Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^18 - 18*x^17 + 12*x^16 + 93*x^15 + 252*x^14 + 756*x^13 + 424*x^12 - 846*x^11 - 3942*x^10 - 12420*x^9 - 20592*x^8 - 30729*x^7 - 35272*x^6 - 30168*x^5 - 18600*x^4 - 8512*x^3 - 2880*x^2 - 768*x - 128)
gp: K = bnfinit(y^21 - 2*y^18 - 18*y^17 + 12*y^16 + 93*y^15 + 252*y^14 + 756*y^13 + 424*y^12 - 846*y^11 - 3942*y^10 - 12420*y^9 - 20592*y^8 - 30729*y^7 - 35272*y^6 - 30168*y^5 - 18600*y^4 - 8512*y^3 - 2880*y^2 - 768*y - 128, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 2*x^18 - 18*x^17 + 12*x^16 + 93*x^15 + 252*x^14 + 756*x^13 + 424*x^12 - 846*x^11 - 3942*x^10 - 12420*x^9 - 20592*x^8 - 30729*x^7 - 35272*x^6 - 30168*x^5 - 18600*x^4 - 8512*x^3 - 2880*x^2 - 768*x - 128);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 2*x^18 - 18*x^17 + 12*x^16 + 93*x^15 + 252*x^14 + 756*x^13 + 424*x^12 - 846*x^11 - 3942*x^10 - 12420*x^9 - 20592*x^8 - 30729*x^7 - 35272*x^6 - 30168*x^5 - 18600*x^4 - 8512*x^3 - 2880*x^2 - 768*x - 128)
\( x^{21} - 2 x^{18} - 18 x^{17} + 12 x^{16} + 93 x^{15} + 252 x^{14} + 756 x^{13} + 424 x^{12} + \cdots - 128 \)
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: | $[3, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-86433900540276528721822666309632\)
\(\medspace = -\,2^{14}\cdot 3^{21}\cdot 11\cdot 71^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(33.17\) | 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: |
\(2\), \(3\), \(11\), \(71\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-2343}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{4}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}+\frac{1}{4}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{17}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{2}a^{12}-\frac{3}{8}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{16}a^{18}-\frac{1}{8}a^{15}-\frac{1}{8}a^{14}-\frac{1}{4}a^{13}-\frac{3}{16}a^{12}-\frac{1}{4}a^{11}+\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{8}a^{8}-\frac{3}{8}a^{7}-\frac{1}{4}a^{6}+\frac{7}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{19}-\frac{1}{16}a^{16}-\frac{1}{16}a^{15}+\frac{3}{8}a^{14}-\frac{3}{32}a^{13}-\frac{1}{8}a^{12}-\frac{3}{8}a^{11}+\frac{1}{4}a^{10}+\frac{1}{16}a^{9}-\frac{3}{16}a^{8}-\frac{1}{8}a^{7}-\frac{1}{2}a^{6}-\frac{9}{32}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{79\!\cdots\!96}a^{20}+\frac{25\!\cdots\!43}{19\!\cdots\!24}a^{19}-\frac{20\!\cdots\!67}{98\!\cdots\!12}a^{18}+\frac{17\!\cdots\!19}{39\!\cdots\!48}a^{17}+\frac{11\!\cdots\!67}{39\!\cdots\!48}a^{16}+\frac{48\!\cdots\!29}{19\!\cdots\!24}a^{15}-\frac{17\!\cdots\!59}{43\!\cdots\!16}a^{14}+\frac{22\!\cdots\!01}{98\!\cdots\!12}a^{13}-\frac{10\!\cdots\!17}{19\!\cdots\!24}a^{12}+\frac{41\!\cdots\!96}{12\!\cdots\!39}a^{11}-\frac{11\!\cdots\!19}{30\!\cdots\!96}a^{10}+\frac{88\!\cdots\!17}{39\!\cdots\!48}a^{9}-\frac{97\!\cdots\!27}{19\!\cdots\!24}a^{8}-\frac{50\!\cdots\!23}{12\!\cdots\!39}a^{7}-\frac{32\!\cdots\!97}{79\!\cdots\!96}a^{6}+\frac{69\!\cdots\!43}{19\!\cdots\!24}a^{5}+\frac{38\!\cdots\!61}{95\!\cdots\!03}a^{4}+\frac{10\!\cdots\!89}{49\!\cdots\!56}a^{3}-\frac{56\!\cdots\!99}{12\!\cdots\!39}a^{2}+\frac{10\!\cdots\!34}{12\!\cdots\!39}a+\frac{36\!\cdots\!09}{12\!\cdots\!39}$
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: | $11$ | 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: |
$\frac{15\!\cdots\!61}{39\!\cdots\!48}a^{20}+\frac{89\!\cdots\!61}{39\!\cdots\!48}a^{19}-\frac{29\!\cdots\!69}{98\!\cdots\!12}a^{18}+\frac{17\!\cdots\!95}{19\!\cdots\!24}a^{17}-\frac{12\!\cdots\!71}{98\!\cdots\!12}a^{16}-\frac{58\!\cdots\!75}{19\!\cdots\!24}a^{15}+\frac{25\!\cdots\!77}{21\!\cdots\!08}a^{14}+\frac{97\!\cdots\!77}{39\!\cdots\!48}a^{13}+\frac{62\!\cdots\!75}{98\!\cdots\!12}a^{12}+\frac{12\!\cdots\!17}{98\!\cdots\!12}a^{11}-\frac{19\!\cdots\!59}{15\!\cdots\!48}a^{10}-\frac{36\!\cdots\!21}{98\!\cdots\!12}a^{9}-\frac{22\!\cdots\!65}{19\!\cdots\!24}a^{8}-\frac{25\!\cdots\!35}{98\!\cdots\!12}a^{7}-\frac{10\!\cdots\!53}{39\!\cdots\!48}a^{6}-\frac{16\!\cdots\!93}{39\!\cdots\!48}a^{5}-\frac{20\!\cdots\!19}{76\!\cdots\!24}a^{4}-\frac{53\!\cdots\!77}{49\!\cdots\!56}a^{3}+\frac{29\!\cdots\!41}{12\!\cdots\!39}a^{2}+\frac{50\!\cdots\!57}{24\!\cdots\!78}a+\frac{14\!\cdots\!14}{12\!\cdots\!39}$, $\frac{58\!\cdots\!63}{79\!\cdots\!96}a^{20}-\frac{17\!\cdots\!91}{39\!\cdots\!48}a^{19}+\frac{36\!\cdots\!73}{98\!\cdots\!12}a^{18}-\frac{68\!\cdots\!27}{39\!\cdots\!48}a^{17}-\frac{48\!\cdots\!77}{39\!\cdots\!48}a^{16}+\frac{78\!\cdots\!29}{49\!\cdots\!56}a^{15}+\frac{24\!\cdots\!03}{43\!\cdots\!16}a^{14}+\frac{60\!\cdots\!03}{39\!\cdots\!48}a^{13}+\frac{94\!\cdots\!15}{19\!\cdots\!24}a^{12}+\frac{55\!\cdots\!95}{98\!\cdots\!12}a^{11}-\frac{17\!\cdots\!53}{30\!\cdots\!96}a^{10}-\frac{10\!\cdots\!31}{39\!\cdots\!48}a^{9}-\frac{76\!\cdots\!25}{98\!\cdots\!12}a^{8}-\frac{10\!\cdots\!11}{98\!\cdots\!12}a^{7}-\frac{13\!\cdots\!39}{79\!\cdots\!96}a^{6}-\frac{69\!\cdots\!49}{39\!\cdots\!48}a^{5}-\frac{13\!\cdots\!92}{95\!\cdots\!03}a^{4}-\frac{19\!\cdots\!01}{24\!\cdots\!78}a^{3}-\frac{42\!\cdots\!60}{12\!\cdots\!39}a^{2}-\frac{26\!\cdots\!07}{24\!\cdots\!78}a-\frac{28\!\cdots\!46}{12\!\cdots\!39}$, $\frac{40\!\cdots\!95}{79\!\cdots\!96}a^{20}-\frac{43\!\cdots\!75}{39\!\cdots\!48}a^{19}-\frac{41\!\cdots\!19}{19\!\cdots\!24}a^{18}-\frac{36\!\cdots\!51}{39\!\cdots\!48}a^{17}-\frac{35\!\cdots\!77}{39\!\cdots\!48}a^{16}+\frac{81\!\cdots\!73}{98\!\cdots\!12}a^{15}+\frac{20\!\cdots\!83}{43\!\cdots\!16}a^{14}+\frac{46\!\cdots\!71}{39\!\cdots\!48}a^{13}+\frac{35\!\cdots\!87}{98\!\cdots\!12}a^{12}+\frac{67\!\cdots\!39}{49\!\cdots\!56}a^{11}-\frac{14\!\cdots\!73}{30\!\cdots\!96}a^{10}-\frac{74\!\cdots\!35}{39\!\cdots\!48}a^{9}-\frac{73\!\cdots\!99}{12\!\cdots\!39}a^{8}-\frac{45\!\cdots\!11}{49\!\cdots\!56}a^{7}-\frac{10\!\cdots\!31}{79\!\cdots\!96}a^{6}-\frac{59\!\cdots\!41}{39\!\cdots\!48}a^{5}-\frac{18\!\cdots\!23}{15\!\cdots\!48}a^{4}-\frac{69\!\cdots\!45}{98\!\cdots\!12}a^{3}-\frac{15\!\cdots\!37}{49\!\cdots\!56}a^{2}-\frac{19\!\cdots\!75}{24\!\cdots\!78}a-\frac{17\!\cdots\!37}{12\!\cdots\!39}$, $\frac{93\!\cdots\!57}{79\!\cdots\!96}a^{20}+\frac{35\!\cdots\!23}{98\!\cdots\!12}a^{19}-\frac{10\!\cdots\!59}{19\!\cdots\!24}a^{18}-\frac{81\!\cdots\!09}{39\!\cdots\!48}a^{17}-\frac{86\!\cdots\!53}{39\!\cdots\!48}a^{16}+\frac{16\!\cdots\!65}{19\!\cdots\!24}a^{15}+\frac{54\!\cdots\!73}{43\!\cdots\!16}a^{14}+\frac{63\!\cdots\!01}{19\!\cdots\!24}a^{13}+\frac{92\!\cdots\!19}{98\!\cdots\!12}a^{12}+\frac{66\!\cdots\!45}{98\!\cdots\!12}a^{11}-\frac{35\!\cdots\!39}{30\!\cdots\!96}a^{10}-\frac{19\!\cdots\!87}{39\!\cdots\!48}a^{9}-\frac{30\!\cdots\!83}{19\!\cdots\!24}a^{8}-\frac{26\!\cdots\!49}{98\!\cdots\!12}a^{7}-\frac{30\!\cdots\!61}{79\!\cdots\!96}a^{6}-\frac{11\!\cdots\!47}{24\!\cdots\!78}a^{5}-\frac{58\!\cdots\!51}{15\!\cdots\!48}a^{4}-\frac{22\!\cdots\!23}{98\!\cdots\!12}a^{3}-\frac{11\!\cdots\!76}{12\!\cdots\!39}a^{2}-\frac{74\!\cdots\!23}{24\!\cdots\!78}a-\frac{85\!\cdots\!66}{12\!\cdots\!39}$, $\frac{26\!\cdots\!69}{39\!\cdots\!48}a^{20}-\frac{26\!\cdots\!73}{24\!\cdots\!78}a^{19}+\frac{32\!\cdots\!79}{49\!\cdots\!56}a^{18}-\frac{26\!\cdots\!31}{19\!\cdots\!24}a^{17}-\frac{20\!\cdots\!89}{19\!\cdots\!24}a^{16}+\frac{26\!\cdots\!07}{98\!\cdots\!12}a^{15}+\frac{83\!\cdots\!65}{21\!\cdots\!08}a^{14}+\frac{76\!\cdots\!93}{98\!\cdots\!12}a^{13}+\frac{30\!\cdots\!15}{98\!\cdots\!12}a^{12}-\frac{37\!\cdots\!49}{98\!\cdots\!12}a^{11}-\frac{85\!\cdots\!39}{15\!\cdots\!48}a^{10}-\frac{29\!\cdots\!79}{19\!\cdots\!24}a^{9}-\frac{48\!\cdots\!05}{98\!\cdots\!12}a^{8}-\frac{14\!\cdots\!91}{49\!\cdots\!56}a^{7}-\frac{24\!\cdots\!81}{39\!\cdots\!48}a^{6}-\frac{15\!\cdots\!01}{49\!\cdots\!56}a^{5}+\frac{11\!\cdots\!89}{19\!\cdots\!06}a^{4}+\frac{50\!\cdots\!49}{98\!\cdots\!12}a^{3}-\frac{32\!\cdots\!15}{12\!\cdots\!39}a^{2}+\frac{27\!\cdots\!98}{12\!\cdots\!39}a-\frac{66\!\cdots\!80}{12\!\cdots\!39}$, $\frac{19\!\cdots\!77}{79\!\cdots\!96}a^{20}-\frac{10\!\cdots\!51}{12\!\cdots\!39}a^{19}+\frac{76\!\cdots\!67}{19\!\cdots\!24}a^{18}-\frac{20\!\cdots\!13}{39\!\cdots\!48}a^{17}-\frac{17\!\cdots\!13}{39\!\cdots\!48}a^{16}+\frac{88\!\cdots\!53}{19\!\cdots\!24}a^{15}+\frac{94\!\cdots\!01}{43\!\cdots\!16}a^{14}+\frac{11\!\cdots\!71}{19\!\cdots\!24}a^{13}+\frac{84\!\cdots\!47}{49\!\cdots\!56}a^{12}+\frac{48\!\cdots\!53}{98\!\cdots\!12}a^{11}-\frac{67\!\cdots\!15}{30\!\cdots\!96}a^{10}-\frac{35\!\cdots\!87}{39\!\cdots\!48}a^{9}-\frac{55\!\cdots\!23}{19\!\cdots\!24}a^{8}-\frac{41\!\cdots\!75}{98\!\cdots\!12}a^{7}-\frac{50\!\cdots\!33}{79\!\cdots\!96}a^{6}-\frac{66\!\cdots\!97}{98\!\cdots\!12}a^{5}-\frac{81\!\cdots\!97}{15\!\cdots\!48}a^{4}-\frac{29\!\cdots\!79}{98\!\cdots\!12}a^{3}-\frac{14\!\cdots\!21}{12\!\cdots\!39}a^{2}-\frac{40\!\cdots\!10}{12\!\cdots\!39}a-\frac{93\!\cdots\!81}{12\!\cdots\!39}$, $\frac{35\!\cdots\!59}{79\!\cdots\!96}a^{20}-\frac{36\!\cdots\!93}{19\!\cdots\!24}a^{19}+\frac{87\!\cdots\!59}{12\!\cdots\!39}a^{18}-\frac{36\!\cdots\!39}{39\!\cdots\!48}a^{17}-\frac{30\!\cdots\!51}{39\!\cdots\!48}a^{16}+\frac{16\!\cdots\!23}{19\!\cdots\!24}a^{15}+\frac{16\!\cdots\!71}{43\!\cdots\!16}a^{14}+\frac{48\!\cdots\!71}{49\!\cdots\!56}a^{13}+\frac{59\!\cdots\!15}{19\!\cdots\!24}a^{12}+\frac{84\!\cdots\!26}{12\!\cdots\!39}a^{11}-\frac{12\!\cdots\!29}{30\!\cdots\!96}a^{10}-\frac{63\!\cdots\!97}{39\!\cdots\!48}a^{9}-\frac{97\!\cdots\!61}{19\!\cdots\!24}a^{8}-\frac{35\!\cdots\!93}{49\!\cdots\!56}a^{7}-\frac{85\!\cdots\!47}{79\!\cdots\!96}a^{6}-\frac{22\!\cdots\!45}{19\!\cdots\!24}a^{5}-\frac{67\!\cdots\!29}{76\!\cdots\!24}a^{4}-\frac{23\!\cdots\!93}{49\!\cdots\!56}a^{3}-\frac{23\!\cdots\!21}{12\!\cdots\!39}a^{2}-\frac{12\!\cdots\!01}{24\!\cdots\!78}a-\frac{16\!\cdots\!46}{12\!\cdots\!39}$, $\frac{25\!\cdots\!87}{79\!\cdots\!96}a^{20}+\frac{33\!\cdots\!31}{39\!\cdots\!48}a^{19}-\frac{12\!\cdots\!89}{19\!\cdots\!24}a^{18}-\frac{26\!\cdots\!03}{39\!\cdots\!48}a^{17}-\frac{28\!\cdots\!29}{39\!\cdots\!48}a^{16}-\frac{12\!\cdots\!04}{12\!\cdots\!39}a^{15}+\frac{22\!\cdots\!47}{43\!\cdots\!16}a^{14}+\frac{60\!\cdots\!61}{39\!\cdots\!48}a^{13}+\frac{19\!\cdots\!53}{49\!\cdots\!56}a^{12}+\frac{30\!\cdots\!11}{49\!\cdots\!56}a^{11}-\frac{11\!\cdots\!29}{30\!\cdots\!96}a^{10}-\frac{88\!\cdots\!27}{39\!\cdots\!48}a^{9}-\frac{66\!\cdots\!63}{98\!\cdots\!12}a^{8}-\frac{73\!\cdots\!75}{49\!\cdots\!56}a^{7}-\frac{15\!\cdots\!47}{79\!\cdots\!96}a^{6}-\frac{10\!\cdots\!95}{39\!\cdots\!48}a^{5}-\frac{35\!\cdots\!41}{15\!\cdots\!48}a^{4}-\frac{12\!\cdots\!55}{98\!\cdots\!12}a^{3}-\frac{59\!\cdots\!83}{12\!\cdots\!39}a^{2}-\frac{38\!\cdots\!09}{24\!\cdots\!78}a-\frac{42\!\cdots\!71}{12\!\cdots\!39}$, $\frac{16\!\cdots\!23}{60\!\cdots\!92}a^{20}-\frac{45\!\cdots\!59}{15\!\cdots\!48}a^{19}+\frac{53\!\cdots\!07}{76\!\cdots\!24}a^{18}-\frac{14\!\cdots\!55}{30\!\cdots\!96}a^{17}-\frac{13\!\cdots\!35}{30\!\cdots\!96}a^{16}+\frac{13\!\cdots\!35}{15\!\cdots\!48}a^{15}+\frac{70\!\cdots\!91}{33\!\cdots\!32}a^{14}+\frac{31\!\cdots\!45}{76\!\cdots\!24}a^{13}+\frac{21\!\cdots\!21}{15\!\cdots\!48}a^{12}-\frac{31\!\cdots\!61}{38\!\cdots\!12}a^{11}-\frac{90\!\cdots\!05}{30\!\cdots\!96}a^{10}-\frac{23\!\cdots\!33}{30\!\cdots\!96}a^{9}-\frac{35\!\cdots\!61}{15\!\cdots\!48}a^{8}-\frac{45\!\cdots\!07}{19\!\cdots\!06}a^{7}-\frac{21\!\cdots\!35}{60\!\cdots\!92}a^{6}-\frac{43\!\cdots\!23}{15\!\cdots\!48}a^{5}-\frac{77\!\cdots\!74}{95\!\cdots\!03}a^{4}+\frac{19\!\cdots\!35}{95\!\cdots\!03}a^{3}+\frac{42\!\cdots\!71}{38\!\cdots\!12}a^{2}+\frac{72\!\cdots\!15}{95\!\cdots\!03}a+\frac{38\!\cdots\!05}{95\!\cdots\!03}$, $\frac{10\!\cdots\!57}{19\!\cdots\!24}a^{20}-\frac{86\!\cdots\!75}{19\!\cdots\!24}a^{19}+\frac{63\!\cdots\!19}{24\!\cdots\!78}a^{18}-\frac{56\!\cdots\!85}{49\!\cdots\!56}a^{17}-\frac{43\!\cdots\!67}{49\!\cdots\!56}a^{16}+\frac{13\!\cdots\!21}{98\!\cdots\!12}a^{15}+\frac{43\!\cdots\!77}{10\!\cdots\!04}a^{14}+\frac{19\!\cdots\!85}{19\!\cdots\!24}a^{13}+\frac{39\!\cdots\!68}{12\!\cdots\!39}a^{12}-\frac{44\!\cdots\!07}{98\!\cdots\!12}a^{11}-\frac{35\!\cdots\!59}{76\!\cdots\!24}a^{10}-\frac{83\!\cdots\!91}{49\!\cdots\!56}a^{9}-\frac{51\!\cdots\!45}{98\!\cdots\!12}a^{8}-\frac{16\!\cdots\!79}{24\!\cdots\!78}a^{7}-\frac{20\!\cdots\!17}{19\!\cdots\!24}a^{6}-\frac{19\!\cdots\!05}{19\!\cdots\!24}a^{5}-\frac{13\!\cdots\!21}{19\!\cdots\!06}a^{4}-\frac{34\!\cdots\!55}{98\!\cdots\!12}a^{3}-\frac{18\!\cdots\!46}{12\!\cdots\!39}a^{2}-\frac{45\!\cdots\!88}{12\!\cdots\!39}a-\frac{14\!\cdots\!09}{12\!\cdots\!39}$, $\frac{21\!\cdots\!07}{39\!\cdots\!48}a^{20}-\frac{13\!\cdots\!21}{39\!\cdots\!48}a^{19}+\frac{16\!\cdots\!89}{19\!\cdots\!24}a^{18}-\frac{22\!\cdots\!31}{19\!\cdots\!24}a^{17}-\frac{88\!\cdots\!41}{98\!\cdots\!12}a^{16}+\frac{24\!\cdots\!09}{19\!\cdots\!24}a^{15}+\frac{97\!\cdots\!27}{21\!\cdots\!08}a^{14}+\frac{42\!\cdots\!59}{39\!\cdots\!48}a^{13}+\frac{64\!\cdots\!49}{19\!\cdots\!24}a^{12}-\frac{66\!\cdots\!23}{98\!\cdots\!12}a^{11}-\frac{82\!\cdots\!37}{15\!\cdots\!48}a^{10}-\frac{18\!\cdots\!65}{98\!\cdots\!12}a^{9}-\frac{10\!\cdots\!65}{19\!\cdots\!24}a^{8}-\frac{35\!\cdots\!83}{49\!\cdots\!56}a^{7}-\frac{41\!\cdots\!59}{39\!\cdots\!48}a^{6}-\frac{39\!\cdots\!23}{39\!\cdots\!48}a^{5}-\frac{97\!\cdots\!21}{15\!\cdots\!48}a^{4}-\frac{94\!\cdots\!61}{49\!\cdots\!56}a^{3}+\frac{28\!\cdots\!85}{49\!\cdots\!56}a^{2}+\frac{37\!\cdots\!40}{12\!\cdots\!39}a+\frac{33\!\cdots\!30}{12\!\cdots\!39}$
| 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: | \( 48578716.598 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 48578716.598 \cdot 1}{2\cdot\sqrt{86433900540276528721822666309632}}\cr\approx \mathstrut & 0.31899420059 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^18 - 18*x^17 + 12*x^16 + 93*x^15 + 252*x^14 + 756*x^13 + 424*x^12 - 846*x^11 - 3942*x^10 - 12420*x^9 - 20592*x^8 - 30729*x^7 - 35272*x^6 - 30168*x^5 - 18600*x^4 - 8512*x^3 - 2880*x^2 - 768*x - 128)
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^18 - 18*x^17 + 12*x^16 + 93*x^15 + 252*x^14 + 756*x^13 + 424*x^12 - 846*x^11 - 3942*x^10 - 12420*x^9 - 20592*x^8 - 30729*x^7 - 35272*x^6 - 30168*x^5 - 18600*x^4 - 8512*x^3 - 2880*x^2 - 768*x - 128, 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^18 - 18*x^17 + 12*x^16 + 93*x^15 + 252*x^14 + 756*x^13 + 424*x^12 - 846*x^11 - 3942*x^10 - 12420*x^9 - 20592*x^8 - 30729*x^7 - 35272*x^6 - 30168*x^5 - 18600*x^4 - 8512*x^3 - 2880*x^2 - 768*x - 128);
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^18 - 18*x^17 + 12*x^16 + 93*x^15 + 252*x^14 + 756*x^13 + 424*x^12 - 846*x^11 - 3942*x^10 - 12420*x^9 - 20592*x^8 - 30729*x^7 - 35272*x^6 - 30168*x^5 - 18600*x^4 - 8512*x^3 - 2880*x^2 - 768*x - 128);
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
$C_3^7:C_2\wr D_7$ (as 21T131):
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 3919104 |
| The 288 conjugacy class representatives for $C_3^7:C_2\wr D_7$ |
| Character table for $C_3^7:C_2\wr D_7$ |
Intermediate fields
| 7.1.357911.1 |
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 42 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 | R | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | $21$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | $21$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.7.0.1 | $x^{7} + x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 2.14.14.20 | $x^{14} + 14 x^{13} + 126 x^{12} + 1168 x^{11} + 9868 x^{10} + 64520 x^{9} + 317384 x^{8} + 1190080 x^{7} + 3447536 x^{6} + 7751584 x^{5} + 13448736 x^{4} + 17658368 x^{3} + 16807744 x^{2} + 10589568 x + 3453824$ | $2$ | $7$ | $14$ | 14T21 | $[2, 2, 2, 2, 2, 2]^{7}$ | |
|
\(3\)
| Deg $21$ | $3$ | $7$ | $21$ | |||
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.6.0.1 | $x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(71\)
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 71.6.3.2 | $x^{6} + 221 x^{4} + 128 x^{3} + 15139 x^{2} - 26752 x + 322815$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |