Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 2*x^19 + 16*x^18 - 8*x^17 - 68*x^16 - 150*x^15 + 164*x^14 + 240*x^13 - 62*x^12 - 256*x^11 + 30*x^10 + 512*x^9 - 436*x^8 - 114*x^7 + 216*x^6 - 44*x^5 - 44*x^4 - 2*x^3 + 20*x^2 + 4*x - 2)
gp: K = bnfinit(y^21 - 2*y^20 - 2*y^19 + 16*y^18 - 8*y^17 - 68*y^16 - 150*y^15 + 164*y^14 + 240*y^13 - 62*y^12 - 256*y^11 + 30*y^10 + 512*y^9 - 436*y^8 - 114*y^7 + 216*y^6 - 44*y^5 - 44*y^4 - 2*y^3 + 20*y^2 + 4*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 2*x^20 - 2*x^19 + 16*x^18 - 8*x^17 - 68*x^16 - 150*x^15 + 164*x^14 + 240*x^13 - 62*x^12 - 256*x^11 + 30*x^10 + 512*x^9 - 436*x^8 - 114*x^7 + 216*x^6 - 44*x^5 - 44*x^4 - 2*x^3 + 20*x^2 + 4*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 2*x^20 - 2*x^19 + 16*x^18 - 8*x^17 - 68*x^16 - 150*x^15 + 164*x^14 + 240*x^13 - 62*x^12 - 256*x^11 + 30*x^10 + 512*x^9 - 436*x^8 - 114*x^7 + 216*x^6 - 44*x^5 - 44*x^4 - 2*x^3 + 20*x^2 + 4*x - 2)
\( x^{21} - 2 x^{20} - 2 x^{19} + 16 x^{18} - 8 x^{17} - 68 x^{16} - 150 x^{15} + 164 x^{14} + 240 x^{13} + \cdots - 2 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[9, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(101010740538307114619974951370752\)
\(\medspace = 2^{20}\cdot 37^{7}\cdot 317^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(33.42\) | 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: | $2^{20/21}37^{1/2}317^{1/2}\approx 209.56836594522878$ | ||
| Ramified primes: |
\(2\), \(37\), \(317\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23}a^{18}+\frac{9}{23}a^{17}+\frac{8}{23}a^{16}+\frac{11}{23}a^{15}-\frac{6}{23}a^{13}+\frac{5}{23}a^{12}-\frac{6}{23}a^{11}-\frac{11}{23}a^{10}+\frac{4}{23}a^{9}-\frac{8}{23}a^{8}+\frac{9}{23}a^{7}-\frac{8}{23}a^{6}+\frac{3}{23}a^{5}+\frac{11}{23}a^{4}-\frac{5}{23}a^{3}+\frac{11}{23}a^{2}+\frac{7}{23}a-\frac{5}{23}$, $\frac{1}{23}a^{19}-\frac{4}{23}a^{17}+\frac{8}{23}a^{16}-\frac{7}{23}a^{15}-\frac{6}{23}a^{14}-\frac{10}{23}a^{13}-\frac{5}{23}a^{12}-\frac{3}{23}a^{11}+\frac{11}{23}a^{10}+\frac{2}{23}a^{9}-\frac{11}{23}a^{8}+\frac{3}{23}a^{7}+\frac{6}{23}a^{6}+\frac{7}{23}a^{5}+\frac{11}{23}a^{4}+\frac{10}{23}a^{3}+\frac{1}{23}a-\frac{1}{23}$, $\frac{1}{77\!\cdots\!23}a^{20}-\frac{99\!\cdots\!98}{77\!\cdots\!23}a^{19}+\frac{32\!\cdots\!83}{77\!\cdots\!23}a^{18}+\frac{20\!\cdots\!85}{77\!\cdots\!23}a^{17}-\frac{10\!\cdots\!26}{77\!\cdots\!23}a^{16}+\frac{41\!\cdots\!31}{77\!\cdots\!23}a^{15}-\frac{23\!\cdots\!39}{77\!\cdots\!23}a^{14}-\frac{27\!\cdots\!97}{77\!\cdots\!23}a^{13}-\frac{19\!\cdots\!75}{77\!\cdots\!23}a^{12}+\frac{11\!\cdots\!68}{77\!\cdots\!23}a^{11}+\frac{10\!\cdots\!98}{77\!\cdots\!23}a^{10}-\frac{47\!\cdots\!01}{33\!\cdots\!01}a^{9}-\frac{31\!\cdots\!80}{77\!\cdots\!23}a^{8}+\frac{10\!\cdots\!86}{77\!\cdots\!23}a^{7}+\frac{38\!\cdots\!62}{77\!\cdots\!23}a^{6}+\frac{19\!\cdots\!68}{77\!\cdots\!23}a^{5}+\frac{27\!\cdots\!54}{77\!\cdots\!23}a^{4}+\frac{47\!\cdots\!97}{77\!\cdots\!23}a^{3}+\frac{31\!\cdots\!15}{77\!\cdots\!23}a^{2}+\frac{30\!\cdots\!26}{77\!\cdots\!23}a+\frac{26\!\cdots\!76}{77\!\cdots\!23}$
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: | $14$ | 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: |
$\frac{39\!\cdots\!28}{77\!\cdots\!23}a^{20}+\frac{12\!\cdots\!39}{77\!\cdots\!23}a^{19}-\frac{35\!\cdots\!52}{77\!\cdots\!23}a^{18}+\frac{32\!\cdots\!46}{77\!\cdots\!23}a^{17}+\frac{25\!\cdots\!43}{77\!\cdots\!23}a^{16}-\frac{23\!\cdots\!15}{77\!\cdots\!23}a^{15}-\frac{84\!\cdots\!86}{33\!\cdots\!01}a^{14}-\frac{34\!\cdots\!67}{77\!\cdots\!23}a^{13}+\frac{15\!\cdots\!79}{77\!\cdots\!23}a^{12}+\frac{55\!\cdots\!74}{77\!\cdots\!23}a^{11}+\frac{27\!\cdots\!63}{77\!\cdots\!23}a^{10}-\frac{32\!\cdots\!21}{77\!\cdots\!23}a^{9}-\frac{17\!\cdots\!22}{77\!\cdots\!23}a^{8}+\frac{54\!\cdots\!30}{77\!\cdots\!23}a^{7}-\frac{29\!\cdots\!99}{77\!\cdots\!23}a^{6}-\frac{12\!\cdots\!72}{77\!\cdots\!23}a^{5}+\frac{16\!\cdots\!76}{77\!\cdots\!23}a^{4}-\frac{32\!\cdots\!01}{77\!\cdots\!23}a^{3}+\frac{67\!\cdots\!44}{77\!\cdots\!23}a^{2}+\frac{46\!\cdots\!22}{77\!\cdots\!23}a+\frac{10\!\cdots\!15}{77\!\cdots\!23}$, $\frac{12\!\cdots\!67}{77\!\cdots\!23}a^{20}-\frac{16\!\cdots\!56}{77\!\cdots\!23}a^{19}-\frac{45\!\cdots\!13}{77\!\cdots\!23}a^{18}+\frac{18\!\cdots\!22}{77\!\cdots\!23}a^{17}+\frac{46\!\cdots\!26}{77\!\cdots\!23}a^{16}-\frac{95\!\cdots\!37}{77\!\cdots\!23}a^{15}-\frac{10\!\cdots\!26}{33\!\cdots\!01}a^{14}+\frac{10\!\cdots\!79}{77\!\cdots\!23}a^{13}+\frac{53\!\cdots\!53}{77\!\cdots\!23}a^{12}+\frac{19\!\cdots\!04}{77\!\cdots\!23}a^{11}-\frac{47\!\cdots\!21}{77\!\cdots\!23}a^{10}-\frac{40\!\cdots\!12}{77\!\cdots\!23}a^{9}+\frac{61\!\cdots\!34}{77\!\cdots\!23}a^{8}+\frac{56\!\cdots\!94}{77\!\cdots\!23}a^{7}-\frac{45\!\cdots\!22}{77\!\cdots\!23}a^{6}+\frac{93\!\cdots\!46}{77\!\cdots\!23}a^{5}+\frac{80\!\cdots\!48}{77\!\cdots\!23}a^{4}+\frac{28\!\cdots\!35}{77\!\cdots\!23}a^{3}-\frac{13\!\cdots\!58}{77\!\cdots\!23}a^{2}-\frac{75\!\cdots\!30}{77\!\cdots\!23}a+\frac{55\!\cdots\!51}{77\!\cdots\!23}$, $\frac{23\!\cdots\!18}{77\!\cdots\!23}a^{20}-\frac{41\!\cdots\!81}{77\!\cdots\!23}a^{19}-\frac{68\!\cdots\!25}{77\!\cdots\!23}a^{18}+\frac{38\!\cdots\!94}{77\!\cdots\!23}a^{17}-\frac{63\!\cdots\!23}{77\!\cdots\!23}a^{16}-\frac{17\!\cdots\!82}{77\!\cdots\!23}a^{15}-\frac{17\!\cdots\!12}{33\!\cdots\!01}a^{14}+\frac{35\!\cdots\!78}{77\!\cdots\!23}a^{13}+\frac{82\!\cdots\!86}{77\!\cdots\!23}a^{12}-\frac{10\!\cdots\!38}{77\!\cdots\!23}a^{11}-\frac{10\!\cdots\!02}{77\!\cdots\!23}a^{10}-\frac{22\!\cdots\!15}{77\!\cdots\!23}a^{9}+\frac{15\!\cdots\!82}{77\!\cdots\!23}a^{8}-\frac{40\!\cdots\!12}{77\!\cdots\!23}a^{7}-\frac{88\!\cdots\!52}{77\!\cdots\!23}a^{6}+\frac{47\!\cdots\!84}{77\!\cdots\!23}a^{5}+\frac{20\!\cdots\!36}{77\!\cdots\!23}a^{4}-\frac{11\!\cdots\!30}{77\!\cdots\!23}a^{3}-\frac{73\!\cdots\!30}{77\!\cdots\!23}a^{2}-\frac{17\!\cdots\!00}{77\!\cdots\!23}a+\frac{22\!\cdots\!73}{77\!\cdots\!23}$, $\frac{16\!\cdots\!51}{33\!\cdots\!01}a^{20}-\frac{17\!\cdots\!74}{33\!\cdots\!01}a^{19}-\frac{47\!\cdots\!10}{33\!\cdots\!01}a^{18}+\frac{14\!\cdots\!43}{33\!\cdots\!01}a^{17}-\frac{59\!\cdots\!44}{33\!\cdots\!01}a^{16}-\frac{38\!\cdots\!60}{33\!\cdots\!01}a^{15}-\frac{34\!\cdots\!18}{33\!\cdots\!01}a^{14}-\frac{38\!\cdots\!38}{33\!\cdots\!01}a^{13}-\frac{50\!\cdots\!58}{33\!\cdots\!01}a^{12}+\frac{65\!\cdots\!24}{33\!\cdots\!01}a^{11}+\frac{14\!\cdots\!79}{33\!\cdots\!01}a^{10}+\frac{52\!\cdots\!96}{33\!\cdots\!01}a^{9}-\frac{69\!\cdots\!52}{33\!\cdots\!01}a^{8}-\frac{12\!\cdots\!95}{33\!\cdots\!01}a^{7}+\frac{15\!\cdots\!46}{33\!\cdots\!01}a^{6}-\frac{24\!\cdots\!80}{33\!\cdots\!01}a^{5}-\frac{81\!\cdots\!36}{33\!\cdots\!01}a^{4}+\frac{84\!\cdots\!00}{33\!\cdots\!01}a^{3}+\frac{14\!\cdots\!72}{33\!\cdots\!01}a^{2}+\frac{94\!\cdots\!32}{33\!\cdots\!01}a-\frac{40\!\cdots\!15}{33\!\cdots\!01}$, $\frac{11\!\cdots\!84}{77\!\cdots\!23}a^{20}+\frac{48\!\cdots\!50}{77\!\cdots\!23}a^{19}-\frac{69\!\cdots\!75}{77\!\cdots\!23}a^{18}+\frac{11\!\cdots\!20}{77\!\cdots\!23}a^{17}+\frac{32\!\cdots\!30}{77\!\cdots\!23}a^{16}-\frac{86\!\cdots\!03}{77\!\cdots\!23}a^{15}-\frac{15\!\cdots\!28}{33\!\cdots\!01}a^{14}-\frac{28\!\cdots\!14}{77\!\cdots\!23}a^{13}+\frac{56\!\cdots\!79}{77\!\cdots\!23}a^{12}+\frac{64\!\cdots\!64}{77\!\cdots\!23}a^{11}-\frac{17\!\cdots\!64}{77\!\cdots\!23}a^{10}-\frac{62\!\cdots\!18}{77\!\cdots\!23}a^{9}+\frac{34\!\cdots\!78}{77\!\cdots\!23}a^{8}+\frac{75\!\cdots\!98}{77\!\cdots\!23}a^{7}-\frac{84\!\cdots\!92}{77\!\cdots\!23}a^{6}-\frac{35\!\cdots\!04}{77\!\cdots\!23}a^{5}+\frac{21\!\cdots\!12}{77\!\cdots\!23}a^{4}-\frac{55\!\cdots\!56}{77\!\cdots\!23}a^{3}-\frac{44\!\cdots\!44}{77\!\cdots\!23}a^{2}+\frac{10\!\cdots\!42}{77\!\cdots\!23}a+\frac{23\!\cdots\!79}{77\!\cdots\!23}$, $\frac{65\!\cdots\!23}{77\!\cdots\!23}a^{20}-\frac{19\!\cdots\!11}{77\!\cdots\!23}a^{19}-\frac{53\!\cdots\!22}{77\!\cdots\!23}a^{18}+\frac{12\!\cdots\!35}{77\!\cdots\!23}a^{17}-\frac{13\!\cdots\!80}{77\!\cdots\!23}a^{16}-\frac{46\!\cdots\!66}{77\!\cdots\!23}a^{15}-\frac{55\!\cdots\!31}{77\!\cdots\!23}a^{14}+\frac{23\!\cdots\!60}{77\!\cdots\!23}a^{13}+\frac{13\!\cdots\!62}{77\!\cdots\!23}a^{12}-\frac{25\!\cdots\!91}{77\!\cdots\!23}a^{11}-\frac{32\!\cdots\!77}{77\!\cdots\!23}a^{10}+\frac{14\!\cdots\!28}{77\!\cdots\!23}a^{9}+\frac{52\!\cdots\!84}{77\!\cdots\!23}a^{8}-\frac{46\!\cdots\!09}{77\!\cdots\!23}a^{7}-\frac{63\!\cdots\!28}{77\!\cdots\!23}a^{6}+\frac{23\!\cdots\!00}{77\!\cdots\!23}a^{5}+\frac{84\!\cdots\!09}{77\!\cdots\!23}a^{4}-\frac{49\!\cdots\!76}{77\!\cdots\!23}a^{3}-\frac{18\!\cdots\!00}{77\!\cdots\!23}a^{2}+\frac{17\!\cdots\!63}{77\!\cdots\!23}a-\frac{62\!\cdots\!11}{77\!\cdots\!23}$, $\frac{78\!\cdots\!94}{77\!\cdots\!23}a^{20}-\frac{47\!\cdots\!17}{77\!\cdots\!23}a^{19}-\frac{53\!\cdots\!31}{77\!\cdots\!23}a^{18}+\frac{11\!\cdots\!40}{77\!\cdots\!23}a^{17}+\frac{16\!\cdots\!39}{77\!\cdots\!23}a^{16}-\frac{82\!\cdots\!11}{77\!\cdots\!23}a^{15}-\frac{20\!\cdots\!11}{77\!\cdots\!23}a^{14}+\frac{67\!\cdots\!84}{77\!\cdots\!23}a^{13}+\frac{71\!\cdots\!53}{77\!\cdots\!23}a^{12}+\frac{25\!\cdots\!57}{77\!\cdots\!23}a^{11}-\frac{77\!\cdots\!84}{77\!\cdots\!23}a^{10}-\frac{76\!\cdots\!54}{77\!\cdots\!23}a^{9}+\frac{60\!\cdots\!19}{77\!\cdots\!23}a^{8}+\frac{72\!\cdots\!02}{77\!\cdots\!23}a^{7}-\frac{90\!\cdots\!84}{77\!\cdots\!23}a^{6}+\frac{22\!\cdots\!47}{77\!\cdots\!23}a^{5}+\frac{35\!\cdots\!16}{77\!\cdots\!23}a^{4}+\frac{99\!\cdots\!62}{33\!\cdots\!01}a^{3}-\frac{39\!\cdots\!57}{77\!\cdots\!23}a^{2}-\frac{18\!\cdots\!76}{33\!\cdots\!01}a-\frac{13\!\cdots\!61}{77\!\cdots\!23}$, $\frac{37\!\cdots\!87}{33\!\cdots\!01}a^{20}-\frac{10\!\cdots\!48}{77\!\cdots\!23}a^{19}-\frac{30\!\cdots\!94}{77\!\cdots\!23}a^{18}+\frac{12\!\cdots\!81}{77\!\cdots\!23}a^{17}+\frac{44\!\cdots\!93}{77\!\cdots\!23}a^{16}-\frac{62\!\cdots\!04}{77\!\cdots\!23}a^{15}-\frac{18\!\cdots\!98}{77\!\cdots\!23}a^{14}+\frac{29\!\cdots\!23}{77\!\cdots\!23}a^{13}+\frac{33\!\cdots\!23}{77\!\cdots\!23}a^{12}+\frac{19\!\cdots\!90}{77\!\cdots\!23}a^{11}-\frac{24\!\cdots\!08}{77\!\cdots\!23}a^{10}-\frac{28\!\cdots\!00}{77\!\cdots\!23}a^{9}+\frac{33\!\cdots\!98}{77\!\cdots\!23}a^{8}+\frac{49\!\cdots\!47}{77\!\cdots\!23}a^{7}-\frac{22\!\cdots\!53}{77\!\cdots\!23}a^{6}+\frac{12\!\cdots\!73}{77\!\cdots\!23}a^{5}+\frac{49\!\cdots\!69}{77\!\cdots\!23}a^{4}+\frac{72\!\cdots\!21}{33\!\cdots\!01}a^{3}-\frac{45\!\cdots\!07}{77\!\cdots\!23}a^{2}-\frac{24\!\cdots\!74}{77\!\cdots\!23}a+\frac{13\!\cdots\!71}{77\!\cdots\!23}$, $\frac{90\!\cdots\!49}{33\!\cdots\!01}a^{20}-\frac{13\!\cdots\!66}{33\!\cdots\!01}a^{19}-\frac{13\!\cdots\!36}{33\!\cdots\!01}a^{18}+\frac{30\!\cdots\!96}{33\!\cdots\!01}a^{17}+\frac{33\!\cdots\!66}{33\!\cdots\!01}a^{16}-\frac{22\!\cdots\!00}{33\!\cdots\!01}a^{15}-\frac{17\!\cdots\!19}{33\!\cdots\!01}a^{14}+\frac{85\!\cdots\!42}{33\!\cdots\!01}a^{13}+\frac{23\!\cdots\!88}{33\!\cdots\!01}a^{12}-\frac{96\!\cdots\!00}{33\!\cdots\!01}a^{11}-\frac{42\!\cdots\!70}{33\!\cdots\!01}a^{10}-\frac{17\!\cdots\!08}{33\!\cdots\!01}a^{9}+\frac{38\!\cdots\!80}{33\!\cdots\!01}a^{8}+\frac{28\!\cdots\!46}{33\!\cdots\!01}a^{7}-\frac{47\!\cdots\!44}{33\!\cdots\!01}a^{6}+\frac{96\!\cdots\!52}{33\!\cdots\!01}a^{5}+\frac{21\!\cdots\!84}{33\!\cdots\!01}a^{4}-\frac{33\!\cdots\!56}{33\!\cdots\!01}a^{3}-\frac{47\!\cdots\!00}{33\!\cdots\!01}a^{2}-\frac{24\!\cdots\!16}{33\!\cdots\!01}a+\frac{22\!\cdots\!03}{33\!\cdots\!01}$, $\frac{62\!\cdots\!61}{77\!\cdots\!23}a^{20}-\frac{59\!\cdots\!52}{77\!\cdots\!23}a^{19}+\frac{26\!\cdots\!00}{77\!\cdots\!23}a^{18}+\frac{40\!\cdots\!28}{77\!\cdots\!23}a^{17}-\frac{27\!\cdots\!17}{77\!\cdots\!23}a^{16}-\frac{51\!\cdots\!17}{77\!\cdots\!23}a^{15}-\frac{14\!\cdots\!30}{77\!\cdots\!23}a^{14}-\frac{19\!\cdots\!33}{77\!\cdots\!23}a^{13}-\frac{24\!\cdots\!87}{77\!\cdots\!23}a^{12}+\frac{45\!\cdots\!13}{77\!\cdots\!23}a^{11}+\frac{81\!\cdots\!49}{77\!\cdots\!23}a^{10}+\frac{11\!\cdots\!05}{77\!\cdots\!23}a^{9}-\frac{64\!\cdots\!52}{77\!\cdots\!23}a^{8}-\frac{58\!\cdots\!80}{77\!\cdots\!23}a^{7}+\frac{98\!\cdots\!37}{77\!\cdots\!23}a^{6}-\frac{25\!\cdots\!95}{77\!\cdots\!23}a^{5}-\frac{49\!\cdots\!50}{77\!\cdots\!23}a^{4}+\frac{94\!\cdots\!17}{77\!\cdots\!23}a^{3}+\frac{12\!\cdots\!92}{77\!\cdots\!23}a^{2}+\frac{39\!\cdots\!46}{77\!\cdots\!23}a-\frac{13\!\cdots\!31}{77\!\cdots\!23}$, $\frac{39\!\cdots\!28}{77\!\cdots\!23}a^{20}+\frac{12\!\cdots\!39}{77\!\cdots\!23}a^{19}-\frac{35\!\cdots\!52}{77\!\cdots\!23}a^{18}+\frac{32\!\cdots\!46}{77\!\cdots\!23}a^{17}+\frac{25\!\cdots\!43}{77\!\cdots\!23}a^{16}-\frac{23\!\cdots\!15}{77\!\cdots\!23}a^{15}-\frac{84\!\cdots\!86}{33\!\cdots\!01}a^{14}-\frac{34\!\cdots\!67}{77\!\cdots\!23}a^{13}+\frac{15\!\cdots\!79}{77\!\cdots\!23}a^{12}+\frac{55\!\cdots\!74}{77\!\cdots\!23}a^{11}+\frac{27\!\cdots\!63}{77\!\cdots\!23}a^{10}-\frac{32\!\cdots\!21}{77\!\cdots\!23}a^{9}-\frac{17\!\cdots\!22}{77\!\cdots\!23}a^{8}+\frac{54\!\cdots\!30}{77\!\cdots\!23}a^{7}-\frac{29\!\cdots\!99}{77\!\cdots\!23}a^{6}-\frac{12\!\cdots\!72}{77\!\cdots\!23}a^{5}+\frac{16\!\cdots\!76}{77\!\cdots\!23}a^{4}-\frac{32\!\cdots\!01}{77\!\cdots\!23}a^{3}+\frac{67\!\cdots\!44}{77\!\cdots\!23}a^{2}-\frac{31\!\cdots\!01}{77\!\cdots\!23}a+\frac{10\!\cdots\!15}{77\!\cdots\!23}$, $\frac{64\!\cdots\!59}{77\!\cdots\!23}a^{20}-\frac{12\!\cdots\!26}{77\!\cdots\!23}a^{19}-\frac{16\!\cdots\!81}{77\!\cdots\!23}a^{18}+\frac{10\!\cdots\!97}{77\!\cdots\!23}a^{17}-\frac{30\!\cdots\!64}{77\!\cdots\!23}a^{16}-\frac{46\!\cdots\!89}{77\!\cdots\!23}a^{15}-\frac{10\!\cdots\!25}{77\!\cdots\!23}a^{14}+\frac{10\!\cdots\!48}{77\!\cdots\!23}a^{13}+\frac{21\!\cdots\!57}{77\!\cdots\!23}a^{12}-\frac{19\!\cdots\!84}{77\!\cdots\!23}a^{11}-\frac{24\!\cdots\!12}{77\!\cdots\!23}a^{10}-\frac{66\!\cdots\!51}{77\!\cdots\!23}a^{9}+\frac{37\!\cdots\!27}{77\!\cdots\!23}a^{8}-\frac{15\!\cdots\!94}{77\!\cdots\!23}a^{7}-\frac{15\!\cdots\!92}{77\!\cdots\!23}a^{6}+\frac{10\!\cdots\!15}{77\!\cdots\!23}a^{5}+\frac{30\!\cdots\!92}{77\!\cdots\!23}a^{4}-\frac{15\!\cdots\!60}{77\!\cdots\!23}a^{3}-\frac{14\!\cdots\!19}{77\!\cdots\!23}a^{2}+\frac{16\!\cdots\!84}{77\!\cdots\!23}a+\frac{26\!\cdots\!17}{77\!\cdots\!23}$, $\frac{12\!\cdots\!73}{77\!\cdots\!23}a^{20}+\frac{86\!\cdots\!57}{77\!\cdots\!23}a^{19}+\frac{37\!\cdots\!10}{77\!\cdots\!23}a^{18}-\frac{34\!\cdots\!34}{77\!\cdots\!23}a^{17}+\frac{71\!\cdots\!69}{77\!\cdots\!23}a^{16}+\frac{21\!\cdots\!64}{77\!\cdots\!23}a^{15}-\frac{35\!\cdots\!26}{33\!\cdots\!01}a^{14}-\frac{34\!\cdots\!08}{77\!\cdots\!23}a^{13}-\frac{39\!\cdots\!29}{77\!\cdots\!23}a^{12}+\frac{28\!\cdots\!45}{77\!\cdots\!23}a^{11}+\frac{87\!\cdots\!84}{77\!\cdots\!23}a^{10}+\frac{39\!\cdots\!55}{77\!\cdots\!23}a^{9}-\frac{47\!\cdots\!16}{77\!\cdots\!23}a^{8}-\frac{11\!\cdots\!43}{33\!\cdots\!01}a^{7}+\frac{51\!\cdots\!00}{77\!\cdots\!23}a^{6}-\frac{34\!\cdots\!67}{77\!\cdots\!23}a^{5}-\frac{36\!\cdots\!08}{77\!\cdots\!23}a^{4}-\frac{71\!\cdots\!50}{77\!\cdots\!23}a^{3}+\frac{25\!\cdots\!21}{77\!\cdots\!23}a^{2}+\frac{35\!\cdots\!13}{77\!\cdots\!23}a-\frac{27\!\cdots\!21}{77\!\cdots\!23}$, $\frac{35\!\cdots\!85}{77\!\cdots\!23}a^{20}-\frac{79\!\cdots\!94}{77\!\cdots\!23}a^{19}-\frac{64\!\cdots\!03}{77\!\cdots\!23}a^{18}+\frac{59\!\cdots\!79}{77\!\cdots\!23}a^{17}-\frac{37\!\cdots\!47}{77\!\cdots\!23}a^{16}-\frac{25\!\cdots\!01}{77\!\cdots\!23}a^{15}-\frac{47\!\cdots\!89}{77\!\cdots\!23}a^{14}+\frac{34\!\cdots\!96}{33\!\cdots\!01}a^{13}+\frac{91\!\cdots\!53}{77\!\cdots\!23}a^{12}-\frac{58\!\cdots\!45}{77\!\cdots\!23}a^{11}-\frac{14\!\cdots\!78}{77\!\cdots\!23}a^{10}+\frac{74\!\cdots\!01}{33\!\cdots\!01}a^{9}+\frac{24\!\cdots\!14}{77\!\cdots\!23}a^{8}-\frac{14\!\cdots\!70}{77\!\cdots\!23}a^{7}-\frac{62\!\cdots\!86}{77\!\cdots\!23}a^{6}+\frac{71\!\cdots\!15}{77\!\cdots\!23}a^{5}+\frac{12\!\cdots\!02}{77\!\cdots\!23}a^{4}-\frac{15\!\cdots\!90}{77\!\cdots\!23}a^{3}-\frac{15\!\cdots\!74}{77\!\cdots\!23}a^{2}+\frac{31\!\cdots\!70}{77\!\cdots\!23}a+\frac{12\!\cdots\!15}{77\!\cdots\!23}$
| 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: | \( 241328139.391 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{6}\cdot 241328139.391 \cdot 1}{2\cdot\sqrt{101010740538307114619974951370752}}\cr\approx \mathstrut & 0.378219017810 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 2*x^19 + 16*x^18 - 8*x^17 - 68*x^16 - 150*x^15 + 164*x^14 + 240*x^13 - 62*x^12 - 256*x^11 + 30*x^10 + 512*x^9 - 436*x^8 - 114*x^7 + 216*x^6 - 44*x^5 - 44*x^4 - 2*x^3 + 20*x^2 + 4*x - 2)
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^21 - 2*x^20 - 2*x^19 + 16*x^18 - 8*x^17 - 68*x^16 - 150*x^15 + 164*x^14 + 240*x^13 - 62*x^12 - 256*x^11 + 30*x^10 + 512*x^9 - 436*x^8 - 114*x^7 + 216*x^6 - 44*x^5 - 44*x^4 - 2*x^3 + 20*x^2 + 4*x - 2, 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^21 - 2*x^20 - 2*x^19 + 16*x^18 - 8*x^17 - 68*x^16 - 150*x^15 + 164*x^14 + 240*x^13 - 62*x^12 - 256*x^11 + 30*x^10 + 512*x^9 - 436*x^8 - 114*x^7 + 216*x^6 - 44*x^5 - 44*x^4 - 2*x^3 + 20*x^2 + 4*x - 2);
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^21 - 2*x^20 - 2*x^19 + 16*x^18 - 8*x^17 - 68*x^16 - 150*x^15 + 164*x^14 + 240*x^13 - 62*x^12 - 256*x^11 + 30*x^10 + 512*x^9 - 436*x^8 - 114*x^7 + 216*x^6 - 44*x^5 - 44*x^4 - 2*x^3 + 20*x^2 + 4*x - 2);
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
$S_3\times \GL(3,2)$ (as 21T27):
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 1008 |
| The 18 conjugacy class representatives for $S_3\times \GL(3,2)$ |
| Character table for $S_3\times \GL(3,2)$ |
Intermediate fields
| 3.3.148.1, 7.3.6431296.1 |
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 24 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
| Arithmetically equvalently siblings: | data not computed |
| Minimal sibling: | 21.9.101010740538307114619974951370752.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $21$ | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.7.0.1}{7} }$ | $21$ | ${\href{/padicField/11.3.0.1}{3} }^{7}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.2.0.1}{2} }^{9}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.7.0.1}{7} }$ | R | ${\href{/padicField/41.3.0.1}{3} }^{7}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $21$ | ${\href{/padicField/53.3.0.1}{3} }^{7}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.21.20.1 | $x^{21} + 2$ | $21$ | $1$ | $20$ | 21T11 | $[\ ]_{21}^{6}$ |
|
\(37\)
| 37.7.0.1 | $x^{7} + 7 x + 35$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 37.14.7.1 | $x^{14} + 6475 x^{13} + 17968384 x^{12} + 27702296825 x^{11} + 25626619053749 x^{10} + 14225028541792250 x^{9} + 4387630775227591619 x^{8} + 580470771655705630995 x^{7} + 162342338683546885633 x^{6} + 19474064268228311560 x^{5} + 1298245444129580597 x^{4} + 152347376246974125 x^{3} + 31188393261851492275 x^{2} + 4209354931095980413020 x + 20435693672323026267417$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ | |
|
\(317\)
| $\Q_{317}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{317}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{317}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |