Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 2*x^19 + 12*x^18 - 25*x^17 + x^16 + 48*x^15 - 91*x^14 + 103*x^13 - 12*x^12 - 110*x^11 + 272*x^10 - 277*x^9 - 78*x^8 + 278*x^7 - 46*x^6 + 11*x^5 - 114*x^4 + 3*x^3 + 93*x^2 - 19*x - 23)
gp: K = bnfinit(y^21 - 2*y^20 - 2*y^19 + 12*y^18 - 25*y^17 + y^16 + 48*y^15 - 91*y^14 + 103*y^13 - 12*y^12 - 110*y^11 + 272*y^10 - 277*y^9 - 78*y^8 + 278*y^7 - 46*y^6 + 11*y^5 - 114*y^4 + 3*y^3 + 93*y^2 - 19*y - 23, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 2*x^20 - 2*x^19 + 12*x^18 - 25*x^17 + x^16 + 48*x^15 - 91*x^14 + 103*x^13 - 12*x^12 - 110*x^11 + 272*x^10 - 277*x^9 - 78*x^8 + 278*x^7 - 46*x^6 + 11*x^5 - 114*x^4 + 3*x^3 + 93*x^2 - 19*x - 23);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 2*x^20 - 2*x^19 + 12*x^18 - 25*x^17 + x^16 + 48*x^15 - 91*x^14 + 103*x^13 - 12*x^12 - 110*x^11 + 272*x^10 - 277*x^9 - 78*x^8 + 278*x^7 - 46*x^6 + 11*x^5 - 114*x^4 + 3*x^3 + 93*x^2 - 19*x - 23)
\( x^{21} - 2 x^{20} - 2 x^{19} + 12 x^{18} - 25 x^{17} + x^{16} + 48 x^{15} - 91 x^{14} + 103 x^{13} + \cdots - 23 \)
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: |
\(18453551447289410297167049873\)
\(\medspace = 71^{3}\cdot 8623^{3}\cdot 283573^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.18\) | 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: | $71^{1/2}8623^{1/2}283573^{2/3}\approx 3377289.469795227$ | ||
| Ramified primes: |
\(71\), \(8623\), \(283573\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{612233}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{51}a^{19}+\frac{7}{51}a^{18}-\frac{2}{17}a^{17}-\frac{1}{51}a^{16}+\frac{11}{51}a^{15}+\frac{14}{51}a^{14}-\frac{2}{51}a^{13}+\frac{8}{17}a^{12}-\frac{2}{17}a^{11}+\frac{3}{17}a^{10}+\frac{16}{51}a^{9}+\frac{1}{3}a^{8}-\frac{23}{51}a^{7}+\frac{4}{51}a^{6}+\frac{19}{51}a^{5}+\frac{10}{51}a^{4}+\frac{1}{51}a^{3}-\frac{10}{51}a^{2}-\frac{1}{51}a-\frac{11}{51}$, $\frac{1}{13\!\cdots\!87}a^{20}+\frac{75\!\cdots\!92}{13\!\cdots\!87}a^{19}-\frac{14\!\cdots\!52}{46\!\cdots\!29}a^{18}+\frac{33\!\cdots\!23}{13\!\cdots\!87}a^{17}+\frac{30\!\cdots\!77}{13\!\cdots\!87}a^{16}+\frac{56\!\cdots\!97}{13\!\cdots\!87}a^{15}+\frac{66\!\cdots\!30}{13\!\cdots\!87}a^{14}+\frac{24\!\cdots\!32}{46\!\cdots\!29}a^{13}+\frac{15\!\cdots\!11}{46\!\cdots\!29}a^{12}+\frac{22\!\cdots\!78}{46\!\cdots\!29}a^{11}+\frac{55\!\cdots\!39}{13\!\cdots\!87}a^{10}+\frac{41\!\cdots\!91}{13\!\cdots\!87}a^{9}-\frac{15\!\cdots\!89}{13\!\cdots\!87}a^{8}+\frac{23\!\cdots\!47}{13\!\cdots\!87}a^{7}-\frac{26\!\cdots\!49}{13\!\cdots\!87}a^{6}-\frac{34\!\cdots\!29}{13\!\cdots\!87}a^{5}+\frac{42\!\cdots\!84}{13\!\cdots\!87}a^{4}-\frac{64\!\cdots\!53}{13\!\cdots\!87}a^{3}+\frac{50\!\cdots\!02}{13\!\cdots\!87}a^{2}-\frac{13\!\cdots\!67}{13\!\cdots\!87}a-\frac{71\!\cdots\!83}{46\!\cdots\!29}$
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: |
$\frac{16\!\cdots\!28}{46\!\cdots\!29}a^{20}-\frac{12\!\cdots\!74}{13\!\cdots\!87}a^{19}-\frac{30\!\cdots\!49}{13\!\cdots\!87}a^{18}+\frac{20\!\cdots\!59}{46\!\cdots\!29}a^{17}-\frac{15\!\cdots\!60}{13\!\cdots\!87}a^{16}+\frac{91\!\cdots\!56}{13\!\cdots\!87}a^{15}+\frac{19\!\cdots\!58}{13\!\cdots\!87}a^{14}-\frac{56\!\cdots\!24}{13\!\cdots\!87}a^{13}+\frac{27\!\cdots\!37}{46\!\cdots\!29}a^{12}-\frac{16\!\cdots\!94}{46\!\cdots\!29}a^{11}-\frac{92\!\cdots\!02}{46\!\cdots\!29}a^{10}+\frac{15\!\cdots\!43}{13\!\cdots\!87}a^{9}-\frac{22\!\cdots\!09}{13\!\cdots\!87}a^{8}+\frac{80\!\cdots\!08}{13\!\cdots\!87}a^{7}+\frac{96\!\cdots\!69}{13\!\cdots\!87}a^{6}-\frac{75\!\cdots\!12}{13\!\cdots\!87}a^{5}+\frac{45\!\cdots\!10}{13\!\cdots\!87}a^{4}-\frac{82\!\cdots\!22}{13\!\cdots\!87}a^{3}+\frac{46\!\cdots\!08}{13\!\cdots\!87}a^{2}+\frac{21\!\cdots\!40}{13\!\cdots\!87}a-\frac{21\!\cdots\!78}{13\!\cdots\!87}$, $\frac{99\!\cdots\!61}{13\!\cdots\!87}a^{20}-\frac{25\!\cdots\!96}{13\!\cdots\!87}a^{19}-\frac{61\!\cdots\!58}{13\!\cdots\!87}a^{18}+\frac{12\!\cdots\!92}{13\!\cdots\!87}a^{17}-\frac{31\!\cdots\!41}{13\!\cdots\!87}a^{16}+\frac{18\!\cdots\!22}{13\!\cdots\!87}a^{15}+\frac{12\!\cdots\!52}{46\!\cdots\!29}a^{14}-\frac{11\!\cdots\!25}{13\!\cdots\!87}a^{13}+\frac{54\!\cdots\!58}{46\!\cdots\!29}a^{12}-\frac{33\!\cdots\!55}{46\!\cdots\!29}a^{11}-\frac{55\!\cdots\!74}{13\!\cdots\!87}a^{10}+\frac{10\!\cdots\!94}{46\!\cdots\!29}a^{9}-\frac{14\!\cdots\!05}{46\!\cdots\!29}a^{8}+\frac{16\!\cdots\!07}{13\!\cdots\!87}a^{7}+\frac{19\!\cdots\!78}{13\!\cdots\!87}a^{6}-\frac{14\!\cdots\!66}{13\!\cdots\!87}a^{5}+\frac{91\!\cdots\!88}{13\!\cdots\!87}a^{4}-\frac{54\!\cdots\!29}{46\!\cdots\!29}a^{3}+\frac{91\!\cdots\!56}{13\!\cdots\!87}a^{2}+\frac{14\!\cdots\!04}{46\!\cdots\!29}a-\frac{42\!\cdots\!26}{13\!\cdots\!87}$, $\frac{10\!\cdots\!23}{13\!\cdots\!87}a^{20}-\frac{25\!\cdots\!20}{13\!\cdots\!87}a^{19}-\frac{68\!\cdots\!75}{13\!\cdots\!87}a^{18}+\frac{12\!\cdots\!53}{13\!\cdots\!87}a^{17}-\frac{31\!\cdots\!28}{13\!\cdots\!87}a^{16}+\frac{17\!\cdots\!69}{13\!\cdots\!87}a^{15}+\frac{12\!\cdots\!66}{46\!\cdots\!29}a^{14}-\frac{11\!\cdots\!41}{13\!\cdots\!87}a^{13}+\frac{54\!\cdots\!30}{46\!\cdots\!29}a^{12}-\frac{33\!\cdots\!73}{46\!\cdots\!29}a^{11}-\frac{53\!\cdots\!38}{13\!\cdots\!87}a^{10}+\frac{10\!\cdots\!19}{46\!\cdots\!29}a^{9}-\frac{14\!\cdots\!33}{46\!\cdots\!29}a^{8}+\frac{15\!\cdots\!42}{13\!\cdots\!87}a^{7}+\frac{18\!\cdots\!85}{13\!\cdots\!87}a^{6}-\frac{14\!\cdots\!29}{13\!\cdots\!87}a^{5}+\frac{93\!\cdots\!27}{13\!\cdots\!87}a^{4}-\frac{53\!\cdots\!73}{46\!\cdots\!29}a^{3}+\frac{89\!\cdots\!38}{13\!\cdots\!87}a^{2}+\frac{13\!\cdots\!18}{46\!\cdots\!29}a-\frac{41\!\cdots\!93}{13\!\cdots\!87}$, $\frac{84\!\cdots\!35}{46\!\cdots\!29}a^{20}-\frac{21\!\cdots\!71}{46\!\cdots\!29}a^{19}-\frac{53\!\cdots\!19}{46\!\cdots\!29}a^{18}+\frac{10\!\cdots\!41}{46\!\cdots\!29}a^{17}-\frac{26\!\cdots\!23}{46\!\cdots\!29}a^{16}+\frac{15\!\cdots\!10}{46\!\cdots\!29}a^{15}+\frac{32\!\cdots\!36}{46\!\cdots\!29}a^{14}-\frac{94\!\cdots\!13}{46\!\cdots\!29}a^{13}+\frac{13\!\cdots\!36}{46\!\cdots\!29}a^{12}-\frac{84\!\cdots\!95}{46\!\cdots\!29}a^{11}-\frac{47\!\cdots\!44}{46\!\cdots\!29}a^{10}+\frac{25\!\cdots\!04}{46\!\cdots\!29}a^{9}-\frac{37\!\cdots\!67}{46\!\cdots\!29}a^{8}+\frac{13\!\cdots\!04}{46\!\cdots\!29}a^{7}+\frac{16\!\cdots\!91}{46\!\cdots\!29}a^{6}-\frac{12\!\cdots\!50}{46\!\cdots\!29}a^{5}+\frac{77\!\cdots\!06}{46\!\cdots\!29}a^{4}-\frac{13\!\cdots\!62}{46\!\cdots\!29}a^{3}+\frac{77\!\cdots\!19}{46\!\cdots\!29}a^{2}+\frac{36\!\cdots\!81}{46\!\cdots\!29}a-\frac{35\!\cdots\!65}{46\!\cdots\!29}$, $\frac{46\!\cdots\!01}{13\!\cdots\!87}a^{20}-\frac{11\!\cdots\!15}{13\!\cdots\!87}a^{19}-\frac{28\!\cdots\!70}{13\!\cdots\!87}a^{18}+\frac{56\!\cdots\!70}{13\!\cdots\!87}a^{17}-\frac{14\!\cdots\!56}{13\!\cdots\!87}a^{16}+\frac{84\!\cdots\!81}{13\!\cdots\!87}a^{15}+\frac{58\!\cdots\!47}{46\!\cdots\!29}a^{14}-\frac{51\!\cdots\!85}{13\!\cdots\!87}a^{13}+\frac{25\!\cdots\!79}{46\!\cdots\!29}a^{12}-\frac{15\!\cdots\!11}{46\!\cdots\!29}a^{11}-\frac{25\!\cdots\!90}{13\!\cdots\!87}a^{10}+\frac{46\!\cdots\!51}{46\!\cdots\!29}a^{9}-\frac{67\!\cdots\!73}{46\!\cdots\!29}a^{8}+\frac{74\!\cdots\!25}{13\!\cdots\!87}a^{7}+\frac{88\!\cdots\!69}{13\!\cdots\!87}a^{6}-\frac{68\!\cdots\!56}{13\!\cdots\!87}a^{5}+\frac{42\!\cdots\!18}{13\!\cdots\!87}a^{4}-\frac{25\!\cdots\!94}{46\!\cdots\!29}a^{3}+\frac{42\!\cdots\!67}{13\!\cdots\!87}a^{2}+\frac{66\!\cdots\!03}{46\!\cdots\!29}a-\frac{19\!\cdots\!17}{13\!\cdots\!87}$, $\frac{95\!\cdots\!13}{13\!\cdots\!87}a^{20}-\frac{24\!\cdots\!63}{13\!\cdots\!87}a^{19}-\frac{49\!\cdots\!52}{13\!\cdots\!87}a^{18}+\frac{11\!\cdots\!05}{13\!\cdots\!87}a^{17}-\frac{30\!\cdots\!24}{13\!\cdots\!87}a^{16}+\frac{18\!\cdots\!16}{13\!\cdots\!87}a^{15}+\frac{12\!\cdots\!56}{46\!\cdots\!29}a^{14}-\frac{10\!\cdots\!26}{13\!\cdots\!87}a^{13}+\frac{53\!\cdots\!01}{46\!\cdots\!29}a^{12}-\frac{33\!\cdots\!88}{46\!\cdots\!29}a^{11}-\frac{52\!\cdots\!68}{13\!\cdots\!87}a^{10}+\frac{97\!\cdots\!47}{46\!\cdots\!29}a^{9}-\frac{14\!\cdots\!82}{46\!\cdots\!29}a^{8}+\frac{16\!\cdots\!45}{13\!\cdots\!87}a^{7}+\frac{18\!\cdots\!29}{13\!\cdots\!87}a^{6}-\frac{15\!\cdots\!58}{13\!\cdots\!87}a^{5}+\frac{86\!\cdots\!93}{13\!\cdots\!87}a^{4}-\frac{53\!\cdots\!97}{46\!\cdots\!29}a^{3}+\frac{94\!\cdots\!69}{13\!\cdots\!87}a^{2}+\frac{13\!\cdots\!91}{46\!\cdots\!29}a-\frac{44\!\cdots\!77}{13\!\cdots\!87}$, $\frac{66\!\cdots\!01}{46\!\cdots\!29}a^{20}-\frac{17\!\cdots\!96}{46\!\cdots\!29}a^{19}-\frac{40\!\cdots\!92}{46\!\cdots\!29}a^{18}+\frac{82\!\cdots\!76}{46\!\cdots\!29}a^{17}-\frac{21\!\cdots\!64}{46\!\cdots\!29}a^{16}+\frac{12\!\cdots\!44}{46\!\cdots\!29}a^{15}+\frac{25\!\cdots\!69}{46\!\cdots\!29}a^{14}-\frac{74\!\cdots\!20}{46\!\cdots\!29}a^{13}+\frac{10\!\cdots\!29}{46\!\cdots\!29}a^{12}-\frac{67\!\cdots\!05}{46\!\cdots\!29}a^{11}-\frac{36\!\cdots\!03}{46\!\cdots\!29}a^{10}+\frac{20\!\cdots\!53}{46\!\cdots\!29}a^{9}-\frac{29\!\cdots\!18}{46\!\cdots\!29}a^{8}+\frac{10\!\cdots\!42}{46\!\cdots\!29}a^{7}+\frac{12\!\cdots\!17}{46\!\cdots\!29}a^{6}-\frac{99\!\cdots\!45}{46\!\cdots\!29}a^{5}+\frac{60\!\cdots\!18}{46\!\cdots\!29}a^{4}-\frac{11\!\cdots\!93}{46\!\cdots\!29}a^{3}+\frac{62\!\cdots\!14}{46\!\cdots\!29}a^{2}+\frac{29\!\cdots\!55}{46\!\cdots\!29}a-\frac{28\!\cdots\!60}{46\!\cdots\!29}$, $\frac{86\!\cdots\!57}{13\!\cdots\!87}a^{20}-\frac{21\!\cdots\!51}{13\!\cdots\!87}a^{19}-\frac{54\!\cdots\!65}{13\!\cdots\!87}a^{18}+\frac{10\!\cdots\!79}{13\!\cdots\!87}a^{17}-\frac{27\!\cdots\!20}{13\!\cdots\!87}a^{16}+\frac{15\!\cdots\!74}{13\!\cdots\!87}a^{15}+\frac{10\!\cdots\!04}{46\!\cdots\!29}a^{14}-\frac{96\!\cdots\!43}{13\!\cdots\!87}a^{13}+\frac{46\!\cdots\!02}{46\!\cdots\!29}a^{12}-\frac{28\!\cdots\!30}{46\!\cdots\!29}a^{11}-\frac{48\!\cdots\!80}{13\!\cdots\!87}a^{10}+\frac{86\!\cdots\!62}{46\!\cdots\!29}a^{9}-\frac{12\!\cdots\!16}{46\!\cdots\!29}a^{8}+\frac{13\!\cdots\!49}{13\!\cdots\!87}a^{7}+\frac{16\!\cdots\!08}{13\!\cdots\!87}a^{6}-\frac{12\!\cdots\!28}{13\!\cdots\!87}a^{5}+\frac{79\!\cdots\!77}{13\!\cdots\!87}a^{4}-\frac{47\!\cdots\!80}{46\!\cdots\!29}a^{3}+\frac{78\!\cdots\!25}{13\!\cdots\!87}a^{2}+\frac{12\!\cdots\!93}{46\!\cdots\!29}a-\frac{36\!\cdots\!33}{13\!\cdots\!87}$, $\frac{22\!\cdots\!49}{46\!\cdots\!29}a^{20}-\frac{17\!\cdots\!87}{13\!\cdots\!87}a^{19}-\frac{42\!\cdots\!20}{13\!\cdots\!87}a^{18}+\frac{28\!\cdots\!27}{46\!\cdots\!29}a^{17}-\frac{21\!\cdots\!44}{13\!\cdots\!87}a^{16}+\frac{12\!\cdots\!59}{13\!\cdots\!87}a^{15}+\frac{26\!\cdots\!00}{13\!\cdots\!87}a^{14}-\frac{76\!\cdots\!96}{13\!\cdots\!87}a^{13}+\frac{37\!\cdots\!38}{46\!\cdots\!29}a^{12}-\frac{23\!\cdots\!81}{46\!\cdots\!29}a^{11}-\frac{12\!\cdots\!96}{46\!\cdots\!29}a^{10}+\frac{20\!\cdots\!69}{13\!\cdots\!87}a^{9}-\frac{30\!\cdots\!93}{13\!\cdots\!87}a^{8}+\frac{11\!\cdots\!62}{13\!\cdots\!87}a^{7}+\frac{13\!\cdots\!65}{13\!\cdots\!87}a^{6}-\frac{10\!\cdots\!36}{13\!\cdots\!87}a^{5}+\frac{62\!\cdots\!28}{13\!\cdots\!87}a^{4}-\frac{11\!\cdots\!66}{13\!\cdots\!87}a^{3}+\frac{62\!\cdots\!69}{13\!\cdots\!87}a^{2}+\frac{30\!\cdots\!10}{13\!\cdots\!87}a-\frac{29\!\cdots\!83}{13\!\cdots\!87}$, $\frac{42\!\cdots\!97}{13\!\cdots\!87}a^{20}-\frac{10\!\cdots\!93}{13\!\cdots\!87}a^{19}-\frac{89\!\cdots\!53}{46\!\cdots\!29}a^{18}+\frac{52\!\cdots\!31}{13\!\cdots\!87}a^{17}-\frac{13\!\cdots\!88}{13\!\cdots\!87}a^{16}+\frac{76\!\cdots\!60}{13\!\cdots\!87}a^{15}+\frac{16\!\cdots\!35}{13\!\cdots\!87}a^{14}-\frac{15\!\cdots\!10}{46\!\cdots\!29}a^{13}+\frac{23\!\cdots\!89}{46\!\cdots\!29}a^{12}-\frac{14\!\cdots\!00}{46\!\cdots\!29}a^{11}-\frac{23\!\cdots\!49}{13\!\cdots\!87}a^{10}+\frac{12\!\cdots\!40}{13\!\cdots\!87}a^{9}-\frac{18\!\cdots\!11}{13\!\cdots\!87}a^{8}+\frac{67\!\cdots\!98}{13\!\cdots\!87}a^{7}+\frac{80\!\cdots\!41}{13\!\cdots\!87}a^{6}-\frac{62\!\cdots\!85}{13\!\cdots\!87}a^{5}+\frac{38\!\cdots\!61}{13\!\cdots\!87}a^{4}-\frac{68\!\cdots\!54}{13\!\cdots\!87}a^{3}+\frac{38\!\cdots\!92}{13\!\cdots\!87}a^{2}+\frac{18\!\cdots\!00}{13\!\cdots\!87}a-\frac{59\!\cdots\!00}{46\!\cdots\!29}$, $\frac{29\!\cdots\!86}{13\!\cdots\!87}a^{20}-\frac{24\!\cdots\!41}{46\!\cdots\!29}a^{19}-\frac{17\!\cdots\!91}{13\!\cdots\!87}a^{18}+\frac{36\!\cdots\!82}{13\!\cdots\!87}a^{17}-\frac{31\!\cdots\!65}{46\!\cdots\!29}a^{16}+\frac{18\!\cdots\!08}{46\!\cdots\!29}a^{15}+\frac{11\!\cdots\!75}{13\!\cdots\!87}a^{14}-\frac{32\!\cdots\!80}{13\!\cdots\!87}a^{13}+\frac{16\!\cdots\!12}{46\!\cdots\!29}a^{12}-\frac{99\!\cdots\!21}{46\!\cdots\!29}a^{11}-\frac{16\!\cdots\!98}{13\!\cdots\!87}a^{10}+\frac{88\!\cdots\!51}{13\!\cdots\!87}a^{9}-\frac{13\!\cdots\!69}{13\!\cdots\!87}a^{8}+\frac{16\!\cdots\!21}{46\!\cdots\!29}a^{7}+\frac{18\!\cdots\!91}{46\!\cdots\!29}a^{6}-\frac{14\!\cdots\!43}{46\!\cdots\!29}a^{5}+\frac{90\!\cdots\!36}{46\!\cdots\!29}a^{4}-\frac{48\!\cdots\!00}{13\!\cdots\!87}a^{3}+\frac{90\!\cdots\!25}{46\!\cdots\!29}a^{2}+\frac{12\!\cdots\!57}{13\!\cdots\!87}a-\frac{12\!\cdots\!78}{13\!\cdots\!87}$, $\frac{50\!\cdots\!94}{13\!\cdots\!87}a^{20}-\frac{12\!\cdots\!31}{13\!\cdots\!87}a^{19}-\frac{10\!\cdots\!24}{46\!\cdots\!29}a^{18}+\frac{62\!\cdots\!25}{13\!\cdots\!87}a^{17}-\frac{15\!\cdots\!64}{13\!\cdots\!87}a^{16}+\frac{90\!\cdots\!09}{13\!\cdots\!87}a^{15}+\frac{19\!\cdots\!88}{13\!\cdots\!87}a^{14}-\frac{18\!\cdots\!13}{46\!\cdots\!29}a^{13}+\frac{27\!\cdots\!66}{46\!\cdots\!29}a^{12}-\frac{16\!\cdots\!79}{46\!\cdots\!29}a^{11}-\frac{28\!\cdots\!76}{13\!\cdots\!87}a^{10}+\frac{15\!\cdots\!59}{13\!\cdots\!87}a^{9}-\frac{22\!\cdots\!62}{13\!\cdots\!87}a^{8}+\frac{80\!\cdots\!05}{13\!\cdots\!87}a^{7}+\frac{96\!\cdots\!25}{13\!\cdots\!87}a^{6}-\frac{74\!\cdots\!83}{13\!\cdots\!87}a^{5}+\frac{46\!\cdots\!44}{13\!\cdots\!87}a^{4}-\frac{82\!\cdots\!50}{13\!\cdots\!87}a^{3}+\frac{45\!\cdots\!77}{13\!\cdots\!87}a^{2}+\frac{21\!\cdots\!21}{13\!\cdots\!87}a-\frac{70\!\cdots\!98}{46\!\cdots\!29}$, $\frac{29\!\cdots\!66}{13\!\cdots\!87}a^{20}-\frac{75\!\cdots\!28}{13\!\cdots\!87}a^{19}-\frac{64\!\cdots\!00}{46\!\cdots\!29}a^{18}+\frac{36\!\cdots\!92}{13\!\cdots\!87}a^{17}-\frac{94\!\cdots\!97}{13\!\cdots\!87}a^{16}+\frac{53\!\cdots\!92}{13\!\cdots\!87}a^{15}+\frac{11\!\cdots\!45}{13\!\cdots\!87}a^{14}-\frac{11\!\cdots\!65}{46\!\cdots\!29}a^{13}+\frac{16\!\cdots\!50}{46\!\cdots\!29}a^{12}-\frac{99\!\cdots\!95}{46\!\cdots\!29}a^{11}-\frac{16\!\cdots\!78}{13\!\cdots\!87}a^{10}+\frac{90\!\cdots\!38}{13\!\cdots\!87}a^{9}-\frac{13\!\cdots\!36}{13\!\cdots\!87}a^{8}+\frac{47\!\cdots\!07}{13\!\cdots\!87}a^{7}+\frac{57\!\cdots\!52}{13\!\cdots\!87}a^{6}-\frac{44\!\cdots\!84}{13\!\cdots\!87}a^{5}+\frac{27\!\cdots\!85}{13\!\cdots\!87}a^{4}-\frac{48\!\cdots\!50}{13\!\cdots\!87}a^{3}+\frac{26\!\cdots\!85}{13\!\cdots\!87}a^{2}+\frac{12\!\cdots\!87}{13\!\cdots\!87}a-\frac{41\!\cdots\!81}{46\!\cdots\!29}$, $\frac{30\!\cdots\!73}{46\!\cdots\!29}a^{20}-\frac{22\!\cdots\!11}{13\!\cdots\!87}a^{19}-\frac{66\!\cdots\!95}{13\!\cdots\!87}a^{18}+\frac{37\!\cdots\!38}{46\!\cdots\!29}a^{17}-\frac{28\!\cdots\!34}{13\!\cdots\!87}a^{16}+\frac{16\!\cdots\!39}{13\!\cdots\!87}a^{15}+\frac{33\!\cdots\!63}{13\!\cdots\!87}a^{14}-\frac{10\!\cdots\!06}{13\!\cdots\!87}a^{13}+\frac{50\!\cdots\!99}{46\!\cdots\!29}a^{12}-\frac{31\!\cdots\!93}{46\!\cdots\!29}a^{11}-\frac{14\!\cdots\!82}{46\!\cdots\!29}a^{10}+\frac{28\!\cdots\!88}{13\!\cdots\!87}a^{9}-\frac{42\!\cdots\!15}{13\!\cdots\!87}a^{8}+\frac{17\!\cdots\!28}{13\!\cdots\!87}a^{7}+\frac{13\!\cdots\!75}{13\!\cdots\!87}a^{6}-\frac{15\!\cdots\!13}{13\!\cdots\!87}a^{5}+\frac{16\!\cdots\!38}{13\!\cdots\!87}a^{4}-\frac{20\!\cdots\!92}{13\!\cdots\!87}a^{3}+\frac{91\!\cdots\!74}{13\!\cdots\!87}a^{2}+\frac{47\!\cdots\!21}{13\!\cdots\!87}a-\frac{59\!\cdots\!23}{13\!\cdots\!87}$
| 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: | \( 1614740.57391 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{6}\cdot 1614740.57391 \cdot 1}{2\cdot\sqrt{18453551447289410297167049873}}\cr\approx \mathstrut & 0.187232809240 \end{aligned}\] (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 + 12*x^18 - 25*x^17 + x^16 + 48*x^15 - 91*x^14 + 103*x^13 - 12*x^12 - 110*x^11 + 272*x^10 - 277*x^9 - 78*x^8 + 278*x^7 - 46*x^6 + 11*x^5 - 114*x^4 + 3*x^3 + 93*x^2 - 19*x - 23)
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 + 12*x^18 - 25*x^17 + x^16 + 48*x^15 - 91*x^14 + 103*x^13 - 12*x^12 - 110*x^11 + 272*x^10 - 277*x^9 - 78*x^8 + 278*x^7 - 46*x^6 + 11*x^5 - 114*x^4 + 3*x^3 + 93*x^2 - 19*x - 23, 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 + 12*x^18 - 25*x^17 + x^16 + 48*x^15 - 91*x^14 + 103*x^13 - 12*x^12 - 110*x^11 + 272*x^10 - 277*x^9 - 78*x^8 + 278*x^7 - 46*x^6 + 11*x^5 - 114*x^4 + 3*x^3 + 93*x^2 - 19*x - 23);
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 + 12*x^18 - 25*x^17 + x^16 + 48*x^15 - 91*x^14 + 103*x^13 - 12*x^12 - 110*x^11 + 272*x^10 - 277*x^9 - 78*x^8 + 278*x^7 - 46*x^6 + 11*x^5 - 114*x^4 + 3*x^3 + 93*x^2 - 19*x - 23);
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
$C_3^7.S_7$ (as 21T139):
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 11022480 |
| The 429 conjugacy class representatives for $C_3^7.S_7$ |
| Character table for $C_3^7.S_7$ |
Intermediate fields
| 7.3.612233.1 |
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 21 sibling: | data not computed |
| 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 | $21$ | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{3}$ | $18{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | $18{,}\,{\href{/padicField/37.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | $15{,}\,{\href{/padicField/47.6.0.1}{6} }$ | $15{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(71\)
| 71.6.3.1 | $x^{6} - 3550 x^{5} + 161634624 x^{4} + 10165888006904 x^{3} + 6625668596 x^{2} - 569794312 x - 22906304$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 71.15.0.1 | $x^{15} + 28 x^{6} + 32 x^{5} + 18 x^{4} + 52 x^{3} + 67 x^{2} + 49 x + 64$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
|
\(8623\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $2$ | $3$ | $3$ | ||||
|
\(283573\)
| Deg $3$ | $3$ | $1$ | $2$ | |||
| Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |