^{19}+\frac{9}{73}a^{18}-\frac{7}{73}a^{17}+\frac{18}{73}a^{16}-\frac{24}{73}a^{15}+\frac{18}{73}a^{14}-\frac{25}{73}a^{13}+\frac{16}{73}a^{12}-\frac{25}{73}a^{11}+\frac{12}{73}a^{10}-\frac{25}{73}a^{9}+\frac{19}{73}a^{8}+\frac{29}{73}a^{7}-\frac{8}{73}a^{6}-\frac{3}{73}a^{5}+\frac{16}{73}a^{4}-\frac{5}{73}a^{3}-\frac{32}{73}a^{2}-\frac{9}{73}a+\frac{18}{73}$, $\frac{1}{89\!\cdots\!71}a^{21}-\frac{10\!\cdots\!20}{89\!\cdots\!71}a^{20}+\frac{94\!\cdots\!71}{89\!\cdots\!71}a^{19}+\frac{32\!\cdots\!20}{12\!\cdots\!27}a^{18}+\frac{36\!\cdots\!25}{89\!\cdots\!71}a^{17}+\frac{16\!\cdots\!80}{89\!\cdots\!71}a^{16}+\frac{17\!\cdots\!63}{89\!\cdots\!71}a^{15}-\frac{30\!\cdots\!93}{89\!\cdots\!71}a^{14}-\frac{18\!\cdots\!29}{89\!\cdots\!71}a^{13}+\frac{42\!\cdots\!79}{89\!\cdots\!71}a^{12}-\frac{24\!\cdots\!84}{89\!\cdots\!71}a^{11}-\frac{34\!\cdots\!10}{89\!\cdots\!71}a^{10}+\frac{10\!\cdots\!77}{89\!\cdots\!71}a^{9}-\frac{70\!\cdots\!77}{89\!\cdots\!71}a^{8}+\frac{10\!\cdots\!57}{89\!\cdots\!71}a^{7}-\frac{77\!\cdots\!31}{24\!\cdots\!83}a^{6}-\frac{77\!\cdots\!41}{89\!\cdots\!71}a^{5}+\frac{14\!\cdots\!31}{89\!\cdots\!71}a^{4}-\frac{44\!\cdots\!96}{89\!\cdots\!71}a^{3}+\frac{11\!\cdots\!54}{89\!\cdots\!71}a^{2}+\frac{20\!\cdots\!65}{89\!\cdots\!71}a-\frac{18\!\cdots\!81}{89\!\cdots\!71}$
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: | $10$ | 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: |
$\frac{68\!\cdots\!95}{89\!\cdots\!71}a^{21}-\frac{28\!\cdots\!12}{89\!\cdots\!71}a^{20}+\frac{12\!\cdots\!70}{89\!\cdots\!71}a^{19}-\frac{34\!\cdots\!74}{89\!\cdots\!71}a^{18}+\frac{89\!\cdots\!02}{89\!\cdots\!71}a^{17}-\frac{17\!\cdots\!69}{89\!\cdots\!71}a^{16}+\frac{34\!\cdots\!67}{89\!\cdots\!71}a^{15}-\frac{55\!\cdots\!93}{89\!\cdots\!71}a^{14}+\frac{88\!\cdots\!24}{89\!\cdots\!71}a^{13}-\frac{11\!\cdots\!68}{89\!\cdots\!71}a^{12}+\frac{16\!\cdots\!70}{89\!\cdots\!71}a^{11}-\frac{18\!\cdots\!37}{89\!\cdots\!71}a^{10}+\frac{22\!\cdots\!20}{89\!\cdots\!71}a^{9}-\frac{20\!\cdots\!78}{89\!\cdots\!71}a^{8}+\frac{20\!\cdots\!39}{89\!\cdots\!71}a^{7}-\frac{41\!\cdots\!87}{24\!\cdots\!83}a^{6}+\frac{12\!\cdots\!00}{89\!\cdots\!71}a^{5}-\frac{86\!\cdots\!77}{89\!\cdots\!71}a^{4}+\frac{58\!\cdots\!31}{89\!\cdots\!71}a^{3}-\frac{22\!\cdots\!85}{89\!\cdots\!71}a^{2}+\frac{12\!\cdots\!08}{89\!\cdots\!71}a-\frac{54\!\cdots\!40}{89\!\cdots\!71}$, $\frac{15\!\cdots\!47}{89\!\cdots\!71}a^{21}-\frac{70\!\cdots\!97}{89\!\cdots\!71}a^{20}+\frac{32\!\cdots\!11}{89\!\cdots\!71}a^{19}-\frac{92\!\cdots\!23}{89\!\cdots\!71}a^{18}+\frac{33\!\cdots\!63}{12\!\cdots\!27}a^{17}-\frac{51\!\cdots\!00}{89\!\cdots\!71}a^{16}+\frac{10\!\cdots\!77}{89\!\cdots\!71}a^{15}-\frac{17\!\cdots\!93}{89\!\cdots\!71}a^{14}+\frac{27\!\cdots\!03}{89\!\cdots\!71}a^{13}-\frac{38\!\cdots\!56}{89\!\cdots\!71}a^{12}+\frac{53\!\cdots\!37}{89\!\cdots\!71}a^{11}-\frac{63\!\cdots\!54}{89\!\cdots\!71}a^{10}+\frac{76\!\cdots\!56}{89\!\cdots\!71}a^{9}-\frac{73\!\cdots\!29}{89\!\cdots\!71}a^{8}+\frac{74\!\cdots\!24}{89\!\cdots\!71}a^{7}-\frac{15\!\cdots\!35}{24\!\cdots\!83}a^{6}+\frac{50\!\cdots\!15}{89\!\cdots\!71}a^{5}-\frac{33\!\cdots\!80}{89\!\cdots\!71}a^{4}+\frac{24\!\cdots\!49}{89\!\cdots\!71}a^{3}-\frac{10\!\cdots\!26}{89\!\cdots\!71}a^{2}+\frac{52\!\cdots\!91}{89\!\cdots\!71}a-\frac{71\!\cdots\!83}{89\!\cdots\!71}$, $\frac{20\!\cdots\!38}{89\!\cdots\!71}a^{21}-\frac{21\!\cdots\!70}{89\!\cdots\!71}a^{20}+\frac{94\!\cdots\!32}{89\!\cdots\!71}a^{19}-\frac{35\!\cdots\!75}{89\!\cdots\!71}a^{18}+\frac{96\!\cdots\!49}{89\!\cdots\!71}a^{17}-\frac{23\!\cdots\!64}{89\!\cdots\!71}a^{16}+\frac{47\!\cdots\!80}{89\!\cdots\!71}a^{15}-\frac{86\!\cdots\!12}{89\!\cdots\!71}a^{14}+\frac{14\!\cdots\!02}{89\!\cdots\!71}a^{13}-\frac{21\!\cdots\!08}{89\!\cdots\!71}a^{12}+\frac{28\!\cdots\!62}{89\!\cdots\!71}a^{11}-\frac{37\!\cdots\!32}{89\!\cdots\!71}a^{10}+\frac{42\!\cdots\!40}{89\!\cdots\!71}a^{9}-\frac{47\!\cdots\!14}{89\!\cdots\!71}a^{8}+\frac{44\!\cdots\!97}{89\!\cdots\!71}a^{7}-\frac{10\!\cdots\!19}{24\!\cdots\!83}a^{6}+\frac{30\!\cdots\!77}{89\!\cdots\!71}a^{5}-\frac{23\!\cdots\!28}{89\!\cdots\!71}a^{4}+\frac{16\!\cdots\!71}{89\!\cdots\!71}a^{3}-\frac{10\!\cdots\!90}{89\!\cdots\!71}a^{2}+\frac{38\!\cdots\!80}{89\!\cdots\!71}a-\frac{13\!\cdots\!19}{89\!\cdots\!71}$, $\frac{63\!\cdots\!79}{89\!\cdots\!71}a^{21}-\frac{28\!\cdots\!35}{89\!\cdots\!71}a^{20}+\frac{13\!\cdots\!77}{89\!\cdots\!71}a^{19}-\frac{39\!\cdots\!24}{89\!\cdots\!71}a^{18}+\frac{10\!\cdots\!47}{89\!\cdots\!71}a^{17}-\frac{22\!\cdots\!32}{89\!\cdots\!71}a^{16}+\frac{45\!\cdots\!93}{89\!\cdots\!71}a^{15}-\frac{77\!\cdots\!08}{89\!\cdots\!71}a^{14}+\frac{12\!\cdots\!11}{89\!\cdots\!71}a^{13}-\frac{17\!\cdots\!76}{89\!\cdots\!71}a^{12}+\frac{24\!\cdots\!09}{89\!\cdots\!71}a^{11}-\frac{28\!\cdots\!55}{89\!\cdots\!71}a^{10}+\frac{35\!\cdots\!96}{89\!\cdots\!71}a^{9}-\frac{32\!\cdots\!25}{89\!\cdots\!71}a^{8}+\frac{34\!\cdots\!81}{89\!\cdots\!71}a^{7}-\frac{65\!\cdots\!20}{24\!\cdots\!83}a^{6}+\frac{23\!\cdots\!03}{89\!\cdots\!71}a^{5}-\frac{12\!\cdots\!34}{89\!\cdots\!71}a^{4}+\frac{11\!\cdots\!13}{89\!\cdots\!71}a^{3}-\frac{38\!\cdots\!14}{89\!\cdots\!71}a^{2}+\frac{24\!\cdots\!22}{89\!\cdots\!71}a+\frac{61\!\cdots\!76}{89\!\cdots\!71}$, $\frac{84\!\cdots\!57}{89\!\cdots\!71}a^{21}-\frac{42\!\cdots\!67}{89\!\cdots\!71}a^{20}+\frac{18\!\cdots\!94}{89\!\cdots\!71}a^{19}-\frac{56\!\cdots\!03}{89\!\cdots\!71}a^{18}+\frac{15\!\cdots\!89}{89\!\cdots\!71}a^{17}-\frac{32\!\cdots\!76}{89\!\cdots\!71}a^{16}+\frac{63\!\cdots\!53}{89\!\cdots\!71}a^{15}-\frac{11\!\cdots\!64}{89\!\cdots\!71}a^{14}+\frac{17\!\cdots\!47}{89\!\cdots\!71}a^{13}-\frac{25\!\cdots\!62}{89\!\cdots\!71}a^{12}+\frac{34\!\cdots\!30}{89\!\cdots\!71}a^{11}-\frac{42\!\cdots\!05}{89\!\cdots\!71}a^{10}+\frac{49\!\cdots\!91}{89\!\cdots\!71}a^{9}-\frac{51\!\cdots\!75}{89\!\cdots\!71}a^{8}+\frac{48\!\cdots\!39}{89\!\cdots\!71}a^{7}-\frac{11\!\cdots\!02}{24\!\cdots\!83}a^{6}+\frac{33\!\cdots\!00}{89\!\cdots\!71}a^{5}-\frac{26\!\cdots\!32}{89\!\cdots\!71}a^{4}+\frac{17\!\cdots\!05}{89\!\cdots\!71}a^{3}-\frac{10\!\cdots\!39}{89\!\cdots\!71}a^{2}+\frac{39\!\cdots\!12}{89\!\cdots\!71}a-\frac{60\!\cdots\!00}{89\!\cdots\!71}$, $\frac{26\!\cdots\!95}{89\!\cdots\!71}a^{21}-\frac{97\!\cdots\!45}{89\!\cdots\!71}a^{20}+\frac{42\!\cdots\!39}{89\!\cdots\!71}a^{19}-\frac{10\!\cdots\!78}{89\!\cdots\!71}a^{18}+\frac{26\!\cdots\!10}{89\!\cdots\!71}a^{17}-\frac{48\!\cdots\!80}{89\!\cdots\!71}a^{16}+\frac{89\!\cdots\!79}{89\!\cdots\!71}a^{15}-\frac{13\!\cdots\!13}{89\!\cdots\!71}a^{14}+\frac{20\!\cdots\!19}{89\!\cdots\!71}a^{13}-\frac{24\!\cdots\!51}{89\!\cdots\!71}a^{12}+\frac{33\!\cdots\!37}{89\!\cdots\!71}a^{11}-\frac{30\!\cdots\!47}{89\!\cdots\!71}a^{10}+\frac{38\!\cdots\!13}{89\!\cdots\!71}a^{9}-\frac{23\!\cdots\!67}{89\!\cdots\!71}a^{8}+\frac{25\!\cdots\!06}{89\!\cdots\!71}a^{7}-\frac{28\!\cdots\!70}{24\!\cdots\!83}a^{6}+\frac{10\!\cdots\!90}{89\!\cdots\!71}a^{5}-\frac{42\!\cdots\!00}{89\!\cdots\!71}a^{4}+\frac{16\!\cdots\!80}{89\!\cdots\!71}a^{3}+\frac{52\!\cdots\!47}{89\!\cdots\!71}a^{2}+\frac{99\!\cdots\!25}{89\!\cdots\!71}a+\frac{72\!\cdots\!04}{89\!\cdots\!71}$, $\frac{10\!\cdots\!36}{89\!\cdots\!71}a^{21}-\frac{18\!\cdots\!40}{89\!\cdots\!71}a^{20}+\frac{71\!\cdots\!89}{89\!\cdots\!71}a^{19}-\frac{31\!\cdots\!28}{89\!\cdots\!71}a^{18}+\frac{83\!\cdots\!80}{89\!\cdots\!71}a^{17}-\frac{21\!\cdots\!22}{89\!\cdots\!71}a^{16}+\frac{41\!\cdots\!45}{89\!\cdots\!71}a^{15}-\frac{78\!\cdots\!36}{89\!\cdots\!71}a^{14}+\frac{12\!\cdots\!17}{89\!\cdots\!71}a^{13}-\frac{19\!\cdots\!75}{89\!\cdots\!71}a^{12}+\frac{24\!\cdots\!14}{89\!\cdots\!71}a^{11}-\frac{34\!\cdots\!88}{89\!\cdots\!71}a^{10}+\frac{35\!\cdots\!36}{89\!\cdots\!71}a^{9}-\frac{43\!\cdots\!08}{89\!\cdots\!71}a^{8}+\frac{35\!\cdots\!57}{89\!\cdots\!71}a^{7}-\frac{97\!\cdots\!70}{24\!\cdots\!83}a^{6}+\frac{22\!\cdots\!66}{89\!\cdots\!71}a^{5}-\frac{20\!\cdots\!15}{89\!\cdots\!71}a^{4}+\frac{12\!\cdots\!42}{89\!\cdots\!71}a^{3}-\frac{73\!\cdots\!96}{89\!\cdots\!71}a^{2}+\frac{11\!\cdots\!58}{89\!\cdots\!71}a-\frac{61\!\cdots\!51}{89\!\cdots\!71}$, $\frac{17\!\cdots\!67}{89\!\cdots\!71}a^{21}-\frac{80\!\cdots\!31}{89\!\cdots\!71}a^{20}+\frac{35\!\cdots\!96}{89\!\cdots\!71}a^{19}-\frac{10\!\cdots\!92}{89\!\cdots\!71}a^{18}+\frac{26\!\cdots\!89}{89\!\cdots\!71}a^{17}-\frac{55\!\cdots\!88}{89\!\cdots\!71}a^{16}+\frac{10\!\cdots\!80}{89\!\cdots\!71}a^{15}-\frac{17\!\cdots\!05}{89\!\cdots\!71}a^{14}+\frac{27\!\cdots\!17}{89\!\cdots\!71}a^{13}-\frac{37\!\cdots\!33}{89\!\cdots\!71}a^{12}+\frac{51\!\cdots\!90}{89\!\cdots\!71}a^{11}-\frac{58\!\cdots\!07}{89\!\cdots\!71}a^{10}+\frac{67\!\cdots\!11}{89\!\cdots\!71}a^{9}-\frac{64\!\cdots\!33}{89\!\cdots\!71}a^{8}+\frac{60\!\cdots\!29}{89\!\cdots\!71}a^{7}-\frac{12\!\cdots\!30}{24\!\cdots\!83}a^{6}+\frac{36\!\cdots\!77}{89\!\cdots\!71}a^{5}-\frac{24\!\cdots\!51}{89\!\cdots\!71}a^{4}+\frac{16\!\cdots\!32}{89\!\cdots\!71}a^{3}-\frac{54\!\cdots\!83}{89\!\cdots\!71}a^{2}+\frac{13\!\cdots\!44}{89\!\cdots\!71}a+\frac{41\!\cdots\!00}{89\!\cdots\!71}$, $\frac{11\!\cdots\!56}{89\!\cdots\!71}a^{21}-\frac{46\!\cdots\!19}{89\!\cdots\!71}a^{20}+\frac{20\!\cdots\!88}{89\!\cdots\!71}a^{19}-\frac{55\!\cdots\!33}{89\!\cdots\!71}a^{18}+\frac{14\!\cdots\!41}{89\!\cdots\!71}a^{17}-\frac{28\!\cdots\!26}{89\!\cdots\!71}a^{16}+\frac{54\!\cdots\!79}{89\!\cdots\!71}a^{15}-\frac{87\!\cdots\!64}{89\!\cdots\!71}a^{14}+\frac{13\!\cdots\!57}{89\!\cdots\!71}a^{13}-\frac{18\!\cdots\!37}{89\!\cdots\!71}a^{12}+\frac{25\!\cdots\!74}{89\!\cdots\!71}a^{11}-\frac{27\!\cdots\!35}{89\!\cdots\!71}a^{10}+\frac{33\!\cdots\!60}{89\!\cdots\!71}a^{9}-\frac{29\!\cdots\!90}{89\!\cdots\!71}a^{8}+\frac{30\!\cdots\!74}{89\!\cdots\!71}a^{7}-\frac{57\!\cdots\!20}{24\!\cdots\!83}a^{6}+\frac{18\!\cdots\!14}{89\!\cdots\!71}a^{5}-\frac{12\!\cdots\!17}{89\!\cdots\!71}a^{4}+\frac{83\!\cdots\!04}{89\!\cdots\!71}a^{3}-\frac{26\!\cdots\!89}{89\!\cdots\!71}a^{2}+\frac{17\!\cdots\!18}{89\!\cdots\!71}a+\frac{44\!\cdots\!46}{89\!\cdots\!71}$, $\frac{65\!\cdots\!98}{89\!\cdots\!71}a^{21}-\frac{32\!\cdots\!75}{89\!\cdots\!71}a^{20}+\frac{14\!\cdots\!73}{89\!\cdots\!71}a^{19}-\frac{43\!\cdots\!49}{89\!\cdots\!71}a^{18}+\frac{11\!\cdots\!18}{89\!\cdots\!71}a^{17}-\frac{25\!\cdots\!07}{89\!\cdots\!71}a^{16}+\frac{49\!\cdots\!10}{89\!\cdots\!71}a^{15}-\frac{86\!\cdots\!51}{89\!\cdots\!71}a^{14}+\frac{13\!\cdots\!83}{89\!\cdots\!71}a^{13}-\frac{20\!\cdots\!35}{89\!\cdots\!71}a^{12}+\frac{27\!\cdots\!60}{89\!\cdots\!71}a^{11}-\frac{34\!\cdots\!64}{89\!\cdots\!71}a^{10}+\frac{40\!\cdots\!12}{89\!\cdots\!71}a^{9}-\frac{41\!\cdots\!69}{89\!\cdots\!71}a^{8}+\frac{41\!\cdots\!19}{89\!\cdots\!71}a^{7}-\frac{95\!\cdots\!05}{24\!\cdots\!83}a^{6}+\frac{29\!\cdots\!59}{89\!\cdots\!71}a^{5}-\frac{20\!\cdots\!32}{89\!\cdots\!71}a^{4}+\frac{21\!\cdots\!81}{12\!\cdots\!27}a^{3}-\frac{74\!\cdots\!72}{89\!\cdots\!71}a^{2}+\frac{43\!\cdots\!48}{89\!\cdots\!71}a-\frac{46\!\cdots\!97}{89\!\cdots\!71}$
| 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: | \( 69206.6233764 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 69206.6233764 \cdot 1}{2\cdot\sqrt{15257581934366008831157764375}}\cr\approx \mathstrut & 0.168792583789 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 + 18*x^20 - 48*x^19 + 125*x^18 - 249*x^17 + 478*x^16 - 766*x^15 + 1216*x^14 - 1618*x^13 + 2242*x^12 - 2454*x^11 + 2995*x^10 - 2616*x^9 + 2692*x^8 - 1937*x^7 + 1684*x^6 - 1102*x^5 + 757*x^4 - 253*x^3 + 163*x^2 + 2*x + 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 - 4*x^21 + 18*x^20 - 48*x^19 + 125*x^18 - 249*x^17 + 478*x^16 - 766*x^15 + 1216*x^14 - 1618*x^13 + 2242*x^12 - 2454*x^11 + 2995*x^10 - 2616*x^9 + 2692*x^8 - 1937*x^7 + 1684*x^6 - 1102*x^5 + 757*x^4 - 253*x^3 + 163*x^2 + 2*x + 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 - 4*x^21 + 18*x^20 - 48*x^19 + 125*x^18 - 249*x^17 + 478*x^16 - 766*x^15 + 1216*x^14 - 1618*x^13 + 2242*x^12 - 2454*x^11 + 2995*x^10 - 2616*x^9 + 2692*x^8 - 1937*x^7 + 1684*x^6 - 1102*x^5 + 757*x^4 - 253*x^3 + 163*x^2 + 2*x + 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 - 4*x^21 + 18*x^20 - 48*x^19 + 125*x^18 - 249*x^17 + 478*x^16 - 766*x^15 + 1216*x^14 - 1618*x^13 + 2242*x^12 - 2454*x^11 + 2995*x^10 - 2616*x^9 + 2692*x^8 - 1937*x^7 + 1684*x^6 - 1102*x^5 + 757*x^4 - 253*x^3 + 163*x^2 + 2*x + 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
$C_2\times A_{11}$ (as 22T46):
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 39916800 |
| The 62 conjugacy class representatives for $C_2\times A_{11}$ |
| Character table for $C_2\times A_{11}$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), 11.3.136113034225.1 |
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 | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | $22$ | R | R | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.11.0.1}{11} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.14.7.1 | $x^{14} + 49 x^{12} + 1029 x^{10} + 12017 x^{8} + 8 x^{7} + 82859 x^{6} - 1176 x^{5} + 352947 x^{4} + 13720 x^{3} + 881203 x^{2} - 19160 x + 794999$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ | |
|
\(83\)
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 83.6.4.1 | $x^{6} + 246 x^{5} + 20178 x^{4} + 552518 x^{3} + 60774 x^{2} + 1674264 x + 45729605$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 83.10.0.1 | $x^{10} + 7 x^{5} + 73 x^{3} + 53 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(127\)
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 127.3.2.1 | $x^{3} + 127$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 127.3.2.1 | $x^{3} + 127$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 127.7.0.1 | $x^{7} + 15 x + 124$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 127.7.0.1 | $x^{7} + 15 x + 124$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |