Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 7*x^20 - 20*x^19 + 38*x^18 - 74*x^17 + 107*x^16 - 167*x^15 + 185*x^14 - 284*x^13 + 308*x^12 - 417*x^11 + 466*x^10 - 462*x^9 + 446*x^8 - 292*x^7 + 170*x^6 - 55*x^5 - 7*x^4 + 11*x^3 - 8*x^2 + x + 1)
gp: K = bnfinit(y^22 - 2*y^21 + 7*y^20 - 20*y^19 + 38*y^18 - 74*y^17 + 107*y^16 - 167*y^15 + 185*y^14 - 284*y^13 + 308*y^12 - 417*y^11 + 466*y^10 - 462*y^9 + 446*y^8 - 292*y^7 + 170*y^6 - 55*y^5 - 7*y^4 + 11*y^3 - 8*y^2 + y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 2*x^21 + 7*x^20 - 20*x^19 + 38*x^18 - 74*x^17 + 107*x^16 - 167*x^15 + 185*x^14 - 284*x^13 + 308*x^12 - 417*x^11 + 466*x^10 - 462*x^9 + 446*x^8 - 292*x^7 + 170*x^6 - 55*x^5 - 7*x^4 + 11*x^3 - 8*x^2 + x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 2*x^21 + 7*x^20 - 20*x^19 + 38*x^18 - 74*x^17 + 107*x^16 - 167*x^15 + 185*x^14 - 284*x^13 + 308*x^12 - 417*x^11 + 466*x^10 - 462*x^9 + 446*x^8 - 292*x^7 + 170*x^6 - 55*x^5 - 7*x^4 + 11*x^3 - 8*x^2 + x + 1)
\( x^{22} - 2 x^{21} + 7 x^{20} - 20 x^{19} + 38 x^{18} - 74 x^{17} + 107 x^{16} - 167 x^{15} + 185 x^{14} + \cdots + 1 \)
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: | $[4, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-1533107758369291702785569375\)
\(\medspace = -\,5^{4}\cdot 7^{4}\cdot 83^{5}\cdot 127^{4}\cdot 997\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.21\) | 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: | $5^{1/2}7^{1/2}83^{5/6}127^{2/3}997^{1/2}\approx 187563.85877253$ | ||
| Ramified primes: |
\(5\), \(7\), \(83\), \(127\), \(997\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-82751}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{49\!\cdots\!55}a^{21}-\frac{24\!\cdots\!48}{49\!\cdots\!55}a^{20}+\frac{45\!\cdots\!74}{98\!\cdots\!31}a^{19}+\frac{17\!\cdots\!68}{98\!\cdots\!31}a^{18}-\frac{13\!\cdots\!82}{49\!\cdots\!55}a^{17}-\frac{16\!\cdots\!87}{49\!\cdots\!55}a^{16}+\frac{42\!\cdots\!99}{49\!\cdots\!55}a^{15}-\frac{17\!\cdots\!16}{49\!\cdots\!55}a^{14}+\frac{77\!\cdots\!16}{49\!\cdots\!55}a^{13}-\frac{39\!\cdots\!46}{98\!\cdots\!31}a^{12}+\frac{63\!\cdots\!58}{49\!\cdots\!55}a^{11}+\frac{81\!\cdots\!52}{98\!\cdots\!31}a^{10}+\frac{10\!\cdots\!01}{49\!\cdots\!55}a^{9}+\frac{78\!\cdots\!32}{49\!\cdots\!55}a^{8}-\frac{18\!\cdots\!21}{49\!\cdots\!55}a^{7}-\frac{23\!\cdots\!26}{49\!\cdots\!55}a^{6}-\frac{24\!\cdots\!79}{49\!\cdots\!55}a^{5}-\frac{11\!\cdots\!16}{49\!\cdots\!55}a^{4}+\frac{24\!\cdots\!29}{49\!\cdots\!55}a^{3}-\frac{41\!\cdots\!63}{49\!\cdots\!55}a^{2}-\frac{38\!\cdots\!28}{98\!\cdots\!31}a+\frac{90\!\cdots\!06}{49\!\cdots\!55}$
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: | $12$ | 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: |
$\frac{43\!\cdots\!84}{98\!\cdots\!31}a^{21}-\frac{10\!\cdots\!91}{98\!\cdots\!31}a^{20}+\frac{32\!\cdots\!99}{98\!\cdots\!31}a^{19}-\frac{10\!\cdots\!50}{98\!\cdots\!31}a^{18}+\frac{19\!\cdots\!85}{98\!\cdots\!31}a^{17}-\frac{38\!\cdots\!06}{98\!\cdots\!31}a^{16}+\frac{60\!\cdots\!52}{98\!\cdots\!31}a^{15}-\frac{91\!\cdots\!86}{98\!\cdots\!31}a^{14}+\frac{11\!\cdots\!54}{98\!\cdots\!31}a^{13}-\frac{15\!\cdots\!87}{98\!\cdots\!31}a^{12}+\frac{20\!\cdots\!44}{98\!\cdots\!31}a^{11}-\frac{23\!\cdots\!13}{98\!\cdots\!31}a^{10}+\frac{30\!\cdots\!33}{98\!\cdots\!31}a^{9}-\frac{29\!\cdots\!36}{98\!\cdots\!31}a^{8}+\frac{30\!\cdots\!49}{98\!\cdots\!31}a^{7}-\frac{24\!\cdots\!60}{98\!\cdots\!31}a^{6}+\frac{15\!\cdots\!45}{98\!\cdots\!31}a^{5}-\frac{10\!\cdots\!53}{98\!\cdots\!31}a^{4}+\frac{31\!\cdots\!00}{98\!\cdots\!31}a^{3}-\frac{84\!\cdots\!56}{98\!\cdots\!31}a^{2}-\frac{95\!\cdots\!32}{98\!\cdots\!31}a+\frac{37\!\cdots\!99}{98\!\cdots\!31}$, $\frac{89\!\cdots\!28}{98\!\cdots\!31}a^{21}-\frac{10\!\cdots\!02}{98\!\cdots\!31}a^{20}+\frac{47\!\cdots\!72}{98\!\cdots\!31}a^{19}-\frac{12\!\cdots\!53}{98\!\cdots\!31}a^{18}+\frac{19\!\cdots\!76}{98\!\cdots\!31}a^{17}-\frac{37\!\cdots\!99}{98\!\cdots\!31}a^{16}+\frac{39\!\cdots\!55}{98\!\cdots\!31}a^{15}-\frac{69\!\cdots\!36}{98\!\cdots\!31}a^{14}+\frac{39\!\cdots\!78}{98\!\cdots\!31}a^{13}-\frac{11\!\cdots\!92}{98\!\cdots\!31}a^{12}+\frac{59\!\cdots\!81}{98\!\cdots\!31}a^{11}-\frac{14\!\cdots\!42}{98\!\cdots\!31}a^{10}+\frac{94\!\cdots\!39}{98\!\cdots\!31}a^{9}-\frac{76\!\cdots\!07}{98\!\cdots\!31}a^{8}+\frac{44\!\cdots\!77}{98\!\cdots\!31}a^{7}+\frac{65\!\cdots\!42}{98\!\cdots\!31}a^{6}-\frac{61\!\cdots\!77}{98\!\cdots\!31}a^{5}+\frac{81\!\cdots\!08}{98\!\cdots\!31}a^{4}-\frac{37\!\cdots\!55}{98\!\cdots\!31}a^{3}+\frac{10\!\cdots\!90}{98\!\cdots\!31}a^{2}+\frac{23\!\cdots\!39}{98\!\cdots\!31}a-\frac{33\!\cdots\!82}{98\!\cdots\!31}$, $\frac{41\!\cdots\!72}{98\!\cdots\!31}a^{21}-\frac{43\!\cdots\!58}{98\!\cdots\!31}a^{20}+\frac{20\!\cdots\!00}{98\!\cdots\!31}a^{19}-\frac{56\!\cdots\!72}{98\!\cdots\!31}a^{18}+\frac{78\!\cdots\!21}{98\!\cdots\!31}a^{17}-\frac{15\!\cdots\!53}{98\!\cdots\!31}a^{16}+\frac{15\!\cdots\!15}{98\!\cdots\!31}a^{15}-\frac{27\!\cdots\!69}{98\!\cdots\!31}a^{14}+\frac{13\!\cdots\!37}{98\!\cdots\!31}a^{13}-\frac{47\!\cdots\!22}{98\!\cdots\!31}a^{12}+\frac{22\!\cdots\!77}{98\!\cdots\!31}a^{11}-\frac{54\!\cdots\!59}{98\!\cdots\!31}a^{10}+\frac{39\!\cdots\!58}{98\!\cdots\!31}a^{9}-\frac{14\!\cdots\!23}{98\!\cdots\!31}a^{8}+\frac{16\!\cdots\!34}{98\!\cdots\!31}a^{7}+\frac{39\!\cdots\!54}{98\!\cdots\!31}a^{6}-\frac{32\!\cdots\!94}{98\!\cdots\!31}a^{5}+\frac{30\!\cdots\!13}{98\!\cdots\!31}a^{4}-\frac{19\!\cdots\!35}{98\!\cdots\!31}a^{3}+\frac{22\!\cdots\!43}{98\!\cdots\!31}a^{2}-\frac{97\!\cdots\!63}{98\!\cdots\!31}a-\frac{72\!\cdots\!43}{98\!\cdots\!31}$, $\frac{98\!\cdots\!41}{98\!\cdots\!31}a^{21}-\frac{17\!\cdots\!42}{98\!\cdots\!31}a^{20}+\frac{61\!\cdots\!34}{98\!\cdots\!31}a^{19}-\frac{17\!\cdots\!69}{98\!\cdots\!31}a^{18}+\frac{31\!\cdots\!61}{98\!\cdots\!31}a^{17}-\frac{59\!\cdots\!64}{98\!\cdots\!31}a^{16}+\frac{78\!\cdots\!30}{98\!\cdots\!31}a^{15}-\frac{12\!\cdots\!81}{98\!\cdots\!31}a^{14}+\frac{11\!\cdots\!34}{98\!\cdots\!31}a^{13}-\frac{19\!\cdots\!02}{98\!\cdots\!31}a^{12}+\frac{19\!\cdots\!26}{98\!\cdots\!31}a^{11}-\frac{27\!\cdots\!07}{98\!\cdots\!31}a^{10}+\frac{28\!\cdots\!96}{98\!\cdots\!31}a^{9}-\frac{25\!\cdots\!23}{98\!\cdots\!31}a^{8}+\frac{21\!\cdots\!20}{98\!\cdots\!31}a^{7}-\frac{82\!\cdots\!58}{98\!\cdots\!31}a^{6}+\frac{17\!\cdots\!78}{98\!\cdots\!31}a^{5}+\frac{44\!\cdots\!51}{98\!\cdots\!31}a^{4}-\frac{39\!\cdots\!97}{98\!\cdots\!31}a^{3}+\frac{15\!\cdots\!38}{98\!\cdots\!31}a^{2}-\frac{41\!\cdots\!41}{98\!\cdots\!31}a-\frac{21\!\cdots\!88}{98\!\cdots\!31}$, $\frac{81\!\cdots\!36}{98\!\cdots\!31}a^{21}-\frac{80\!\cdots\!83}{98\!\cdots\!31}a^{20}+\frac{43\!\cdots\!09}{98\!\cdots\!31}a^{19}-\frac{11\!\cdots\!10}{98\!\cdots\!31}a^{18}+\frac{16\!\cdots\!56}{98\!\cdots\!31}a^{17}-\frac{35\!\cdots\!38}{98\!\cdots\!31}a^{16}+\frac{38\!\cdots\!40}{98\!\cdots\!31}a^{15}-\frac{71\!\cdots\!72}{98\!\cdots\!31}a^{14}+\frac{45\!\cdots\!40}{98\!\cdots\!31}a^{13}-\frac{13\!\cdots\!64}{98\!\cdots\!31}a^{12}+\frac{69\!\cdots\!59}{98\!\cdots\!31}a^{11}-\frac{17\!\cdots\!54}{98\!\cdots\!31}a^{10}+\frac{12\!\cdots\!43}{98\!\cdots\!31}a^{9}-\frac{11\!\cdots\!08}{98\!\cdots\!31}a^{8}+\frac{11\!\cdots\!46}{98\!\cdots\!31}a^{7}-\frac{84\!\cdots\!54}{98\!\cdots\!31}a^{6}+\frac{15\!\cdots\!28}{98\!\cdots\!31}a^{5}+\frac{28\!\cdots\!40}{98\!\cdots\!31}a^{4}-\frac{15\!\cdots\!14}{98\!\cdots\!31}a^{3}+\frac{50\!\cdots\!80}{98\!\cdots\!31}a^{2}-\frac{39\!\cdots\!92}{98\!\cdots\!31}a-\frac{12\!\cdots\!23}{98\!\cdots\!31}$, $\frac{45\!\cdots\!79}{49\!\cdots\!55}a^{21}-\frac{14\!\cdots\!72}{49\!\cdots\!55}a^{20}+\frac{84\!\cdots\!29}{98\!\cdots\!31}a^{19}-\frac{25\!\cdots\!47}{98\!\cdots\!31}a^{18}+\frac{27\!\cdots\!42}{49\!\cdots\!55}a^{17}-\frac{53\!\cdots\!98}{49\!\cdots\!55}a^{16}+\frac{87\!\cdots\!11}{49\!\cdots\!55}a^{15}-\frac{13\!\cdots\!64}{49\!\cdots\!55}a^{14}+\frac{17\!\cdots\!69}{49\!\cdots\!55}a^{13}-\frac{44\!\cdots\!31}{98\!\cdots\!31}a^{12}+\frac{29\!\cdots\!47}{49\!\cdots\!55}a^{11}-\frac{69\!\cdots\!52}{98\!\cdots\!31}a^{10}+\frac{43\!\cdots\!89}{49\!\cdots\!55}a^{9}-\frac{45\!\cdots\!67}{49\!\cdots\!55}a^{8}+\frac{44\!\cdots\!11}{49\!\cdots\!55}a^{7}-\frac{37\!\cdots\!49}{49\!\cdots\!55}a^{6}+\frac{23\!\cdots\!39}{49\!\cdots\!55}a^{5}-\frac{12\!\cdots\!79}{49\!\cdots\!55}a^{4}+\frac{32\!\cdots\!56}{49\!\cdots\!55}a^{3}-\frac{29\!\cdots\!12}{49\!\cdots\!55}a^{2}-\frac{10\!\cdots\!31}{98\!\cdots\!31}a+\frac{29\!\cdots\!44}{49\!\cdots\!55}$, $\frac{23\!\cdots\!84}{49\!\cdots\!55}a^{21}-\frac{38\!\cdots\!97}{49\!\cdots\!55}a^{20}+\frac{30\!\cdots\!27}{98\!\cdots\!31}a^{19}-\frac{83\!\cdots\!45}{98\!\cdots\!31}a^{18}+\frac{76\!\cdots\!67}{49\!\cdots\!55}a^{17}-\frac{15\!\cdots\!78}{49\!\cdots\!55}a^{16}+\frac{19\!\cdots\!76}{49\!\cdots\!55}a^{15}-\frac{32\!\cdots\!89}{49\!\cdots\!55}a^{14}+\frac{31\!\cdots\!74}{49\!\cdots\!55}a^{13}-\frac{11\!\cdots\!33}{98\!\cdots\!31}a^{12}+\frac{50\!\cdots\!92}{49\!\cdots\!55}a^{11}-\frac{16\!\cdots\!90}{98\!\cdots\!31}a^{10}+\frac{76\!\cdots\!14}{49\!\cdots\!55}a^{9}-\frac{81\!\cdots\!62}{49\!\cdots\!55}a^{8}+\frac{71\!\cdots\!11}{49\!\cdots\!55}a^{7}-\frac{37\!\cdots\!09}{49\!\cdots\!55}a^{6}+\frac{21\!\cdots\!94}{49\!\cdots\!55}a^{5}+\frac{29\!\cdots\!31}{49\!\cdots\!55}a^{4}-\frac{41\!\cdots\!64}{49\!\cdots\!55}a^{3}+\frac{24\!\cdots\!63}{49\!\cdots\!55}a^{2}-\frac{22\!\cdots\!79}{98\!\cdots\!31}a-\frac{10\!\cdots\!46}{49\!\cdots\!55}$, $\frac{21\!\cdots\!34}{49\!\cdots\!55}a^{21}-\frac{49\!\cdots\!67}{49\!\cdots\!55}a^{20}+\frac{31\!\cdots\!80}{98\!\cdots\!31}a^{19}-\frac{94\!\cdots\!29}{98\!\cdots\!31}a^{18}+\frac{94\!\cdots\!02}{49\!\cdots\!55}a^{17}-\frac{18\!\cdots\!88}{49\!\cdots\!55}a^{16}+\frac{28\!\cdots\!51}{49\!\cdots\!55}a^{15}-\frac{43\!\cdots\!74}{49\!\cdots\!55}a^{14}+\frac{53\!\cdots\!64}{49\!\cdots\!55}a^{13}-\frac{15\!\cdots\!68}{98\!\cdots\!31}a^{12}+\frac{91\!\cdots\!07}{49\!\cdots\!55}a^{11}-\frac{23\!\cdots\!36}{98\!\cdots\!31}a^{10}+\frac{13\!\cdots\!59}{49\!\cdots\!55}a^{9}-\frac{14\!\cdots\!37}{49\!\cdots\!55}a^{8}+\frac{13\!\cdots\!01}{49\!\cdots\!55}a^{7}-\frac{11\!\cdots\!89}{49\!\cdots\!55}a^{6}+\frac{73\!\cdots\!54}{49\!\cdots\!55}a^{5}-\frac{41\!\cdots\!59}{49\!\cdots\!55}a^{4}+\frac{15\!\cdots\!16}{49\!\cdots\!55}a^{3}-\frac{35\!\cdots\!97}{49\!\cdots\!55}a^{2}+\frac{10\!\cdots\!28}{98\!\cdots\!31}a+\frac{70\!\cdots\!64}{49\!\cdots\!55}$, $\frac{14\!\cdots\!16}{49\!\cdots\!55}a^{21}-\frac{38\!\cdots\!28}{49\!\cdots\!55}a^{20}+\frac{19\!\cdots\!92}{98\!\cdots\!31}a^{19}-\frac{63\!\cdots\!75}{98\!\cdots\!31}a^{18}+\frac{59\!\cdots\!38}{49\!\cdots\!55}a^{17}-\frac{10\!\cdots\!97}{49\!\cdots\!55}a^{16}+\frac{15\!\cdots\!69}{49\!\cdots\!55}a^{15}-\frac{21\!\cdots\!36}{49\!\cdots\!55}a^{14}+\frac{25\!\cdots\!36}{49\!\cdots\!55}a^{13}-\frac{63\!\cdots\!16}{98\!\cdots\!31}a^{12}+\frac{46\!\cdots\!13}{49\!\cdots\!55}a^{11}-\frac{89\!\cdots\!26}{98\!\cdots\!31}a^{10}+\frac{64\!\cdots\!86}{49\!\cdots\!55}a^{9}-\frac{49\!\cdots\!83}{49\!\cdots\!55}a^{8}+\frac{44\!\cdots\!24}{49\!\cdots\!55}a^{7}-\frac{30\!\cdots\!71}{49\!\cdots\!55}a^{6}+\frac{96\!\cdots\!66}{49\!\cdots\!55}a^{5}-\frac{38\!\cdots\!66}{49\!\cdots\!55}a^{4}-\frac{33\!\cdots\!26}{49\!\cdots\!55}a^{3}+\frac{76\!\cdots\!37}{49\!\cdots\!55}a^{2}+\frac{19\!\cdots\!30}{98\!\cdots\!31}a+\frac{82\!\cdots\!51}{49\!\cdots\!55}$, $\frac{50\!\cdots\!91}{49\!\cdots\!55}a^{21}-\frac{11\!\cdots\!38}{49\!\cdots\!55}a^{20}+\frac{70\!\cdots\!13}{98\!\cdots\!31}a^{19}-\frac{20\!\cdots\!21}{98\!\cdots\!31}a^{18}+\frac{19\!\cdots\!63}{49\!\cdots\!55}a^{17}-\frac{37\!\cdots\!97}{49\!\cdots\!55}a^{16}+\frac{53\!\cdots\!09}{49\!\cdots\!55}a^{15}-\frac{80\!\cdots\!41}{49\!\cdots\!55}a^{14}+\frac{90\!\cdots\!66}{49\!\cdots\!55}a^{13}-\frac{26\!\cdots\!37}{98\!\cdots\!31}a^{12}+\frac{15\!\cdots\!48}{49\!\cdots\!55}a^{11}-\frac{38\!\cdots\!20}{98\!\cdots\!31}a^{10}+\frac{22\!\cdots\!81}{49\!\cdots\!55}a^{9}-\frac{21\!\cdots\!43}{49\!\cdots\!55}a^{8}+\frac{19\!\cdots\!74}{49\!\cdots\!55}a^{7}-\frac{12\!\cdots\!56}{49\!\cdots\!55}a^{6}+\frac{53\!\cdots\!01}{49\!\cdots\!55}a^{5}-\frac{90\!\cdots\!21}{49\!\cdots\!55}a^{4}-\frac{66\!\cdots\!66}{49\!\cdots\!55}a^{3}+\frac{51\!\cdots\!72}{49\!\cdots\!55}a^{2}-\frac{32\!\cdots\!20}{98\!\cdots\!31}a-\frac{14\!\cdots\!24}{49\!\cdots\!55}$, $\frac{10\!\cdots\!33}{49\!\cdots\!55}a^{21}-\frac{22\!\cdots\!59}{49\!\cdots\!55}a^{20}+\frac{15\!\cdots\!45}{98\!\cdots\!31}a^{19}-\frac{44\!\cdots\!22}{98\!\cdots\!31}a^{18}+\frac{43\!\cdots\!94}{49\!\cdots\!55}a^{17}-\frac{85\!\cdots\!06}{49\!\cdots\!55}a^{16}+\frac{12\!\cdots\!02}{49\!\cdots\!55}a^{15}-\frac{19\!\cdots\!43}{49\!\cdots\!55}a^{14}+\frac{23\!\cdots\!88}{49\!\cdots\!55}a^{13}-\frac{69\!\cdots\!04}{98\!\cdots\!31}a^{12}+\frac{38\!\cdots\!49}{49\!\cdots\!55}a^{11}-\frac{10\!\cdots\!17}{98\!\cdots\!31}a^{10}+\frac{58\!\cdots\!88}{49\!\cdots\!55}a^{9}-\frac{60\!\cdots\!24}{49\!\cdots\!55}a^{8}+\frac{59\!\cdots\!22}{49\!\cdots\!55}a^{7}-\frac{43\!\cdots\!08}{49\!\cdots\!55}a^{6}+\frac{27\!\cdots\!78}{49\!\cdots\!55}a^{5}-\frac{12\!\cdots\!38}{49\!\cdots\!55}a^{4}+\frac{32\!\cdots\!92}{49\!\cdots\!55}a^{3}-\frac{12\!\cdots\!59}{49\!\cdots\!55}a^{2}-\frac{10\!\cdots\!33}{98\!\cdots\!31}a+\frac{18\!\cdots\!93}{49\!\cdots\!55}$, $\frac{49\!\cdots\!64}{98\!\cdots\!31}a^{21}-\frac{65\!\cdots\!77}{98\!\cdots\!31}a^{20}+\frac{24\!\cdots\!63}{98\!\cdots\!31}a^{19}-\frac{69\!\cdots\!15}{98\!\cdots\!31}a^{18}+\frac{10\!\cdots\!41}{98\!\cdots\!31}a^{17}-\frac{17\!\cdots\!14}{98\!\cdots\!31}a^{16}+\frac{17\!\cdots\!30}{98\!\cdots\!31}a^{15}-\frac{26\!\cdots\!33}{98\!\cdots\!31}a^{14}+\frac{71\!\cdots\!75}{98\!\cdots\!31}a^{13}-\frac{35\!\cdots\!50}{98\!\cdots\!31}a^{12}+\frac{11\!\cdots\!99}{98\!\cdots\!31}a^{11}-\frac{32\!\cdots\!47}{98\!\cdots\!31}a^{10}+\frac{12\!\cdots\!54}{98\!\cdots\!31}a^{9}+\frac{27\!\cdots\!78}{98\!\cdots\!31}a^{8}-\frac{50\!\cdots\!35}{98\!\cdots\!31}a^{7}+\frac{10\!\cdots\!50}{98\!\cdots\!31}a^{6}-\frac{10\!\cdots\!99}{98\!\cdots\!31}a^{5}+\frac{83\!\cdots\!81}{98\!\cdots\!31}a^{4}-\frac{40\!\cdots\!76}{98\!\cdots\!31}a^{3}+\frac{79\!\cdots\!53}{98\!\cdots\!31}a^{2}+\frac{22\!\cdots\!37}{98\!\cdots\!31}a-\frac{28\!\cdots\!02}{98\!\cdots\!31}$
| 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: | \( 56909.7191275 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{9}\cdot 56909.7191275 \cdot 1}{2\cdot\sqrt{1533107758369291702785569375}}\cr\approx \mathstrut & 0.177463443250 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 7*x^20 - 20*x^19 + 38*x^18 - 74*x^17 + 107*x^16 - 167*x^15 + 185*x^14 - 284*x^13 + 308*x^12 - 417*x^11 + 466*x^10 - 462*x^9 + 446*x^8 - 292*x^7 + 170*x^6 - 55*x^5 - 7*x^4 + 11*x^3 - 8*x^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 - 2*x^21 + 7*x^20 - 20*x^19 + 38*x^18 - 74*x^17 + 107*x^16 - 167*x^15 + 185*x^14 - 284*x^13 + 308*x^12 - 417*x^11 + 466*x^10 - 462*x^9 + 446*x^8 - 292*x^7 + 170*x^6 - 55*x^5 - 7*x^4 + 11*x^3 - 8*x^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 - 2*x^21 + 7*x^20 - 20*x^19 + 38*x^18 - 74*x^17 + 107*x^16 - 167*x^15 + 185*x^14 - 284*x^13 + 308*x^12 - 417*x^11 + 466*x^10 - 462*x^9 + 446*x^8 - 292*x^7 + 170*x^6 - 55*x^5 - 7*x^4 + 11*x^3 - 8*x^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 - 2*x^21 + 7*x^20 - 20*x^19 + 38*x^18 - 74*x^17 + 107*x^16 - 167*x^15 + 185*x^14 - 284*x^13 + 308*x^12 - 417*x^11 + 466*x^10 - 462*x^9 + 446*x^8 - 292*x^7 + 170*x^6 - 55*x^5 - 7*x^4 + 11*x^3 - 8*x^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
$C_2^{11}.A_{11}$ (as 22T52):
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 40874803200 |
| The 400 conjugacy class representatives for $C_2^{11}.A_{11}$ are not computed |
| Character table for $C_2^{11}.A_{11}$ is not computed |
Intermediate fields
| 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]
Sibling fields
| Degree 44 sibling: | 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 | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | R | R | $22$ | $22$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.11.0.1}{11} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.12.0.1 | $x^{12} + x^{7} + x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(7\)
| 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 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.14.0.1 | $x^{14} + 5 x^{7} + 6 x^{5} + 2 x^{4} + 3 x^{2} + 6 x + 3$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(83\)
| 83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 83.4.0.1 | $x^{4} + 4 x^{2} + 42 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 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 | $x^{3} + 127$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 127.14.0.1 | $x^{14} - x + 29$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(997\)
| $\Q_{997}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{997}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{997}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{997}$ | $x$ | $1$ | $1$ | $0$ | 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |