Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 22*x^20 - 44*x^19 + 176*x^18 - 242*x^17 + 506*x^16 - 726*x^15 + 528*x^14 - 5214*x^13 + 3894*x^12 + 316*x^11 - 23716*x^10 - 10472*x^9 + 82962*x^8 + 101090*x^7 - 82456*x^6 - 170368*x^5 - 12100*x^4 + 75504*x^3 + 37510*x^2 + 6776*x + 242)
gp: K = bnfinit(y^22 + 22*y^20 - 44*y^19 + 176*y^18 - 242*y^17 + 506*y^16 - 726*y^15 + 528*y^14 - 5214*y^13 + 3894*y^12 + 316*y^11 - 23716*y^10 - 10472*y^9 + 82962*y^8 + 101090*y^7 - 82456*y^6 - 170368*y^5 - 12100*y^4 + 75504*y^3 + 37510*y^2 + 6776*y + 242, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 22*x^20 - 44*x^19 + 176*x^18 - 242*x^17 + 506*x^16 - 726*x^15 + 528*x^14 - 5214*x^13 + 3894*x^12 + 316*x^11 - 23716*x^10 - 10472*x^9 + 82962*x^8 + 101090*x^7 - 82456*x^6 - 170368*x^5 - 12100*x^4 + 75504*x^3 + 37510*x^2 + 6776*x + 242);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 22*x^20 - 44*x^19 + 176*x^18 - 242*x^17 + 506*x^16 - 726*x^15 + 528*x^14 - 5214*x^13 + 3894*x^12 + 316*x^11 - 23716*x^10 - 10472*x^9 + 82962*x^8 + 101090*x^7 - 82456*x^6 - 170368*x^5 - 12100*x^4 + 75504*x^3 + 37510*x^2 + 6776*x + 242)
\( x^{22} + 22 x^{20} - 44 x^{19} + 176 x^{18} - 242 x^{17} + 506 x^{16} - 726 x^{15} + 528 x^{14} + \cdots + 242 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[6, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(6172475179135960522642404800603172634624\)
\(\medspace = 2^{28}\cdot 7^{10}\cdot 11^{22}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(64.37\) | 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: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{11}a^{17}-\frac{3}{11}a^{6}$, $\frac{1}{11}a^{18}-\frac{3}{11}a^{7}$, $\frac{1}{11}a^{19}-\frac{3}{11}a^{8}$, $\frac{1}{11}a^{20}-\frac{3}{11}a^{9}$, $\frac{1}{17\!\cdots\!23}a^{21}-\frac{28\!\cdots\!73}{17\!\cdots\!23}a^{20}-\frac{35\!\cdots\!49}{17\!\cdots\!23}a^{19}+\frac{43\!\cdots\!55}{17\!\cdots\!23}a^{18}-\frac{70\!\cdots\!03}{17\!\cdots\!23}a^{17}-\frac{43\!\cdots\!27}{15\!\cdots\!93}a^{16}+\frac{73\!\cdots\!67}{15\!\cdots\!93}a^{15}-\frac{63\!\cdots\!74}{15\!\cdots\!93}a^{14}+\frac{47\!\cdots\!75}{15\!\cdots\!93}a^{13}-\frac{11\!\cdots\!05}{15\!\cdots\!93}a^{12}-\frac{41\!\cdots\!68}{15\!\cdots\!93}a^{11}+\frac{38\!\cdots\!51}{17\!\cdots\!23}a^{10}-\frac{17\!\cdots\!06}{17\!\cdots\!23}a^{9}-\frac{66\!\cdots\!67}{17\!\cdots\!23}a^{8}-\frac{10\!\cdots\!29}{17\!\cdots\!23}a^{7}+\frac{68\!\cdots\!90}{17\!\cdots\!23}a^{6}+\frac{36\!\cdots\!70}{15\!\cdots\!93}a^{5}-\frac{75\!\cdots\!07}{15\!\cdots\!93}a^{4}+\frac{26\!\cdots\!53}{15\!\cdots\!93}a^{3}-\frac{42\!\cdots\!81}{15\!\cdots\!93}a^{2}-\frac{61\!\cdots\!20}{15\!\cdots\!93}a-\frac{18\!\cdots\!59}{15\!\cdots\!93}$
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: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{12\!\cdots\!08}{17\!\cdots\!23}a^{21}-\frac{10\!\cdots\!18}{17\!\cdots\!23}a^{20}+\frac{28\!\cdots\!54}{17\!\cdots\!23}a^{19}-\frac{78\!\cdots\!25}{17\!\cdots\!23}a^{18}+\frac{27\!\cdots\!04}{17\!\cdots\!23}a^{17}-\frac{47\!\cdots\!95}{15\!\cdots\!93}a^{16}+\frac{91\!\cdots\!28}{15\!\cdots\!93}a^{15}-\frac{15\!\cdots\!37}{15\!\cdots\!93}a^{14}+\frac{16\!\cdots\!94}{15\!\cdots\!93}a^{13}-\frac{70\!\cdots\!79}{15\!\cdots\!93}a^{12}+\frac{99\!\cdots\!32}{15\!\cdots\!93}a^{11}-\frac{68\!\cdots\!40}{17\!\cdots\!23}a^{10}-\frac{25\!\cdots\!63}{17\!\cdots\!23}a^{9}+\frac{95\!\cdots\!14}{17\!\cdots\!23}a^{8}+\frac{10\!\cdots\!60}{17\!\cdots\!23}a^{7}+\frac{48\!\cdots\!02}{17\!\cdots\!23}a^{6}-\frac{14\!\cdots\!42}{15\!\cdots\!93}a^{5}-\frac{98\!\cdots\!49}{15\!\cdots\!93}a^{4}+\frac{67\!\cdots\!58}{15\!\cdots\!93}a^{3}+\frac{47\!\cdots\!61}{15\!\cdots\!93}a^{2}+\frac{11\!\cdots\!90}{15\!\cdots\!93}a+\frac{61\!\cdots\!77}{15\!\cdots\!93}$, $\frac{35\!\cdots\!12}{17\!\cdots\!23}a^{21}-\frac{57\!\cdots\!85}{17\!\cdots\!23}a^{20}+\frac{76\!\cdots\!82}{17\!\cdots\!23}a^{19}-\frac{28\!\cdots\!33}{17\!\cdots\!23}a^{18}+\frac{83\!\cdots\!40}{17\!\cdots\!23}a^{17}-\frac{17\!\cdots\!94}{15\!\cdots\!93}a^{16}+\frac{28\!\cdots\!60}{15\!\cdots\!93}a^{15}-\frac{56\!\cdots\!67}{15\!\cdots\!93}a^{14}+\frac{58\!\cdots\!56}{15\!\cdots\!93}a^{13}-\frac{22\!\cdots\!04}{15\!\cdots\!93}a^{12}+\frac{42\!\cdots\!76}{15\!\cdots\!93}a^{11}-\frac{18\!\cdots\!38}{17\!\cdots\!23}a^{10}-\frac{57\!\cdots\!73}{17\!\cdots\!23}a^{9}+\frac{84\!\cdots\!72}{17\!\cdots\!23}a^{8}+\frac{41\!\cdots\!60}{17\!\cdots\!23}a^{7}+\frac{35\!\cdots\!92}{17\!\cdots\!23}a^{6}-\frac{76\!\cdots\!16}{15\!\cdots\!93}a^{5}-\frac{50\!\cdots\!73}{15\!\cdots\!93}a^{4}+\frac{40\!\cdots\!94}{15\!\cdots\!93}a^{3}+\frac{36\!\cdots\!41}{15\!\cdots\!93}a^{2}+\frac{11\!\cdots\!82}{15\!\cdots\!93}a+\frac{11\!\cdots\!65}{15\!\cdots\!93}$, $\frac{39\!\cdots\!81}{10\!\cdots\!19}a^{21}+\frac{11\!\cdots\!01}{10\!\cdots\!19}a^{20}+\frac{13\!\cdots\!35}{10\!\cdots\!19}a^{19}+\frac{10\!\cdots\!90}{10\!\cdots\!19}a^{18}+\frac{14\!\cdots\!72}{10\!\cdots\!19}a^{17}-\frac{55\!\cdots\!14}{91\!\cdots\!29}a^{16}+\frac{77\!\cdots\!31}{91\!\cdots\!29}a^{15}-\frac{52\!\cdots\!62}{91\!\cdots\!29}a^{14}+\frac{19\!\cdots\!60}{91\!\cdots\!29}a^{13}-\frac{38\!\cdots\!55}{91\!\cdots\!29}a^{12}-\frac{12\!\cdots\!96}{91\!\cdots\!29}a^{11}-\frac{23\!\cdots\!54}{10\!\cdots\!19}a^{10}-\frac{30\!\cdots\!11}{10\!\cdots\!19}a^{9}-\frac{42\!\cdots\!11}{10\!\cdots\!19}a^{8}-\frac{10\!\cdots\!52}{10\!\cdots\!19}a^{7}-\frac{92\!\cdots\!50}{10\!\cdots\!19}a^{6}+\frac{32\!\cdots\!02}{91\!\cdots\!29}a^{5}+\frac{42\!\cdots\!01}{91\!\cdots\!29}a^{4}-\frac{68\!\cdots\!64}{91\!\cdots\!29}a^{3}-\frac{40\!\cdots\!57}{91\!\cdots\!29}a^{2}-\frac{18\!\cdots\!46}{91\!\cdots\!29}a-\frac{30\!\cdots\!95}{91\!\cdots\!29}$, $\frac{69\!\cdots\!59}{17\!\cdots\!23}a^{21}-\frac{26\!\cdots\!75}{17\!\cdots\!23}a^{20}+\frac{14\!\cdots\!52}{15\!\cdots\!93}a^{19}-\frac{36\!\cdots\!74}{17\!\cdots\!23}a^{18}+\frac{14\!\cdots\!18}{17\!\cdots\!23}a^{17}-\frac{21\!\cdots\!16}{15\!\cdots\!93}a^{16}+\frac{46\!\cdots\!48}{15\!\cdots\!93}a^{15}-\frac{69\!\cdots\!84}{15\!\cdots\!93}a^{14}+\frac{79\!\cdots\!62}{15\!\cdots\!93}a^{13}-\frac{37\!\cdots\!86}{15\!\cdots\!93}a^{12}+\frac{41\!\cdots\!78}{15\!\cdots\!93}a^{11}-\frac{35\!\cdots\!98}{17\!\cdots\!23}a^{10}-\frac{14\!\cdots\!41}{17\!\cdots\!23}a^{9}-\frac{24\!\cdots\!02}{15\!\cdots\!93}a^{8}+\frac{49\!\cdots\!64}{17\!\cdots\!23}a^{7}+\frac{40\!\cdots\!50}{17\!\cdots\!23}a^{6}-\frac{48\!\cdots\!64}{15\!\cdots\!93}a^{5}-\frac{49\!\cdots\!55}{15\!\cdots\!93}a^{4}+\frac{18\!\cdots\!48}{15\!\cdots\!93}a^{3}+\frac{12\!\cdots\!17}{15\!\cdots\!93}a^{2}+\frac{17\!\cdots\!30}{15\!\cdots\!93}a-\frac{28\!\cdots\!45}{15\!\cdots\!93}$, $\frac{67\!\cdots\!05}{10\!\cdots\!19}a^{21}-\frac{10\!\cdots\!76}{91\!\cdots\!29}a^{20}+\frac{14\!\cdots\!34}{10\!\cdots\!19}a^{19}-\frac{32\!\cdots\!10}{10\!\cdots\!19}a^{18}+\frac{12\!\cdots\!96}{10\!\cdots\!19}a^{17}-\frac{17\!\cdots\!69}{91\!\cdots\!29}a^{16}+\frac{34\!\cdots\!47}{91\!\cdots\!29}a^{15}-\frac{52\!\cdots\!84}{91\!\cdots\!29}a^{14}+\frac{43\!\cdots\!23}{91\!\cdots\!29}a^{13}-\frac{33\!\cdots\!50}{91\!\cdots\!29}a^{12}+\frac{30\!\cdots\!38}{91\!\cdots\!29}a^{11}-\frac{37\!\cdots\!06}{10\!\cdots\!19}a^{10}-\frac{14\!\cdots\!99}{91\!\cdots\!29}a^{9}-\frac{48\!\cdots\!45}{10\!\cdots\!19}a^{8}+\frac{57\!\cdots\!52}{10\!\cdots\!19}a^{7}+\frac{62\!\cdots\!03}{10\!\cdots\!19}a^{6}-\frac{59\!\cdots\!00}{91\!\cdots\!29}a^{5}-\frac{10\!\cdots\!15}{91\!\cdots\!29}a^{4}+\frac{22\!\cdots\!56}{91\!\cdots\!29}a^{3}+\frac{50\!\cdots\!99}{91\!\cdots\!29}a^{2}+\frac{22\!\cdots\!64}{91\!\cdots\!29}a+\frac{36\!\cdots\!33}{91\!\cdots\!29}$, $\frac{26\!\cdots\!54}{10\!\cdots\!19}a^{21}-\frac{15\!\cdots\!52}{10\!\cdots\!19}a^{20}+\frac{58\!\cdots\!26}{10\!\cdots\!19}a^{19}-\frac{15\!\cdots\!47}{10\!\cdots\!19}a^{18}+\frac{53\!\cdots\!23}{10\!\cdots\!19}a^{17}-\frac{85\!\cdots\!12}{91\!\cdots\!29}a^{16}+\frac{15\!\cdots\!54}{91\!\cdots\!29}a^{15}-\frac{26\!\cdots\!18}{91\!\cdots\!29}a^{14}+\frac{23\!\cdots\!46}{91\!\cdots\!29}a^{13}-\frac{13\!\cdots\!60}{91\!\cdots\!29}a^{12}+\frac{17\!\cdots\!64}{91\!\cdots\!29}a^{11}-\frac{43\!\cdots\!51}{10\!\cdots\!19}a^{10}-\frac{57\!\cdots\!40}{10\!\cdots\!19}a^{9}+\frac{73\!\cdots\!86}{10\!\cdots\!19}a^{8}+\frac{25\!\cdots\!88}{10\!\cdots\!19}a^{7}+\frac{17\!\cdots\!18}{10\!\cdots\!19}a^{6}-\frac{34\!\cdots\!38}{91\!\cdots\!29}a^{5}-\frac{39\!\cdots\!49}{91\!\cdots\!29}a^{4}+\frac{98\!\cdots\!14}{91\!\cdots\!29}a^{3}+\frac{23\!\cdots\!14}{91\!\cdots\!29}a^{2}+\frac{90\!\cdots\!18}{91\!\cdots\!29}a+\frac{12\!\cdots\!51}{91\!\cdots\!29}$, $\frac{36\!\cdots\!16}{10\!\cdots\!19}a^{21}+\frac{40\!\cdots\!15}{10\!\cdots\!19}a^{20}+\frac{80\!\cdots\!79}{10\!\cdots\!19}a^{19}-\frac{15\!\cdots\!50}{10\!\cdots\!19}a^{18}+\frac{64\!\cdots\!33}{10\!\cdots\!19}a^{17}-\frac{80\!\cdots\!94}{91\!\cdots\!29}a^{16}+\frac{18\!\cdots\!43}{91\!\cdots\!29}a^{15}-\frac{27\!\cdots\!01}{91\!\cdots\!29}a^{14}+\frac{27\!\cdots\!56}{91\!\cdots\!29}a^{13}-\frac{19\!\cdots\!73}{91\!\cdots\!29}a^{12}+\frac{14\!\cdots\!22}{91\!\cdots\!29}a^{11}-\frac{56\!\cdots\!95}{10\!\cdots\!19}a^{10}-\frac{70\!\cdots\!85}{10\!\cdots\!19}a^{9}-\frac{69\!\cdots\!18}{10\!\cdots\!19}a^{8}+\frac{30\!\cdots\!06}{10\!\cdots\!19}a^{7}+\frac{41\!\cdots\!02}{10\!\cdots\!19}a^{6}-\frac{19\!\cdots\!14}{91\!\cdots\!29}a^{5}-\frac{63\!\cdots\!07}{91\!\cdots\!29}a^{4}-\frac{17\!\cdots\!14}{91\!\cdots\!29}a^{3}+\frac{31\!\cdots\!25}{91\!\cdots\!29}a^{2}+\frac{17\!\cdots\!16}{91\!\cdots\!29}a+\frac{91\!\cdots\!17}{91\!\cdots\!29}$, $\frac{36\!\cdots\!25}{17\!\cdots\!23}a^{21}-\frac{50\!\cdots\!55}{17\!\cdots\!23}a^{20}+\frac{81\!\cdots\!16}{17\!\cdots\!23}a^{19}-\frac{17\!\cdots\!29}{17\!\cdots\!23}a^{18}+\frac{68\!\cdots\!91}{17\!\cdots\!23}a^{17}-\frac{90\!\cdots\!63}{15\!\cdots\!93}a^{16}+\frac{19\!\cdots\!10}{15\!\cdots\!93}a^{15}-\frac{27\!\cdots\!54}{15\!\cdots\!93}a^{14}+\frac{24\!\cdots\!99}{15\!\cdots\!93}a^{13}-\frac{17\!\cdots\!37}{15\!\cdots\!93}a^{12}+\frac{15\!\cdots\!36}{15\!\cdots\!93}a^{11}-\frac{52\!\cdots\!99}{17\!\cdots\!23}a^{10}-\frac{85\!\cdots\!63}{17\!\cdots\!23}a^{9}-\frac{27\!\cdots\!07}{17\!\cdots\!23}a^{8}+\frac{28\!\cdots\!60}{17\!\cdots\!23}a^{7}+\frac{30\!\cdots\!21}{17\!\cdots\!23}a^{6}-\frac{27\!\cdots\!26}{15\!\cdots\!93}a^{5}-\frac{43\!\cdots\!66}{15\!\cdots\!93}a^{4}+\frac{52\!\cdots\!94}{15\!\cdots\!93}a^{3}+\frac{18\!\cdots\!39}{15\!\cdots\!93}a^{2}+\frac{46\!\cdots\!04}{15\!\cdots\!93}a+\frac{30\!\cdots\!15}{15\!\cdots\!93}$, $\frac{18\!\cdots\!50}{17\!\cdots\!23}a^{21}+\frac{24\!\cdots\!89}{17\!\cdots\!23}a^{20}+\frac{42\!\cdots\!20}{17\!\cdots\!23}a^{19}-\frac{22\!\cdots\!26}{17\!\cdots\!23}a^{18}+\frac{26\!\cdots\!66}{17\!\cdots\!23}a^{17}-\frac{35\!\cdots\!59}{15\!\cdots\!93}a^{16}+\frac{62\!\cdots\!11}{15\!\cdots\!93}a^{15}-\frac{14\!\cdots\!61}{15\!\cdots\!93}a^{14}+\frac{16\!\cdots\!03}{15\!\cdots\!93}a^{13}-\frac{77\!\cdots\!35}{15\!\cdots\!93}a^{12}-\frac{46\!\cdots\!24}{15\!\cdots\!93}a^{11}-\frac{14\!\cdots\!99}{17\!\cdots\!23}a^{10}-\frac{47\!\cdots\!58}{17\!\cdots\!23}a^{9}-\frac{82\!\cdots\!72}{17\!\cdots\!23}a^{8}+\frac{65\!\cdots\!26}{17\!\cdots\!23}a^{7}+\frac{28\!\cdots\!21}{17\!\cdots\!23}a^{6}+\frac{15\!\cdots\!18}{15\!\cdots\!93}a^{5}-\frac{14\!\cdots\!50}{15\!\cdots\!93}a^{4}-\frac{18\!\cdots\!32}{15\!\cdots\!93}a^{3}-\frac{50\!\cdots\!48}{15\!\cdots\!93}a^{2}-\frac{20\!\cdots\!36}{15\!\cdots\!93}a+\frac{11\!\cdots\!69}{15\!\cdots\!93}$, $\frac{22\!\cdots\!33}{10\!\cdots\!19}a^{21}+\frac{20\!\cdots\!19}{10\!\cdots\!19}a^{20}+\frac{46\!\cdots\!75}{10\!\cdots\!19}a^{19}-\frac{52\!\cdots\!55}{10\!\cdots\!19}a^{18}+\frac{24\!\cdots\!38}{10\!\cdots\!19}a^{17}-\frac{28\!\cdots\!80}{91\!\cdots\!29}a^{16}+\frac{36\!\cdots\!45}{91\!\cdots\!29}a^{15}+\frac{39\!\cdots\!32}{91\!\cdots\!29}a^{14}-\frac{23\!\cdots\!28}{91\!\cdots\!29}a^{13}-\frac{64\!\cdots\!62}{91\!\cdots\!29}a^{12}-\frac{57\!\cdots\!24}{91\!\cdots\!29}a^{11}+\frac{27\!\cdots\!34}{10\!\cdots\!19}a^{10}-\frac{74\!\cdots\!63}{10\!\cdots\!19}a^{9}-\frac{53\!\cdots\!09}{10\!\cdots\!19}a^{8}+\frac{21\!\cdots\!76}{10\!\cdots\!19}a^{7}+\frac{41\!\cdots\!60}{10\!\cdots\!19}a^{6}-\frac{16\!\cdots\!66}{91\!\cdots\!29}a^{5}-\frac{66\!\cdots\!48}{91\!\cdots\!29}a^{4}-\frac{14\!\cdots\!16}{91\!\cdots\!29}a^{3}+\frac{34\!\cdots\!59}{91\!\cdots\!29}a^{2}+\frac{17\!\cdots\!62}{91\!\cdots\!29}a+\frac{13\!\cdots\!77}{91\!\cdots\!29}$, $\frac{28\!\cdots\!01}{17\!\cdots\!23}a^{21}+\frac{34\!\cdots\!21}{17\!\cdots\!23}a^{20}+\frac{66\!\cdots\!34}{17\!\cdots\!23}a^{19}-\frac{45\!\cdots\!62}{17\!\cdots\!23}a^{18}+\frac{44\!\cdots\!52}{17\!\cdots\!23}a^{17}-\frac{13\!\cdots\!12}{15\!\cdots\!93}a^{16}+\frac{11\!\cdots\!10}{15\!\cdots\!93}a^{15}-\frac{51\!\cdots\!86}{15\!\cdots\!93}a^{14}+\frac{65\!\cdots\!77}{15\!\cdots\!93}a^{13}-\frac{12\!\cdots\!33}{15\!\cdots\!93}a^{12}-\frac{51\!\cdots\!46}{15\!\cdots\!93}a^{11}-\frac{49\!\cdots\!22}{17\!\cdots\!23}a^{10}-\frac{73\!\cdots\!12}{17\!\cdots\!23}a^{9}-\frac{11\!\cdots\!28}{17\!\cdots\!23}a^{8}+\frac{99\!\cdots\!61}{17\!\cdots\!23}a^{7}+\frac{41\!\cdots\!74}{17\!\cdots\!23}a^{6}+\frac{22\!\cdots\!66}{15\!\cdots\!93}a^{5}-\frac{18\!\cdots\!52}{15\!\cdots\!93}a^{4}-\frac{26\!\cdots\!24}{15\!\cdots\!93}a^{3}-\frac{10\!\cdots\!80}{15\!\cdots\!93}a^{2}-\frac{16\!\cdots\!58}{15\!\cdots\!93}a-\frac{54\!\cdots\!47}{15\!\cdots\!93}$, $\frac{16\!\cdots\!05}{15\!\cdots\!93}a^{21}-\frac{13\!\cdots\!53}{17\!\cdots\!23}a^{20}+\frac{40\!\cdots\!96}{17\!\cdots\!23}a^{19}-\frac{11\!\cdots\!92}{17\!\cdots\!23}a^{18}+\frac{39\!\cdots\!32}{17\!\cdots\!23}a^{17}-\frac{66\!\cdots\!28}{15\!\cdots\!93}a^{16}+\frac{13\!\cdots\!82}{15\!\cdots\!93}a^{15}-\frac{21\!\cdots\!95}{15\!\cdots\!93}a^{14}+\frac{23\!\cdots\!59}{15\!\cdots\!93}a^{13}-\frac{10\!\cdots\!72}{15\!\cdots\!93}a^{12}+\frac{13\!\cdots\!73}{15\!\cdots\!93}a^{11}-\frac{93\!\cdots\!47}{15\!\cdots\!93}a^{10}-\frac{36\!\cdots\!02}{17\!\cdots\!23}a^{9}+\frac{90\!\cdots\!92}{17\!\cdots\!23}a^{8}+\frac{14\!\cdots\!01}{17\!\cdots\!23}a^{7}+\frac{72\!\cdots\!67}{17\!\cdots\!23}a^{6}-\frac{19\!\cdots\!48}{15\!\cdots\!93}a^{5}-\frac{13\!\cdots\!47}{15\!\cdots\!93}a^{4}+\frac{97\!\cdots\!36}{15\!\cdots\!93}a^{3}+\frac{60\!\cdots\!92}{15\!\cdots\!93}a^{2}+\frac{98\!\cdots\!79}{15\!\cdots\!93}a-\frac{64\!\cdots\!03}{15\!\cdots\!93}$, $\frac{45\!\cdots\!49}{15\!\cdots\!93}a^{21}-\frac{23\!\cdots\!93}{17\!\cdots\!23}a^{20}+\frac{11\!\cdots\!46}{17\!\cdots\!23}a^{19}-\frac{22\!\cdots\!75}{17\!\cdots\!23}a^{18}+\frac{89\!\cdots\!36}{17\!\cdots\!23}a^{17}-\frac{11\!\cdots\!66}{15\!\cdots\!93}a^{16}+\frac{23\!\cdots\!55}{15\!\cdots\!93}a^{15}-\frac{34\!\cdots\!39}{15\!\cdots\!93}a^{14}+\frac{25\!\cdots\!67}{15\!\cdots\!93}a^{13}-\frac{23\!\cdots\!37}{15\!\cdots\!93}a^{12}+\frac{18\!\cdots\!64}{15\!\cdots\!93}a^{11}+\frac{56\!\cdots\!80}{15\!\cdots\!93}a^{10}-\frac{11\!\cdots\!45}{17\!\cdots\!23}a^{9}-\frac{47\!\cdots\!36}{17\!\cdots\!23}a^{8}+\frac{41\!\cdots\!52}{17\!\cdots\!23}a^{7}+\frac{48\!\cdots\!67}{17\!\cdots\!23}a^{6}-\frac{39\!\cdots\!85}{15\!\cdots\!93}a^{5}-\frac{76\!\cdots\!21}{15\!\cdots\!93}a^{4}-\frac{19\!\cdots\!55}{15\!\cdots\!93}a^{3}+\frac{34\!\cdots\!55}{15\!\cdots\!93}a^{2}+\frac{15\!\cdots\!65}{15\!\cdots\!93}a+\frac{23\!\cdots\!91}{15\!\cdots\!93}$
| 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: | \( 629593387255 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 629593387255 \cdot 1}{2\cdot\sqrt{6172475179135960522642404800603172634624}}\cr\approx \mathstrut & 0.622901543425920 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 22*x^20 - 44*x^19 + 176*x^18 - 242*x^17 + 506*x^16 - 726*x^15 + 528*x^14 - 5214*x^13 + 3894*x^12 + 316*x^11 - 23716*x^10 - 10472*x^9 + 82962*x^8 + 101090*x^7 - 82456*x^6 - 170368*x^5 - 12100*x^4 + 75504*x^3 + 37510*x^2 + 6776*x + 242)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^22 + 22*x^20 - 44*x^19 + 176*x^18 - 242*x^17 + 506*x^16 - 726*x^15 + 528*x^14 - 5214*x^13 + 3894*x^12 + 316*x^11 - 23716*x^10 - 10472*x^9 + 82962*x^8 + 101090*x^7 - 82456*x^6 - 170368*x^5 - 12100*x^4 + 75504*x^3 + 37510*x^2 + 6776*x + 242, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^22 + 22*x^20 - 44*x^19 + 176*x^18 - 242*x^17 + 506*x^16 - 726*x^15 + 528*x^14 - 5214*x^13 + 3894*x^12 + 316*x^11 - 23716*x^10 - 10472*x^9 + 82962*x^8 + 101090*x^7 - 82456*x^6 - 170368*x^5 - 12100*x^4 + 75504*x^3 + 37510*x^2 + 6776*x + 242);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 22*x^20 - 44*x^19 + 176*x^18 - 242*x^17 + 506*x^16 - 726*x^15 + 528*x^14 - 5214*x^13 + 3894*x^12 + 316*x^11 - 23716*x^10 - 10472*x^9 + 82962*x^8 + 101090*x^7 - 82456*x^6 - 170368*x^5 - 12100*x^4 + 75504*x^3 + 37510*x^2 + 6776*x + 242);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_2^{10}.F_{11}$ (as 22T34):
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 112640 |
| The 44 conjugacy class representatives for $C_2^{10}.F_{11}$ |
| Character table for $C_2^{10}.F_{11}$ is not computed |
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 | $20{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ | $20{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.5.0.1}{5} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $22$ | $22$ | $1$ | $28$ | |||
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.10.5.1 | $x^{10} + 2401 x^{2} - 67228$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 7.10.5.1 | $x^{10} + 2401 x^{2} - 67228$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(11\)
| 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |
| 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |