Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 + 2*x^19 - 4*x^18 + 10*x^17 - 9*x^16 + 70*x^15 - 167*x^14 + 91*x^13 + 110*x^12 - 252*x^11 + 50*x^10 + 456*x^9 - 519*x^8 + x^7 + 487*x^6 - 548*x^5 + 272*x^4 - 48*x^3 + x^2 - 3*x + 1)
gp: K = bnfinit(y^21 - 3*y^20 + 2*y^19 - 4*y^18 + 10*y^17 - 9*y^16 + 70*y^15 - 167*y^14 + 91*y^13 + 110*y^12 - 252*y^11 + 50*y^10 + 456*y^9 - 519*y^8 + y^7 + 487*y^6 - 548*y^5 + 272*y^4 - 48*y^3 + y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 3*x^20 + 2*x^19 - 4*x^18 + 10*x^17 - 9*x^16 + 70*x^15 - 167*x^14 + 91*x^13 + 110*x^12 - 252*x^11 + 50*x^10 + 456*x^9 - 519*x^8 + x^7 + 487*x^6 - 548*x^5 + 272*x^4 - 48*x^3 + x^2 - 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 3*x^20 + 2*x^19 - 4*x^18 + 10*x^17 - 9*x^16 + 70*x^15 - 167*x^14 + 91*x^13 + 110*x^12 - 252*x^11 + 50*x^10 + 456*x^9 - 519*x^8 + x^7 + 487*x^6 - 548*x^5 + 272*x^4 - 48*x^3 + x^2 - 3*x + 1)
\( x^{21} - 3 x^{20} + 2 x^{19} - 4 x^{18} + 10 x^{17} - 9 x^{16} + 70 x^{15} - 167 x^{14} + 91 x^{13} + \cdots + 1 \)
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: |
\(-188012010050973537326115059\)
\(\medspace = -\,59^{8}\cdot 10859^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.83\) | 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: | $59^{1/2}10859^{1/2}\approx 800.4255118372977$ | ||
| Ramified primes: |
\(59\), \(10859\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-10859}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{53\!\cdots\!19}a^{20}-\frac{17\!\cdots\!57}{53\!\cdots\!19}a^{19}+\frac{96\!\cdots\!51}{53\!\cdots\!19}a^{18}+\frac{16\!\cdots\!88}{53\!\cdots\!19}a^{17}-\frac{32\!\cdots\!60}{53\!\cdots\!19}a^{16}+\frac{46\!\cdots\!54}{53\!\cdots\!19}a^{15}-\frac{13\!\cdots\!42}{31\!\cdots\!07}a^{14}+\frac{18\!\cdots\!75}{53\!\cdots\!19}a^{13}-\frac{21\!\cdots\!63}{53\!\cdots\!19}a^{12}-\frac{92\!\cdots\!93}{53\!\cdots\!19}a^{11}+\frac{69\!\cdots\!23}{53\!\cdots\!19}a^{10}-\frac{15\!\cdots\!86}{31\!\cdots\!07}a^{9}+\frac{62\!\cdots\!26}{17\!\cdots\!49}a^{8}-\frac{65\!\cdots\!49}{53\!\cdots\!19}a^{7}+\frac{82\!\cdots\!44}{53\!\cdots\!19}a^{6}+\frac{10\!\cdots\!86}{53\!\cdots\!19}a^{5}-\frac{21\!\cdots\!75}{53\!\cdots\!19}a^{4}-\frac{76\!\cdots\!01}{53\!\cdots\!19}a^{3}-\frac{24\!\cdots\!66}{53\!\cdots\!19}a^{2}+\frac{11\!\cdots\!25}{53\!\cdots\!19}a-\frac{39\!\cdots\!96}{53\!\cdots\!19}$
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: |
$\frac{31\!\cdots\!82}{53\!\cdots\!19}a^{20}-\frac{64\!\cdots\!48}{53\!\cdots\!19}a^{19}-\frac{93\!\cdots\!02}{53\!\cdots\!19}a^{18}-\frac{10\!\cdots\!43}{53\!\cdots\!19}a^{17}+\frac{20\!\cdots\!26}{53\!\cdots\!19}a^{16}-\frac{45\!\cdots\!41}{53\!\cdots\!19}a^{15}+\frac{12\!\cdots\!78}{31\!\cdots\!07}a^{14}-\frac{32\!\cdots\!97}{53\!\cdots\!19}a^{13}-\frac{96\!\cdots\!23}{53\!\cdots\!19}a^{12}+\frac{39\!\cdots\!13}{53\!\cdots\!19}a^{11}-\frac{43\!\cdots\!25}{53\!\cdots\!19}a^{10}-\frac{22\!\cdots\!20}{31\!\cdots\!07}a^{9}+\frac{41\!\cdots\!56}{17\!\cdots\!49}a^{8}-\frac{36\!\cdots\!23}{53\!\cdots\!19}a^{7}-\frac{80\!\cdots\!96}{53\!\cdots\!19}a^{6}+\frac{10\!\cdots\!93}{53\!\cdots\!19}a^{5}-\frac{55\!\cdots\!92}{53\!\cdots\!19}a^{4}-\frac{86\!\cdots\!59}{53\!\cdots\!19}a^{3}+\frac{10\!\cdots\!33}{53\!\cdots\!19}a^{2}+\frac{15\!\cdots\!30}{53\!\cdots\!19}a-\frac{21\!\cdots\!09}{53\!\cdots\!19}$, $\frac{90\!\cdots\!93}{53\!\cdots\!19}a^{20}-\frac{27\!\cdots\!20}{53\!\cdots\!19}a^{19}+\frac{21\!\cdots\!11}{53\!\cdots\!19}a^{18}-\frac{39\!\cdots\!47}{53\!\cdots\!19}a^{17}+\frac{91\!\cdots\!01}{53\!\cdots\!19}a^{16}-\frac{95\!\cdots\!51}{53\!\cdots\!19}a^{15}+\frac{37\!\cdots\!53}{31\!\cdots\!07}a^{14}-\frac{15\!\cdots\!31}{53\!\cdots\!19}a^{13}+\frac{10\!\cdots\!18}{53\!\cdots\!19}a^{12}+\frac{84\!\cdots\!69}{53\!\cdots\!19}a^{11}-\frac{24\!\cdots\!95}{53\!\cdots\!19}a^{10}+\frac{42\!\cdots\!63}{31\!\cdots\!07}a^{9}+\frac{12\!\cdots\!79}{17\!\cdots\!49}a^{8}-\frac{52\!\cdots\!10}{53\!\cdots\!19}a^{7}+\frac{81\!\cdots\!92}{53\!\cdots\!19}a^{6}+\frac{45\!\cdots\!88}{53\!\cdots\!19}a^{5}-\frac{55\!\cdots\!52}{53\!\cdots\!19}a^{4}+\frac{31\!\cdots\!37}{53\!\cdots\!19}a^{3}-\frac{71\!\cdots\!55}{53\!\cdots\!19}a^{2}-\frac{52\!\cdots\!09}{53\!\cdots\!19}a-\frac{60\!\cdots\!28}{53\!\cdots\!19}$, $\frac{51\!\cdots\!17}{53\!\cdots\!19}a^{20}-\frac{11\!\cdots\!13}{53\!\cdots\!19}a^{19}+\frac{12\!\cdots\!27}{53\!\cdots\!19}a^{18}-\frac{19\!\cdots\!48}{53\!\cdots\!19}a^{17}+\frac{36\!\cdots\!77}{53\!\cdots\!19}a^{16}-\frac{17\!\cdots\!80}{53\!\cdots\!19}a^{15}+\frac{20\!\cdots\!64}{31\!\cdots\!07}a^{14}-\frac{59\!\cdots\!73}{53\!\cdots\!19}a^{13}-\frac{19\!\cdots\!71}{53\!\cdots\!19}a^{12}+\frac{58\!\cdots\!14}{53\!\cdots\!19}a^{11}-\frac{84\!\cdots\!76}{53\!\cdots\!19}a^{10}-\frac{24\!\cdots\!10}{31\!\cdots\!07}a^{9}+\frac{66\!\cdots\!45}{17\!\cdots\!49}a^{8}-\frac{10\!\cdots\!33}{53\!\cdots\!19}a^{7}-\frac{90\!\cdots\!34}{53\!\cdots\!19}a^{6}+\frac{18\!\cdots\!69}{53\!\cdots\!19}a^{5}-\frac{13\!\cdots\!79}{53\!\cdots\!19}a^{4}+\frac{30\!\cdots\!44}{53\!\cdots\!19}a^{3}+\frac{45\!\cdots\!63}{53\!\cdots\!19}a^{2}+\frac{25\!\cdots\!08}{53\!\cdots\!19}a-\frac{15\!\cdots\!21}{53\!\cdots\!19}$, $\frac{68\!\cdots\!40}{53\!\cdots\!19}a^{20}-\frac{15\!\cdots\!03}{53\!\cdots\!19}a^{19}+\frac{22\!\cdots\!67}{53\!\cdots\!19}a^{18}-\frac{26\!\cdots\!33}{53\!\cdots\!19}a^{17}+\frac{49\!\cdots\!52}{53\!\cdots\!19}a^{16}-\frac{25\!\cdots\!83}{53\!\cdots\!19}a^{15}+\frac{27\!\cdots\!60}{31\!\cdots\!07}a^{14}-\frac{80\!\cdots\!92}{53\!\cdots\!19}a^{13}+\frac{30\!\cdots\!67}{53\!\cdots\!19}a^{12}+\frac{75\!\cdots\!29}{53\!\cdots\!19}a^{11}-\frac{11\!\cdots\!66}{53\!\cdots\!19}a^{10}-\frac{29\!\cdots\!28}{31\!\cdots\!07}a^{9}+\frac{87\!\cdots\!70}{17\!\cdots\!49}a^{8}-\frac{15\!\cdots\!65}{53\!\cdots\!19}a^{7}-\frac{10\!\cdots\!93}{53\!\cdots\!19}a^{6}+\frac{24\!\cdots\!46}{53\!\cdots\!19}a^{5}-\frac{19\!\cdots\!51}{53\!\cdots\!19}a^{4}+\frac{52\!\cdots\!01}{53\!\cdots\!19}a^{3}-\frac{30\!\cdots\!76}{53\!\cdots\!19}a^{2}+\frac{11\!\cdots\!03}{53\!\cdots\!19}a+\frac{51\!\cdots\!88}{53\!\cdots\!19}$, $\frac{40\!\cdots\!13}{88\!\cdots\!19}a^{20}-\frac{10\!\cdots\!69}{88\!\cdots\!19}a^{19}+\frac{38\!\cdots\!76}{88\!\cdots\!19}a^{18}-\frac{14\!\cdots\!33}{88\!\cdots\!19}a^{17}+\frac{34\!\cdots\!25}{88\!\cdots\!19}a^{16}-\frac{22\!\cdots\!90}{88\!\cdots\!19}a^{15}+\frac{15\!\cdots\!54}{52\!\cdots\!07}a^{14}-\frac{56\!\cdots\!78}{88\!\cdots\!19}a^{13}+\frac{14\!\cdots\!55}{88\!\cdots\!19}a^{12}+\frac{50\!\cdots\!93}{88\!\cdots\!19}a^{11}-\frac{81\!\cdots\!77}{88\!\cdots\!19}a^{10}-\frac{74\!\cdots\!66}{52\!\cdots\!07}a^{9}+\frac{57\!\cdots\!00}{28\!\cdots\!49}a^{8}-\frac{13\!\cdots\!90}{88\!\cdots\!19}a^{7}-\frac{56\!\cdots\!94}{88\!\cdots\!19}a^{6}+\frac{17\!\cdots\!62}{88\!\cdots\!19}a^{5}-\frac{15\!\cdots\!25}{88\!\cdots\!19}a^{4}+\frac{47\!\cdots\!05}{88\!\cdots\!19}a^{3}+\frac{15\!\cdots\!25}{88\!\cdots\!19}a^{2}+\frac{23\!\cdots\!53}{88\!\cdots\!19}a-\frac{10\!\cdots\!08}{88\!\cdots\!19}$, $\frac{16\!\cdots\!46}{53\!\cdots\!19}a^{20}-\frac{45\!\cdots\!39}{53\!\cdots\!19}a^{19}+\frac{20\!\cdots\!39}{53\!\cdots\!19}a^{18}-\frac{58\!\cdots\!46}{53\!\cdots\!19}a^{17}+\frac{15\!\cdots\!33}{53\!\cdots\!19}a^{16}-\frac{10\!\cdots\!49}{53\!\cdots\!19}a^{15}+\frac{65\!\cdots\!52}{31\!\cdots\!07}a^{14}-\frac{24\!\cdots\!97}{53\!\cdots\!19}a^{13}+\frac{80\!\cdots\!20}{53\!\cdots\!19}a^{12}+\frac{21\!\cdots\!08}{53\!\cdots\!19}a^{11}-\frac{35\!\cdots\!71}{53\!\cdots\!19}a^{10}-\frac{11\!\cdots\!64}{31\!\cdots\!07}a^{9}+\frac{24\!\cdots\!42}{17\!\cdots\!49}a^{8}-\frac{64\!\cdots\!83}{53\!\cdots\!19}a^{7}-\frac{19\!\cdots\!79}{53\!\cdots\!19}a^{6}+\frac{76\!\cdots\!43}{53\!\cdots\!19}a^{5}-\frac{68\!\cdots\!98}{53\!\cdots\!19}a^{4}+\frac{24\!\cdots\!95}{53\!\cdots\!19}a^{3}-\frac{15\!\cdots\!70}{53\!\cdots\!19}a^{2}+\frac{72\!\cdots\!42}{53\!\cdots\!19}a-\frac{50\!\cdots\!26}{53\!\cdots\!19}$, $\frac{28\!\cdots\!91}{53\!\cdots\!19}a^{20}-\frac{84\!\cdots\!09}{53\!\cdots\!19}a^{19}+\frac{45\!\cdots\!06}{53\!\cdots\!19}a^{18}-\frac{93\!\cdots\!67}{53\!\cdots\!19}a^{17}+\frac{28\!\cdots\!39}{53\!\cdots\!19}a^{16}-\frac{20\!\cdots\!42}{53\!\cdots\!19}a^{15}+\frac{11\!\cdots\!73}{31\!\cdots\!07}a^{14}-\frac{46\!\cdots\!72}{53\!\cdots\!19}a^{13}+\frac{18\!\cdots\!60}{53\!\cdots\!19}a^{12}+\frac{39\!\cdots\!80}{53\!\cdots\!19}a^{11}-\frac{67\!\cdots\!39}{53\!\cdots\!19}a^{10}+\frac{25\!\cdots\!06}{31\!\cdots\!07}a^{9}+\frac{45\!\cdots\!28}{17\!\cdots\!49}a^{8}-\frac{13\!\cdots\!49}{53\!\cdots\!19}a^{7}-\frac{33\!\cdots\!88}{53\!\cdots\!19}a^{6}+\frac{14\!\cdots\!60}{53\!\cdots\!19}a^{5}-\frac{13\!\cdots\!82}{53\!\cdots\!19}a^{4}+\frac{48\!\cdots\!56}{53\!\cdots\!19}a^{3}+\frac{14\!\cdots\!63}{53\!\cdots\!19}a^{2}-\frac{19\!\cdots\!05}{53\!\cdots\!19}a-\frac{13\!\cdots\!14}{53\!\cdots\!19}$, $\frac{49\!\cdots\!43}{53\!\cdots\!19}a^{20}-\frac{12\!\cdots\!78}{53\!\cdots\!19}a^{19}+\frac{40\!\cdots\!14}{53\!\cdots\!19}a^{18}-\frac{19\!\cdots\!29}{53\!\cdots\!19}a^{17}+\frac{39\!\cdots\!04}{53\!\cdots\!19}a^{16}-\frac{25\!\cdots\!30}{53\!\cdots\!19}a^{15}+\frac{19\!\cdots\!07}{31\!\cdots\!07}a^{14}-\frac{65\!\cdots\!94}{53\!\cdots\!19}a^{13}+\frac{14\!\cdots\!04}{53\!\cdots\!19}a^{12}+\frac{56\!\cdots\!41}{53\!\cdots\!19}a^{11}-\frac{97\!\cdots\!18}{53\!\cdots\!19}a^{10}-\frac{11\!\cdots\!61}{31\!\cdots\!07}a^{9}+\frac{67\!\cdots\!44}{17\!\cdots\!49}a^{8}-\frac{15\!\cdots\!32}{53\!\cdots\!19}a^{7}-\frac{58\!\cdots\!81}{53\!\cdots\!19}a^{6}+\frac{20\!\cdots\!18}{53\!\cdots\!19}a^{5}-\frac{17\!\cdots\!67}{53\!\cdots\!19}a^{4}+\frac{60\!\cdots\!20}{53\!\cdots\!19}a^{3}-\frac{38\!\cdots\!48}{53\!\cdots\!19}a^{2}-\frac{10\!\cdots\!17}{53\!\cdots\!19}a-\frac{10\!\cdots\!07}{53\!\cdots\!19}$, $\frac{15\!\cdots\!29}{53\!\cdots\!19}a^{20}-\frac{40\!\cdots\!13}{53\!\cdots\!19}a^{19}+\frac{15\!\cdots\!33}{53\!\cdots\!19}a^{18}-\frac{54\!\cdots\!38}{53\!\cdots\!19}a^{17}+\frac{13\!\cdots\!89}{53\!\cdots\!19}a^{16}-\frac{89\!\cdots\!06}{53\!\cdots\!19}a^{15}+\frac{60\!\cdots\!48}{31\!\cdots\!07}a^{14}-\frac{21\!\cdots\!53}{53\!\cdots\!19}a^{13}+\frac{59\!\cdots\!61}{53\!\cdots\!19}a^{12}+\frac{19\!\cdots\!53}{53\!\cdots\!19}a^{11}-\frac{31\!\cdots\!15}{53\!\cdots\!19}a^{10}-\frac{22\!\cdots\!88}{31\!\cdots\!07}a^{9}+\frac{22\!\cdots\!32}{17\!\cdots\!49}a^{8}-\frac{54\!\cdots\!26}{53\!\cdots\!19}a^{7}-\frac{20\!\cdots\!41}{53\!\cdots\!19}a^{6}+\frac{67\!\cdots\!61}{53\!\cdots\!19}a^{5}-\frac{59\!\cdots\!76}{53\!\cdots\!19}a^{4}+\frac{19\!\cdots\!81}{53\!\cdots\!19}a^{3}+\frac{28\!\cdots\!28}{53\!\cdots\!19}a^{2}+\frac{35\!\cdots\!26}{53\!\cdots\!19}a-\frac{46\!\cdots\!87}{53\!\cdots\!19}$, $\frac{43\!\cdots\!19}{53\!\cdots\!19}a^{20}-\frac{98\!\cdots\!09}{53\!\cdots\!19}a^{19}+\frac{17\!\cdots\!67}{53\!\cdots\!19}a^{18}-\frac{17\!\cdots\!93}{53\!\cdots\!19}a^{17}+\frac{31\!\cdots\!08}{53\!\cdots\!19}a^{16}-\frac{17\!\cdots\!96}{53\!\cdots\!19}a^{15}+\frac{17\!\cdots\!19}{31\!\cdots\!07}a^{14}-\frac{51\!\cdots\!28}{53\!\cdots\!19}a^{13}+\frac{42\!\cdots\!05}{53\!\cdots\!19}a^{12}+\frac{45\!\cdots\!63}{53\!\cdots\!19}a^{11}-\frac{73\!\cdots\!05}{53\!\cdots\!19}a^{10}-\frac{16\!\cdots\!43}{31\!\cdots\!07}a^{9}+\frac{54\!\cdots\!60}{17\!\cdots\!49}a^{8}-\frac{10\!\cdots\!75}{53\!\cdots\!19}a^{7}-\frac{57\!\cdots\!97}{53\!\cdots\!19}a^{6}+\frac{15\!\cdots\!95}{53\!\cdots\!19}a^{5}-\frac{12\!\cdots\!66}{53\!\cdots\!19}a^{4}+\frac{41\!\cdots\!91}{53\!\cdots\!19}a^{3}-\frac{80\!\cdots\!81}{53\!\cdots\!19}a^{2}+\frac{30\!\cdots\!37}{53\!\cdots\!19}a-\frac{12\!\cdots\!73}{53\!\cdots\!19}$, $\frac{13\!\cdots\!55}{53\!\cdots\!19}a^{20}-\frac{36\!\cdots\!30}{53\!\cdots\!19}a^{19}+\frac{15\!\cdots\!33}{53\!\cdots\!19}a^{18}-\frac{49\!\cdots\!79}{53\!\cdots\!19}a^{17}+\frac{12\!\cdots\!94}{53\!\cdots\!19}a^{16}-\frac{85\!\cdots\!07}{53\!\cdots\!19}a^{15}+\frac{54\!\cdots\!79}{31\!\cdots\!07}a^{14}-\frac{19\!\cdots\!25}{53\!\cdots\!19}a^{13}+\frac{62\!\cdots\!60}{53\!\cdots\!19}a^{12}+\frac{17\!\cdots\!28}{53\!\cdots\!19}a^{11}-\frac{29\!\cdots\!32}{53\!\cdots\!19}a^{10}-\frac{13\!\cdots\!70}{31\!\cdots\!07}a^{9}+\frac{19\!\cdots\!14}{17\!\cdots\!49}a^{8}-\frac{51\!\cdots\!34}{53\!\cdots\!19}a^{7}-\frac{16\!\cdots\!91}{53\!\cdots\!19}a^{6}+\frac{61\!\cdots\!83}{53\!\cdots\!19}a^{5}-\frac{55\!\cdots\!16}{53\!\cdots\!19}a^{4}+\frac{19\!\cdots\!39}{53\!\cdots\!19}a^{3}-\frac{30\!\cdots\!33}{53\!\cdots\!19}a^{2}+\frac{92\!\cdots\!47}{53\!\cdots\!19}a-\frac{36\!\cdots\!12}{53\!\cdots\!19}$
| 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: | \( 47990.3401891 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 47990.3401891 \cdot 1}{2\cdot\sqrt{188012010050973537326115059}}\cr\approx \mathstrut & 0.213668174125 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 + 2*x^19 - 4*x^18 + 10*x^17 - 9*x^16 + 70*x^15 - 167*x^14 + 91*x^13 + 110*x^12 - 252*x^11 + 50*x^10 + 456*x^9 - 519*x^8 + x^7 + 487*x^6 - 548*x^5 + 272*x^4 - 48*x^3 + x^2 - 3*x + 1)
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 - 3*x^20 + 2*x^19 - 4*x^18 + 10*x^17 - 9*x^16 + 70*x^15 - 167*x^14 + 91*x^13 + 110*x^12 - 252*x^11 + 50*x^10 + 456*x^9 - 519*x^8 + x^7 + 487*x^6 - 548*x^5 + 272*x^4 - 48*x^3 + x^2 - 3*x + 1, 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 - 3*x^20 + 2*x^19 - 4*x^18 + 10*x^17 - 9*x^16 + 70*x^15 - 167*x^14 + 91*x^13 + 110*x^12 - 252*x^11 + 50*x^10 + 456*x^9 - 519*x^8 + x^7 + 487*x^6 - 548*x^5 + 272*x^4 - 48*x^3 + x^2 - 3*x + 1);
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 - 3*x^20 + 2*x^19 - 4*x^18 + 10*x^17 - 9*x^16 + 70*x^15 - 167*x^14 + 91*x^13 + 110*x^12 - 252*x^11 + 50*x^10 + 456*x^9 - 519*x^8 + x^7 + 487*x^6 - 548*x^5 + 272*x^4 - 48*x^3 + x^2 - 3*x + 1);
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
$S_3\times S_7$ (as 21T74):
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 30240 |
| The 45 conjugacy class representatives for $S_3\times S_7$ |
| Character table for $S_3\times S_7$ is not computed |
Intermediate fields
| 3.1.59.1, 7.3.640681.1 |
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 | ${\href{/padicField/2.14.0.1}{14} }{,}\,{\href{/padicField/2.7.0.1}{7} }$ | $21$ | $21$ | $15{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.5.0.1}{5} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{6}$ | $21$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{9}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $21$ | R |
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 |
|---|---|---|---|---|---|---|---|
|
\(59\)
| 59.2.1.1 | $x^{2} + 118$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 59.4.2.1 | $x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 59.5.0.1 | $x^{5} + 8 x + 57$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 59.10.5.2 | $x^{10} + 295 x^{8} + 34826 x^{6} + 114 x^{5} + 2049070 x^{4} - 67260 x^{3} + 60308389 x^{2} + 1985082 x + 718217388$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(10859\)
| 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |