Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 - 24*x^20 + 59*x^19 + 273*x^18 - 436*x^17 - 1924*x^16 + 1356*x^15 + 5010*x^14 + 14999*x^13 - 9512*x^12 - 114771*x^11 + 18405*x^10 + 290616*x^9 + 69522*x^8 - 375325*x^7 - 259853*x^6 + 293924*x^5 + 260945*x^4 - 133418*x^3 - 93266*x^2 + 21174*x + 11623)
gp: K = bnfinit(y^22 - 3*y^21 - 24*y^20 + 59*y^19 + 273*y^18 - 436*y^17 - 1924*y^16 + 1356*y^15 + 5010*y^14 + 14999*y^13 - 9512*y^12 - 114771*y^11 + 18405*y^10 + 290616*y^9 + 69522*y^8 - 375325*y^7 - 259853*y^6 + 293924*y^5 + 260945*y^4 - 133418*y^3 - 93266*y^2 + 21174*y + 11623, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 3*x^21 - 24*x^20 + 59*x^19 + 273*x^18 - 436*x^17 - 1924*x^16 + 1356*x^15 + 5010*x^14 + 14999*x^13 - 9512*x^12 - 114771*x^11 + 18405*x^10 + 290616*x^9 + 69522*x^8 - 375325*x^7 - 259853*x^6 + 293924*x^5 + 260945*x^4 - 133418*x^3 - 93266*x^2 + 21174*x + 11623);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 3*x^21 - 24*x^20 + 59*x^19 + 273*x^18 - 436*x^17 - 1924*x^16 + 1356*x^15 + 5010*x^14 + 14999*x^13 - 9512*x^12 - 114771*x^11 + 18405*x^10 + 290616*x^9 + 69522*x^8 - 375325*x^7 - 259853*x^6 + 293924*x^5 + 260945*x^4 - 133418*x^3 - 93266*x^2 + 21174*x + 11623)
\( x^{22} - 3 x^{21} - 24 x^{20} + 59 x^{19} + 273 x^{18} - 436 x^{17} - 1924 x^{16} + 1356 x^{15} + \cdots + 11623 \)
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: | $[14, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(22661033510180079603495293971842498241\)
\(\medspace = 1297^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(49.88\) | 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: | $1297^{3/4}\approx 216.12498794754728$ | ||
| Ramified primes: |
\(1297\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5}a^{16}-\frac{1}{5}a^{15}-\frac{1}{5}a^{13}-\frac{2}{5}a^{12}-\frac{1}{5}a^{11}-\frac{1}{5}a^{10}+\frac{2}{5}a^{9}-\frac{2}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{3}+\frac{2}{5}$, $\frac{1}{5}a^{17}-\frac{1}{5}a^{15}-\frac{1}{5}a^{14}+\frac{2}{5}a^{13}+\frac{2}{5}a^{12}-\frac{2}{5}a^{11}+\frac{1}{5}a^{10}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{18}-\frac{2}{5}a^{15}+\frac{2}{5}a^{14}+\frac{1}{5}a^{13}+\frac{1}{5}a^{12}-\frac{1}{5}a^{10}+\frac{1}{5}a^{9}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{19}+\frac{1}{5}a^{14}-\frac{1}{5}a^{13}+\frac{1}{5}a^{12}+\frac{2}{5}a^{11}-\frac{1}{5}a^{10}-\frac{2}{5}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{20}+\frac{1}{5}a^{15}-\frac{1}{5}a^{14}+\frac{1}{5}a^{13}+\frac{2}{5}a^{12}-\frac{1}{5}a^{11}-\frac{2}{5}a^{9}-\frac{2}{5}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{68\!\cdots\!25}a^{21}+\frac{47\!\cdots\!88}{68\!\cdots\!25}a^{20}-\frac{42\!\cdots\!11}{68\!\cdots\!25}a^{19}-\frac{38\!\cdots\!07}{68\!\cdots\!25}a^{18}+\frac{42\!\cdots\!36}{68\!\cdots\!25}a^{17}+\frac{90\!\cdots\!64}{13\!\cdots\!85}a^{16}+\frac{25\!\cdots\!11}{68\!\cdots\!25}a^{15}-\frac{14\!\cdots\!83}{68\!\cdots\!25}a^{14}+\frac{19\!\cdots\!07}{68\!\cdots\!25}a^{13}-\frac{92\!\cdots\!59}{61\!\cdots\!75}a^{12}+\frac{34\!\cdots\!79}{68\!\cdots\!25}a^{11}+\frac{56\!\cdots\!78}{61\!\cdots\!75}a^{10}+\frac{17\!\cdots\!38}{68\!\cdots\!25}a^{9}+\frac{33\!\cdots\!44}{68\!\cdots\!25}a^{8}-\frac{27\!\cdots\!89}{68\!\cdots\!25}a^{7}-\frac{17\!\cdots\!44}{68\!\cdots\!25}a^{6}-\frac{31\!\cdots\!37}{68\!\cdots\!25}a^{5}+\frac{21\!\cdots\!32}{68\!\cdots\!25}a^{4}-\frac{60\!\cdots\!08}{68\!\cdots\!25}a^{3}-\frac{52\!\cdots\!91}{61\!\cdots\!75}a^{2}-\frac{21\!\cdots\!77}{68\!\cdots\!25}a-\frac{20\!\cdots\!33}{68\!\cdots\!25}$
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
$C_{2}$, which has order $2$ (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: | $17$ | 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: |
$\frac{35\!\cdots\!01}{68\!\cdots\!25}a^{21}-\frac{81\!\cdots\!82}{68\!\cdots\!25}a^{20}-\frac{91\!\cdots\!01}{68\!\cdots\!25}a^{19}+\frac{14\!\cdots\!98}{68\!\cdots\!25}a^{18}+\frac{10\!\cdots\!96}{68\!\cdots\!25}a^{17}-\frac{15\!\cdots\!19}{13\!\cdots\!85}a^{16}-\frac{72\!\cdots\!89}{68\!\cdots\!25}a^{15}-\frac{50\!\cdots\!03}{68\!\cdots\!25}a^{14}+\frac{16\!\cdots\!67}{68\!\cdots\!25}a^{13}+\frac{59\!\cdots\!91}{61\!\cdots\!75}a^{12}+\frac{15\!\cdots\!64}{68\!\cdots\!25}a^{11}-\frac{35\!\cdots\!12}{61\!\cdots\!75}a^{10}-\frac{21\!\cdots\!72}{68\!\cdots\!25}a^{9}+\frac{83\!\cdots\!14}{68\!\cdots\!25}a^{8}+\frac{82\!\cdots\!86}{68\!\cdots\!25}a^{7}-\frac{64\!\cdots\!99}{68\!\cdots\!25}a^{6}-\frac{12\!\cdots\!12}{68\!\cdots\!25}a^{5}+\frac{36\!\cdots\!02}{68\!\cdots\!25}a^{4}+\frac{80\!\cdots\!92}{68\!\cdots\!25}a^{3}+\frac{11\!\cdots\!49}{61\!\cdots\!75}a^{2}-\frac{15\!\cdots\!07}{68\!\cdots\!25}a-\frac{32\!\cdots\!43}{68\!\cdots\!25}$, $\frac{15\!\cdots\!33}{68\!\cdots\!25}a^{21}-\frac{60\!\cdots\!26}{68\!\cdots\!25}a^{20}-\frac{33\!\cdots\!33}{68\!\cdots\!25}a^{19}+\frac{12\!\cdots\!99}{68\!\cdots\!25}a^{18}+\frac{34\!\cdots\!53}{68\!\cdots\!25}a^{17}-\frac{20\!\cdots\!08}{13\!\cdots\!85}a^{16}-\frac{24\!\cdots\!17}{68\!\cdots\!25}a^{15}+\frac{45\!\cdots\!41}{68\!\cdots\!25}a^{14}+\frac{59\!\cdots\!31}{68\!\cdots\!25}a^{13}+\frac{15\!\cdots\!08}{61\!\cdots\!75}a^{12}-\frac{35\!\cdots\!08}{68\!\cdots\!25}a^{11}-\frac{15\!\cdots\!71}{61\!\cdots\!75}a^{10}+\frac{17\!\cdots\!69}{68\!\cdots\!25}a^{9}+\frac{42\!\cdots\!17}{68\!\cdots\!25}a^{8}-\frac{23\!\cdots\!37}{68\!\cdots\!25}a^{7}-\frac{66\!\cdots\!07}{68\!\cdots\!25}a^{6}-\frac{80\!\cdots\!96}{68\!\cdots\!25}a^{5}+\frac{73\!\cdots\!61}{68\!\cdots\!25}a^{4}+\frac{10\!\cdots\!66}{68\!\cdots\!25}a^{3}-\frac{37\!\cdots\!93}{61\!\cdots\!75}a^{2}-\frac{11\!\cdots\!61}{68\!\cdots\!25}a+\frac{66\!\cdots\!46}{68\!\cdots\!25}$, $\frac{18\!\cdots\!24}{68\!\cdots\!25}a^{21}-\frac{36\!\cdots\!03}{68\!\cdots\!25}a^{20}-\frac{49\!\cdots\!79}{68\!\cdots\!25}a^{19}+\frac{60\!\cdots\!02}{68\!\cdots\!25}a^{18}+\frac{59\!\cdots\!74}{68\!\cdots\!25}a^{17}-\frac{45\!\cdots\!51}{13\!\cdots\!85}a^{16}-\frac{40\!\cdots\!41}{68\!\cdots\!25}a^{15}-\frac{14\!\cdots\!87}{68\!\cdots\!25}a^{14}+\frac{89\!\cdots\!93}{68\!\cdots\!25}a^{13}+\frac{33\!\cdots\!99}{61\!\cdots\!75}a^{12}+\frac{19\!\cdots\!21}{68\!\cdots\!25}a^{11}-\frac{18\!\cdots\!63}{61\!\cdots\!75}a^{10}-\frac{17\!\cdots\!58}{68\!\cdots\!25}a^{9}+\frac{42\!\cdots\!21}{68\!\cdots\!25}a^{8}+\frac{56\!\cdots\!19}{68\!\cdots\!25}a^{7}-\frac{23\!\cdots\!91}{68\!\cdots\!25}a^{6}-\frac{80\!\cdots\!88}{68\!\cdots\!25}a^{5}-\frac{16\!\cdots\!27}{68\!\cdots\!25}a^{4}+\frac{48\!\cdots\!38}{68\!\cdots\!25}a^{3}+\frac{16\!\cdots\!71}{61\!\cdots\!75}a^{2}-\frac{83\!\cdots\!53}{68\!\cdots\!25}a-\frac{36\!\cdots\!12}{68\!\cdots\!25}$, $\frac{94\!\cdots\!52}{68\!\cdots\!25}a^{21}-\frac{20\!\cdots\!04}{68\!\cdots\!25}a^{20}-\frac{24\!\cdots\!77}{68\!\cdots\!25}a^{19}+\frac{35\!\cdots\!81}{68\!\cdots\!25}a^{18}+\frac{28\!\cdots\!17}{68\!\cdots\!25}a^{17}-\frac{33\!\cdots\!19}{13\!\cdots\!85}a^{16}-\frac{19\!\cdots\!08}{68\!\cdots\!25}a^{15}-\frac{37\!\cdots\!71}{68\!\cdots\!25}a^{14}+\frac{43\!\cdots\!84}{68\!\cdots\!25}a^{13}+\frac{16\!\cdots\!22}{61\!\cdots\!75}a^{12}+\frac{59\!\cdots\!98}{68\!\cdots\!25}a^{11}-\frac{93\!\cdots\!04}{61\!\cdots\!75}a^{10}-\frac{69\!\cdots\!14}{68\!\cdots\!25}a^{9}+\frac{21\!\cdots\!03}{68\!\cdots\!25}a^{8}+\frac{24\!\cdots\!77}{68\!\cdots\!25}a^{7}-\frac{14\!\cdots\!83}{68\!\cdots\!25}a^{6}-\frac{37\!\cdots\!09}{68\!\cdots\!25}a^{5}-\frac{34\!\cdots\!96}{68\!\cdots\!25}a^{4}+\frac{22\!\cdots\!59}{68\!\cdots\!25}a^{3}+\frac{58\!\cdots\!63}{61\!\cdots\!75}a^{2}-\frac{42\!\cdots\!79}{68\!\cdots\!25}a-\frac{15\!\cdots\!51}{68\!\cdots\!25}$, $\frac{58\!\cdots\!23}{68\!\cdots\!25}a^{21}-\frac{19\!\cdots\!56}{68\!\cdots\!25}a^{20}-\frac{13\!\cdots\!13}{68\!\cdots\!25}a^{19}+\frac{37\!\cdots\!34}{68\!\cdots\!25}a^{18}+\frac{14\!\cdots\!38}{68\!\cdots\!25}a^{17}-\frac{11\!\cdots\!85}{27\!\cdots\!57}a^{16}-\frac{10\!\cdots\!42}{68\!\cdots\!25}a^{15}+\frac{10\!\cdots\!66}{68\!\cdots\!25}a^{14}+\frac{23\!\cdots\!91}{68\!\cdots\!25}a^{13}+\frac{75\!\cdots\!83}{61\!\cdots\!75}a^{12}-\frac{71\!\cdots\!33}{68\!\cdots\!25}a^{11}-\frac{57\!\cdots\!21}{61\!\cdots\!75}a^{10}+\frac{27\!\cdots\!84}{68\!\cdots\!25}a^{9}+\frac{14\!\cdots\!72}{68\!\cdots\!25}a^{8}-\frac{13\!\cdots\!62}{68\!\cdots\!25}a^{7}-\frac{18\!\cdots\!12}{68\!\cdots\!25}a^{6}-\frac{83\!\cdots\!71}{68\!\cdots\!25}a^{5}+\frac{15\!\cdots\!36}{68\!\cdots\!25}a^{4}+\frac{70\!\cdots\!61}{68\!\cdots\!25}a^{3}-\frac{71\!\cdots\!33}{61\!\cdots\!75}a^{2}-\frac{79\!\cdots\!51}{68\!\cdots\!25}a+\frac{99\!\cdots\!91}{68\!\cdots\!25}$, $\frac{16\!\cdots\!77}{68\!\cdots\!25}a^{21}-\frac{45\!\cdots\!79}{68\!\cdots\!25}a^{20}-\frac{41\!\cdots\!22}{68\!\cdots\!25}a^{19}+\frac{83\!\cdots\!96}{68\!\cdots\!25}a^{18}+\frac{47\!\cdots\!22}{68\!\cdots\!25}a^{17}-\frac{10\!\cdots\!68}{13\!\cdots\!85}a^{16}-\frac{32\!\cdots\!48}{68\!\cdots\!25}a^{15}+\frac{94\!\cdots\!84}{68\!\cdots\!25}a^{14}+\frac{75\!\cdots\!74}{68\!\cdots\!25}a^{13}+\frac{25\!\cdots\!92}{61\!\cdots\!75}a^{12}-\frac{39\!\cdots\!57}{68\!\cdots\!25}a^{11}-\frac{16\!\cdots\!49}{61\!\cdots\!75}a^{10}-\frac{34\!\cdots\!14}{68\!\cdots\!25}a^{9}+\frac{40\!\cdots\!93}{68\!\cdots\!25}a^{8}+\frac{25\!\cdots\!67}{68\!\cdots\!25}a^{7}-\frac{40\!\cdots\!08}{68\!\cdots\!25}a^{6}-\frac{48\!\cdots\!24}{68\!\cdots\!25}a^{5}+\frac{20\!\cdots\!29}{68\!\cdots\!25}a^{4}+\frac{32\!\cdots\!54}{68\!\cdots\!25}a^{3}-\frac{43\!\cdots\!22}{61\!\cdots\!75}a^{2}-\frac{68\!\cdots\!54}{68\!\cdots\!25}a+\frac{10\!\cdots\!94}{68\!\cdots\!25}$, $\frac{35\!\cdots\!01}{68\!\cdots\!25}a^{21}-\frac{81\!\cdots\!82}{68\!\cdots\!25}a^{20}-\frac{91\!\cdots\!01}{68\!\cdots\!25}a^{19}+\frac{14\!\cdots\!98}{68\!\cdots\!25}a^{18}+\frac{10\!\cdots\!96}{68\!\cdots\!25}a^{17}-\frac{15\!\cdots\!19}{13\!\cdots\!85}a^{16}-\frac{72\!\cdots\!89}{68\!\cdots\!25}a^{15}-\frac{50\!\cdots\!03}{68\!\cdots\!25}a^{14}+\frac{16\!\cdots\!67}{68\!\cdots\!25}a^{13}+\frac{59\!\cdots\!91}{61\!\cdots\!75}a^{12}+\frac{15\!\cdots\!64}{68\!\cdots\!25}a^{11}-\frac{35\!\cdots\!12}{61\!\cdots\!75}a^{10}-\frac{21\!\cdots\!72}{68\!\cdots\!25}a^{9}+\frac{83\!\cdots\!14}{68\!\cdots\!25}a^{8}+\frac{82\!\cdots\!86}{68\!\cdots\!25}a^{7}-\frac{64\!\cdots\!99}{68\!\cdots\!25}a^{6}-\frac{12\!\cdots\!12}{68\!\cdots\!25}a^{5}+\frac{36\!\cdots\!02}{68\!\cdots\!25}a^{4}+\frac{80\!\cdots\!92}{68\!\cdots\!25}a^{3}+\frac{11\!\cdots\!49}{61\!\cdots\!75}a^{2}-\frac{15\!\cdots\!07}{68\!\cdots\!25}a-\frac{39\!\cdots\!68}{68\!\cdots\!25}$, $\frac{46\!\cdots\!02}{68\!\cdots\!25}a^{21}-\frac{18\!\cdots\!84}{68\!\cdots\!25}a^{20}-\frac{97\!\cdots\!22}{68\!\cdots\!25}a^{19}+\frac{37\!\cdots\!61}{68\!\cdots\!25}a^{18}+\frac{99\!\cdots\!02}{68\!\cdots\!25}a^{17}-\frac{12\!\cdots\!68}{27\!\cdots\!57}a^{16}-\frac{69\!\cdots\!53}{68\!\cdots\!25}a^{15}+\frac{13\!\cdots\!89}{68\!\cdots\!25}a^{14}+\frac{16\!\cdots\!04}{68\!\cdots\!25}a^{13}+\frac{47\!\cdots\!02}{61\!\cdots\!75}a^{12}-\frac{10\!\cdots\!27}{68\!\cdots\!25}a^{11}-\frac{44\!\cdots\!24}{61\!\cdots\!75}a^{10}+\frac{53\!\cdots\!66}{68\!\cdots\!25}a^{9}+\frac{12\!\cdots\!83}{68\!\cdots\!25}a^{8}-\frac{71\!\cdots\!43}{68\!\cdots\!25}a^{7}-\frac{18\!\cdots\!03}{68\!\cdots\!25}a^{6}+\frac{31\!\cdots\!26}{68\!\cdots\!25}a^{5}+\frac{20\!\cdots\!29}{68\!\cdots\!25}a^{4}+\frac{22\!\cdots\!54}{68\!\cdots\!25}a^{3}-\frac{10\!\cdots\!77}{61\!\cdots\!75}a^{2}+\frac{21\!\cdots\!41}{68\!\cdots\!25}a+\frac{17\!\cdots\!14}{68\!\cdots\!25}$, $\frac{18\!\cdots\!24}{68\!\cdots\!25}a^{21}-\frac{36\!\cdots\!03}{68\!\cdots\!25}a^{20}-\frac{49\!\cdots\!79}{68\!\cdots\!25}a^{19}+\frac{60\!\cdots\!02}{68\!\cdots\!25}a^{18}+\frac{59\!\cdots\!74}{68\!\cdots\!25}a^{17}-\frac{45\!\cdots\!51}{13\!\cdots\!85}a^{16}-\frac{40\!\cdots\!41}{68\!\cdots\!25}a^{15}-\frac{14\!\cdots\!87}{68\!\cdots\!25}a^{14}+\frac{89\!\cdots\!93}{68\!\cdots\!25}a^{13}+\frac{33\!\cdots\!99}{61\!\cdots\!75}a^{12}+\frac{19\!\cdots\!21}{68\!\cdots\!25}a^{11}-\frac{18\!\cdots\!63}{61\!\cdots\!75}a^{10}-\frac{17\!\cdots\!58}{68\!\cdots\!25}a^{9}+\frac{42\!\cdots\!21}{68\!\cdots\!25}a^{8}+\frac{56\!\cdots\!19}{68\!\cdots\!25}a^{7}-\frac{23\!\cdots\!91}{68\!\cdots\!25}a^{6}-\frac{80\!\cdots\!88}{68\!\cdots\!25}a^{5}-\frac{16\!\cdots\!27}{68\!\cdots\!25}a^{4}+\frac{48\!\cdots\!38}{68\!\cdots\!25}a^{3}+\frac{16\!\cdots\!71}{61\!\cdots\!75}a^{2}-\frac{83\!\cdots\!53}{68\!\cdots\!25}a-\frac{43\!\cdots\!37}{68\!\cdots\!25}$, $\frac{25\!\cdots\!81}{68\!\cdots\!25}a^{21}-\frac{35\!\cdots\!62}{68\!\cdots\!25}a^{20}-\frac{72\!\cdots\!26}{68\!\cdots\!25}a^{19}+\frac{45\!\cdots\!03}{68\!\cdots\!25}a^{18}+\frac{88\!\cdots\!21}{68\!\cdots\!25}a^{17}+\frac{24\!\cdots\!48}{13\!\cdots\!85}a^{16}-\frac{60\!\cdots\!89}{68\!\cdots\!25}a^{15}-\frac{50\!\cdots\!83}{68\!\cdots\!25}a^{14}+\frac{13\!\cdots\!47}{68\!\cdots\!25}a^{13}+\frac{53\!\cdots\!46}{61\!\cdots\!75}a^{12}+\frac{51\!\cdots\!14}{68\!\cdots\!25}a^{11}-\frac{25\!\cdots\!07}{61\!\cdots\!75}a^{10}-\frac{41\!\cdots\!32}{68\!\cdots\!25}a^{9}+\frac{53\!\cdots\!89}{68\!\cdots\!25}a^{8}+\frac{11\!\cdots\!76}{68\!\cdots\!25}a^{7}-\frac{83\!\cdots\!24}{68\!\cdots\!25}a^{6}-\frac{14\!\cdots\!02}{68\!\cdots\!25}a^{5}-\frac{69\!\cdots\!28}{68\!\cdots\!25}a^{4}+\frac{83\!\cdots\!52}{68\!\cdots\!25}a^{3}+\frac{52\!\cdots\!44}{61\!\cdots\!75}a^{2}-\frac{17\!\cdots\!42}{68\!\cdots\!25}a-\frac{10\!\cdots\!23}{68\!\cdots\!25}$, $\frac{16\!\cdots\!53}{13\!\cdots\!85}a^{21}-\frac{14\!\cdots\!81}{27\!\cdots\!57}a^{20}-\frac{34\!\cdots\!19}{13\!\cdots\!85}a^{19}+\frac{14\!\cdots\!91}{13\!\cdots\!85}a^{18}+\frac{33\!\cdots\!63}{13\!\cdots\!85}a^{17}-\frac{12\!\cdots\!17}{13\!\cdots\!85}a^{16}-\frac{23\!\cdots\!03}{13\!\cdots\!85}a^{15}+\frac{61\!\cdots\!58}{13\!\cdots\!85}a^{14}+\frac{57\!\cdots\!12}{13\!\cdots\!85}a^{13}+\frac{26\!\cdots\!13}{24\!\cdots\!87}a^{12}-\frac{45\!\cdots\!69}{13\!\cdots\!85}a^{11}-\frac{31\!\cdots\!87}{24\!\cdots\!87}a^{10}+\frac{52\!\cdots\!75}{27\!\cdots\!57}a^{9}+\frac{45\!\cdots\!53}{13\!\cdots\!85}a^{8}-\frac{47\!\cdots\!69}{13\!\cdots\!85}a^{7}-\frac{78\!\cdots\!89}{13\!\cdots\!85}a^{6}+\frac{27\!\cdots\!98}{13\!\cdots\!85}a^{5}+\frac{10\!\cdots\!91}{13\!\cdots\!85}a^{4}-\frac{33\!\cdots\!71}{13\!\cdots\!85}a^{3}-\frac{62\!\cdots\!34}{12\!\cdots\!35}a^{2}+\frac{68\!\cdots\!60}{27\!\cdots\!57}a+\frac{11\!\cdots\!52}{13\!\cdots\!85}$, $\frac{63\!\cdots\!44}{68\!\cdots\!25}a^{21}-\frac{21\!\cdots\!58}{68\!\cdots\!25}a^{20}-\frac{14\!\cdots\!89}{68\!\cdots\!25}a^{19}+\frac{43\!\cdots\!37}{68\!\cdots\!25}a^{18}+\frac{15\!\cdots\!69}{68\!\cdots\!25}a^{17}-\frac{67\!\cdots\!74}{13\!\cdots\!85}a^{16}-\frac{10\!\cdots\!96}{68\!\cdots\!25}a^{15}+\frac{12\!\cdots\!03}{68\!\cdots\!25}a^{14}+\frac{24\!\cdots\!98}{68\!\cdots\!25}a^{13}+\frac{76\!\cdots\!04}{61\!\cdots\!75}a^{12}-\frac{93\!\cdots\!74}{68\!\cdots\!25}a^{11}-\frac{61\!\cdots\!48}{61\!\cdots\!75}a^{10}+\frac{41\!\cdots\!02}{68\!\cdots\!25}a^{9}+\frac{15\!\cdots\!16}{68\!\cdots\!25}a^{8}-\frac{28\!\cdots\!06}{68\!\cdots\!25}a^{7}-\frac{21\!\cdots\!46}{68\!\cdots\!25}a^{6}-\frac{59\!\cdots\!13}{68\!\cdots\!25}a^{5}+\frac{20\!\cdots\!88}{68\!\cdots\!25}a^{4}+\frac{57\!\cdots\!98}{68\!\cdots\!25}a^{3}-\frac{99\!\cdots\!59}{61\!\cdots\!75}a^{2}-\frac{19\!\cdots\!83}{68\!\cdots\!25}a+\frac{14\!\cdots\!38}{68\!\cdots\!25}$, $\frac{60\!\cdots\!29}{68\!\cdots\!25}a^{21}+\frac{14\!\cdots\!22}{68\!\cdots\!25}a^{20}-\frac{23\!\cdots\!94}{68\!\cdots\!25}a^{19}-\frac{42\!\cdots\!03}{68\!\cdots\!25}a^{18}+\frac{32\!\cdots\!19}{68\!\cdots\!25}a^{17}+\frac{12\!\cdots\!51}{13\!\cdots\!85}a^{16}-\frac{21\!\cdots\!61}{68\!\cdots\!25}a^{15}-\frac{52\!\cdots\!02}{68\!\cdots\!25}a^{14}+\frac{44\!\cdots\!73}{68\!\cdots\!25}a^{13}+\frac{20\!\cdots\!04}{61\!\cdots\!75}a^{12}+\frac{49\!\cdots\!46}{68\!\cdots\!25}a^{11}-\frac{66\!\cdots\!88}{61\!\cdots\!75}a^{10}-\frac{32\!\cdots\!88}{68\!\cdots\!25}a^{9}+\frac{10\!\cdots\!36}{68\!\cdots\!25}a^{8}+\frac{74\!\cdots\!99}{68\!\cdots\!25}a^{7}+\frac{25\!\cdots\!79}{68\!\cdots\!25}a^{6}-\frac{77\!\cdots\!53}{68\!\cdots\!25}a^{5}-\frac{67\!\cdots\!42}{68\!\cdots\!25}a^{4}+\frac{39\!\cdots\!58}{68\!\cdots\!25}a^{3}+\frac{38\!\cdots\!76}{61\!\cdots\!75}a^{2}-\frac{83\!\cdots\!78}{68\!\cdots\!25}a-\frac{65\!\cdots\!57}{68\!\cdots\!25}$, $\frac{15\!\cdots\!48}{13\!\cdots\!85}a^{21}+\frac{19\!\cdots\!09}{27\!\cdots\!57}a^{20}-\frac{14\!\cdots\!43}{27\!\cdots\!57}a^{19}-\frac{51\!\cdots\!98}{27\!\cdots\!57}a^{18}+\frac{11\!\cdots\!97}{13\!\cdots\!85}a^{17}+\frac{32\!\cdots\!59}{13\!\cdots\!85}a^{16}-\frac{74\!\cdots\!71}{13\!\cdots\!85}a^{15}-\frac{24\!\cdots\!14}{13\!\cdots\!85}a^{14}+\frac{15\!\cdots\!34}{13\!\cdots\!85}a^{13}+\frac{77\!\cdots\!54}{12\!\cdots\!35}a^{12}+\frac{43\!\cdots\!76}{27\!\cdots\!57}a^{11}-\frac{36\!\cdots\!42}{24\!\cdots\!87}a^{10}-\frac{15\!\cdots\!24}{13\!\cdots\!85}a^{9}+\frac{17\!\cdots\!23}{13\!\cdots\!85}a^{8}+\frac{35\!\cdots\!99}{13\!\cdots\!85}a^{7}+\frac{14\!\cdots\!47}{13\!\cdots\!85}a^{6}-\frac{37\!\cdots\!06}{13\!\cdots\!85}a^{5}-\frac{34\!\cdots\!62}{13\!\cdots\!85}a^{4}+\frac{20\!\cdots\!14}{13\!\cdots\!85}a^{3}+\frac{23\!\cdots\!37}{12\!\cdots\!35}a^{2}-\frac{63\!\cdots\!96}{13\!\cdots\!85}a-\frac{50\!\cdots\!49}{13\!\cdots\!85}$, $\frac{18\!\cdots\!72}{68\!\cdots\!25}a^{21}-\frac{50\!\cdots\!89}{68\!\cdots\!25}a^{20}-\frac{44\!\cdots\!47}{68\!\cdots\!25}a^{19}+\frac{96\!\cdots\!66}{68\!\cdots\!25}a^{18}+\frac{50\!\cdots\!72}{68\!\cdots\!25}a^{17}-\frac{26\!\cdots\!43}{27\!\cdots\!57}a^{16}-\frac{34\!\cdots\!38}{68\!\cdots\!25}a^{15}+\frac{14\!\cdots\!79}{68\!\cdots\!25}a^{14}+\frac{82\!\cdots\!29}{68\!\cdots\!25}a^{13}+\frac{26\!\cdots\!32}{61\!\cdots\!75}a^{12}-\frac{79\!\cdots\!92}{68\!\cdots\!25}a^{11}-\frac{18\!\cdots\!04}{61\!\cdots\!75}a^{10}-\frac{13\!\cdots\!99}{68\!\cdots\!25}a^{9}+\frac{45\!\cdots\!48}{68\!\cdots\!25}a^{8}+\frac{22\!\cdots\!92}{68\!\cdots\!25}a^{7}-\frac{47\!\cdots\!18}{68\!\cdots\!25}a^{6}-\frac{49\!\cdots\!99}{68\!\cdots\!25}a^{5}+\frac{25\!\cdots\!74}{68\!\cdots\!25}a^{4}+\frac{35\!\cdots\!89}{68\!\cdots\!25}a^{3}-\frac{64\!\cdots\!72}{61\!\cdots\!75}a^{2}-\frac{57\!\cdots\!54}{68\!\cdots\!25}a+\frac{43\!\cdots\!74}{68\!\cdots\!25}$, $\frac{63\!\cdots\!34}{68\!\cdots\!25}a^{21}-\frac{11\!\cdots\!03}{68\!\cdots\!25}a^{20}-\frac{16\!\cdots\!59}{68\!\cdots\!25}a^{19}+\frac{18\!\cdots\!22}{68\!\cdots\!25}a^{18}+\frac{20\!\cdots\!04}{68\!\cdots\!25}a^{17}-\frac{10\!\cdots\!27}{13\!\cdots\!85}a^{16}-\frac{13\!\cdots\!06}{68\!\cdots\!25}a^{15}-\frac{65\!\cdots\!22}{68\!\cdots\!25}a^{14}+\frac{31\!\cdots\!88}{68\!\cdots\!25}a^{13}+\frac{11\!\cdots\!44}{61\!\cdots\!75}a^{12}+\frac{74\!\cdots\!16}{68\!\cdots\!25}a^{11}-\frac{63\!\cdots\!28}{61\!\cdots\!75}a^{10}-\frac{69\!\cdots\!63}{68\!\cdots\!25}a^{9}+\frac{14\!\cdots\!16}{68\!\cdots\!25}a^{8}+\frac{22\!\cdots\!59}{68\!\cdots\!25}a^{7}-\frac{71\!\cdots\!31}{68\!\cdots\!25}a^{6}-\frac{30\!\cdots\!58}{68\!\cdots\!25}a^{5}-\frac{85\!\cdots\!87}{68\!\cdots\!25}a^{4}+\frac{18\!\cdots\!23}{68\!\cdots\!25}a^{3}+\frac{80\!\cdots\!66}{61\!\cdots\!75}a^{2}-\frac{34\!\cdots\!13}{68\!\cdots\!25}a-\frac{18\!\cdots\!12}{68\!\cdots\!25}$, $\frac{65\!\cdots\!24}{13\!\cdots\!85}a^{21}-\frac{11\!\cdots\!64}{13\!\cdots\!85}a^{20}-\frac{34\!\cdots\!72}{27\!\cdots\!57}a^{19}+\frac{15\!\cdots\!69}{13\!\cdots\!85}a^{18}+\frac{20\!\cdots\!88}{13\!\cdots\!85}a^{17}-\frac{11\!\cdots\!67}{27\!\cdots\!57}a^{16}-\frac{26\!\cdots\!52}{27\!\cdots\!57}a^{15}-\frac{10\!\cdots\!22}{13\!\cdots\!85}a^{14}+\frac{25\!\cdots\!93}{13\!\cdots\!85}a^{13}+\frac{13\!\cdots\!01}{12\!\cdots\!35}a^{12}+\frac{97\!\cdots\!52}{13\!\cdots\!85}a^{11}-\frac{11\!\cdots\!02}{24\!\cdots\!87}a^{10}-\frac{87\!\cdots\!74}{13\!\cdots\!85}a^{9}+\frac{10\!\cdots\!23}{13\!\cdots\!85}a^{8}+\frac{28\!\cdots\!63}{13\!\cdots\!85}a^{7}+\frac{16\!\cdots\!87}{13\!\cdots\!85}a^{6}-\frac{33\!\cdots\!87}{13\!\cdots\!85}a^{5}-\frac{21\!\cdots\!66}{13\!\cdots\!85}a^{4}+\frac{15\!\cdots\!13}{13\!\cdots\!85}a^{3}+\frac{17\!\cdots\!26}{12\!\cdots\!35}a^{2}-\frac{27\!\cdots\!52}{13\!\cdots\!85}a-\frac{36\!\cdots\!84}{13\!\cdots\!85}$
| 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: | \( 76552602113.1 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 76552602113.1 \cdot 2}{2\cdot\sqrt{22661033510180079603495293971842498241}}\cr\approx \mathstrut & 0.410638357079 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 - 24*x^20 + 59*x^19 + 273*x^18 - 436*x^17 - 1924*x^16 + 1356*x^15 + 5010*x^14 + 14999*x^13 - 9512*x^12 - 114771*x^11 + 18405*x^10 + 290616*x^9 + 69522*x^8 - 375325*x^7 - 259853*x^6 + 293924*x^5 + 260945*x^4 - 133418*x^3 - 93266*x^2 + 21174*x + 11623)
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 - 3*x^21 - 24*x^20 + 59*x^19 + 273*x^18 - 436*x^17 - 1924*x^16 + 1356*x^15 + 5010*x^14 + 14999*x^13 - 9512*x^12 - 114771*x^11 + 18405*x^10 + 290616*x^9 + 69522*x^8 - 375325*x^7 - 259853*x^6 + 293924*x^5 + 260945*x^4 - 133418*x^3 - 93266*x^2 + 21174*x + 11623, 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 - 3*x^21 - 24*x^20 + 59*x^19 + 273*x^18 - 436*x^17 - 1924*x^16 + 1356*x^15 + 5010*x^14 + 14999*x^13 - 9512*x^12 - 114771*x^11 + 18405*x^10 + 290616*x^9 + 69522*x^8 - 375325*x^7 - 259853*x^6 + 293924*x^5 + 260945*x^4 - 133418*x^3 - 93266*x^2 + 21174*x + 11623);
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 - 3*x^21 - 24*x^20 + 59*x^19 + 273*x^18 - 436*x^17 - 1924*x^16 + 1356*x^15 + 5010*x^14 + 14999*x^13 - 9512*x^12 - 114771*x^11 + 18405*x^10 + 290616*x^9 + 69522*x^8 - 375325*x^7 - 259853*x^6 + 293924*x^5 + 260945*x^4 - 133418*x^3 - 93266*x^2 + 21174*x + 11623);
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
$C_2^{10}.D_{11}$ (as 22T30):
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 22528 |
| The 100 conjugacy class representatives for $C_2^{10}.D_{11}$ are not computed |
| Character table for $C_2^{10}.D_{11}$ is not computed |
Intermediate fields
| 11.11.3670285774226257.1 |
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 siblings: | data not computed |
| Degree 44 siblings: | data not computed |
| Minimal sibling: | 22.14.17471883970840462300304775614373553.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.11.0.1}{11} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(1297\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $4$ | $1$ | $3$ |