Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 7*x^19 + 6*x^18 + 3*x^17 - 30*x^16 + 72*x^15 + 134*x^14 - 49*x^13 - 195*x^12 - 375*x^11 + 190*x^10 + 878*x^9 - 233*x^8 - 491*x^7 + 177*x^6 + 32*x^5 - 47*x^4 + 21*x^3 + 10*x^2 - 4*x - 1)
gp: K = bnfinit(y^21 - y^20 - 7*y^19 + 6*y^18 + 3*y^17 - 30*y^16 + 72*y^15 + 134*y^14 - 49*y^13 - 195*y^12 - 375*y^11 + 190*y^10 + 878*y^9 - 233*y^8 - 491*y^7 + 177*y^6 + 32*y^5 - 47*y^4 + 21*y^3 + 10*y^2 - 4*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - x^20 - 7*x^19 + 6*x^18 + 3*x^17 - 30*x^16 + 72*x^15 + 134*x^14 - 49*x^13 - 195*x^12 - 375*x^11 + 190*x^10 + 878*x^9 - 233*x^8 - 491*x^7 + 177*x^6 + 32*x^5 - 47*x^4 + 21*x^3 + 10*x^2 - 4*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - x^20 - 7*x^19 + 6*x^18 + 3*x^17 - 30*x^16 + 72*x^15 + 134*x^14 - 49*x^13 - 195*x^12 - 375*x^11 + 190*x^10 + 878*x^9 - 233*x^8 - 491*x^7 + 177*x^6 + 32*x^5 - 47*x^4 + 21*x^3 + 10*x^2 - 4*x - 1)
\( x^{21} - x^{20} - 7 x^{19} + 6 x^{18} + 3 x^{17} - 30 x^{16} + 72 x^{15} + 134 x^{14} - 49 x^{13} + \cdots - 1 \)
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: |
\(-31095511042786085990206459319\)
\(\medspace = -\,7^{14}\cdot 71^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.74\) | 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: | $7^{2/3}71^{1/2}\approx 30.833857978493015$ | ||
| Ramified primes: |
\(7\), \(71\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-71}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{13}a^{19}+\frac{3}{13}a^{18}-\frac{4}{13}a^{17}+\frac{2}{13}a^{16}-\frac{5}{13}a^{15}-\frac{3}{13}a^{14}+\frac{1}{13}a^{13}-\frac{4}{13}a^{12}+\frac{4}{13}a^{11}+\frac{5}{13}a^{9}+\frac{2}{13}a^{8}-\frac{4}{13}a^{7}+\frac{6}{13}a^{6}-\frac{2}{13}a^{5}-\frac{2}{13}a^{4}+\frac{3}{13}a^{3}-\frac{4}{13}a^{2}+\frac{4}{13}a-\frac{3}{13}$, $\frac{1}{32\!\cdots\!43}a^{20}-\frac{40\!\cdots\!63}{32\!\cdots\!43}a^{19}-\frac{63\!\cdots\!42}{32\!\cdots\!43}a^{18}-\frac{14\!\cdots\!63}{32\!\cdots\!43}a^{17}-\frac{12\!\cdots\!41}{32\!\cdots\!43}a^{16}-\frac{11\!\cdots\!31}{32\!\cdots\!43}a^{15}-\frac{33\!\cdots\!45}{32\!\cdots\!43}a^{14}+\frac{58\!\cdots\!94}{32\!\cdots\!43}a^{13}+\frac{97\!\cdots\!24}{32\!\cdots\!43}a^{12}+\frac{37\!\cdots\!11}{24\!\cdots\!11}a^{11}-\frac{56\!\cdots\!57}{32\!\cdots\!43}a^{10}-\frac{13\!\cdots\!67}{32\!\cdots\!43}a^{9}-\frac{12\!\cdots\!17}{32\!\cdots\!43}a^{8}+\frac{15\!\cdots\!58}{32\!\cdots\!43}a^{7}+\frac{13\!\cdots\!54}{32\!\cdots\!43}a^{6}+\frac{13\!\cdots\!14}{32\!\cdots\!43}a^{5}-\frac{13\!\cdots\!31}{32\!\cdots\!43}a^{4}+\frac{13\!\cdots\!34}{32\!\cdots\!43}a^{3}+\frac{74\!\cdots\!90}{32\!\cdots\!43}a^{2}+\frac{38\!\cdots\!33}{32\!\cdots\!43}a-\frac{19\!\cdots\!17}{24\!\cdots\!11}$
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: |
$\frac{57\!\cdots\!09}{32\!\cdots\!43}a^{20}-\frac{22\!\cdots\!65}{32\!\cdots\!43}a^{19}-\frac{41\!\cdots\!12}{32\!\cdots\!43}a^{18}+\frac{93\!\cdots\!59}{32\!\cdots\!43}a^{17}+\frac{24\!\cdots\!16}{32\!\cdots\!43}a^{16}-\frac{15\!\cdots\!68}{32\!\cdots\!43}a^{15}+\frac{31\!\cdots\!16}{32\!\cdots\!43}a^{14}+\frac{96\!\cdots\!15}{32\!\cdots\!43}a^{13}+\frac{28\!\cdots\!31}{32\!\cdots\!43}a^{12}-\frac{99\!\cdots\!61}{32\!\cdots\!43}a^{11}-\frac{27\!\cdots\!32}{32\!\cdots\!43}a^{10}-\frac{51\!\cdots\!18}{32\!\cdots\!43}a^{9}+\frac{48\!\cdots\!55}{32\!\cdots\!43}a^{8}+\frac{16\!\cdots\!62}{32\!\cdots\!43}a^{7}-\frac{20\!\cdots\!59}{32\!\cdots\!43}a^{6}-\frac{35\!\cdots\!30}{32\!\cdots\!43}a^{5}+\frac{11\!\cdots\!11}{32\!\cdots\!43}a^{4}-\frac{18\!\cdots\!01}{32\!\cdots\!43}a^{3}-\frac{21\!\cdots\!40}{32\!\cdots\!43}a^{2}+\frac{66\!\cdots\!68}{32\!\cdots\!43}a+\frac{16\!\cdots\!77}{32\!\cdots\!43}$, $\frac{65\!\cdots\!81}{32\!\cdots\!43}a^{20}-\frac{13\!\cdots\!48}{32\!\cdots\!43}a^{19}-\frac{48\!\cdots\!66}{32\!\cdots\!43}a^{18}+\frac{19\!\cdots\!60}{32\!\cdots\!43}a^{17}+\frac{31\!\cdots\!07}{32\!\cdots\!43}a^{16}-\frac{17\!\cdots\!90}{32\!\cdots\!43}a^{15}+\frac{32\!\cdots\!11}{32\!\cdots\!43}a^{14}+\frac{11\!\cdots\!26}{32\!\cdots\!43}a^{13}+\frac{51\!\cdots\!18}{32\!\cdots\!43}a^{12}-\frac{10\!\cdots\!62}{32\!\cdots\!43}a^{11}-\frac{33\!\cdots\!89}{32\!\cdots\!43}a^{10}-\frac{11\!\cdots\!97}{32\!\cdots\!43}a^{9}+\frac{54\!\cdots\!59}{32\!\cdots\!43}a^{8}+\frac{27\!\cdots\!01}{32\!\cdots\!43}a^{7}-\frac{22\!\cdots\!77}{32\!\cdots\!43}a^{6}-\frac{70\!\cdots\!08}{32\!\cdots\!43}a^{5}+\frac{21\!\cdots\!95}{32\!\cdots\!43}a^{4}-\frac{24\!\cdots\!00}{32\!\cdots\!43}a^{3}-\frac{57\!\cdots\!42}{32\!\cdots\!43}a^{2}+\frac{84\!\cdots\!84}{32\!\cdots\!43}a+\frac{20\!\cdots\!05}{32\!\cdots\!43}$, $\frac{11\!\cdots\!87}{32\!\cdots\!43}a^{20}-\frac{41\!\cdots\!38}{32\!\cdots\!43}a^{19}-\frac{87\!\cdots\!95}{32\!\cdots\!43}a^{18}+\frac{11\!\cdots\!00}{24\!\cdots\!11}a^{17}+\frac{49\!\cdots\!11}{32\!\cdots\!43}a^{16}-\frac{32\!\cdots\!81}{32\!\cdots\!43}a^{15}+\frac{64\!\cdots\!17}{32\!\cdots\!43}a^{14}+\frac{20\!\cdots\!24}{32\!\cdots\!43}a^{13}+\frac{71\!\cdots\!65}{32\!\cdots\!43}a^{12}-\frac{19\!\cdots\!62}{32\!\cdots\!43}a^{11}-\frac{57\!\cdots\!51}{32\!\cdots\!43}a^{10}-\frac{14\!\cdots\!15}{32\!\cdots\!43}a^{9}+\frac{98\!\cdots\!35}{32\!\cdots\!43}a^{8}+\frac{36\!\cdots\!47}{32\!\cdots\!43}a^{7}-\frac{38\!\cdots\!46}{32\!\cdots\!43}a^{6}-\frac{57\!\cdots\!00}{32\!\cdots\!43}a^{5}+\frac{18\!\cdots\!54}{32\!\cdots\!43}a^{4}-\frac{42\!\cdots\!08}{32\!\cdots\!43}a^{3}-\frac{35\!\cdots\!23}{32\!\cdots\!43}a^{2}+\frac{11\!\cdots\!74}{32\!\cdots\!43}a+\frac{28\!\cdots\!75}{32\!\cdots\!43}$, $\frac{63\!\cdots\!63}{32\!\cdots\!43}a^{20}-\frac{27\!\cdots\!83}{32\!\cdots\!43}a^{19}-\frac{45\!\cdots\!18}{32\!\cdots\!43}a^{18}+\frac{11\!\cdots\!45}{32\!\cdots\!43}a^{17}+\frac{24\!\cdots\!75}{32\!\cdots\!43}a^{16}-\frac{17\!\cdots\!12}{32\!\cdots\!43}a^{15}+\frac{35\!\cdots\!51}{32\!\cdots\!43}a^{14}+\frac{10\!\cdots\!63}{32\!\cdots\!43}a^{13}+\frac{29\!\cdots\!17}{32\!\cdots\!43}a^{12}-\frac{10\!\cdots\!61}{32\!\cdots\!43}a^{11}-\frac{29\!\cdots\!55}{32\!\cdots\!43}a^{10}-\frac{51\!\cdots\!30}{32\!\cdots\!43}a^{9}+\frac{52\!\cdots\!35}{32\!\cdots\!43}a^{8}+\frac{15\!\cdots\!64}{32\!\cdots\!43}a^{7}-\frac{21\!\cdots\!67}{32\!\cdots\!43}a^{6}-\frac{14\!\cdots\!64}{32\!\cdots\!43}a^{5}+\frac{10\!\cdots\!45}{32\!\cdots\!43}a^{4}-\frac{22\!\cdots\!30}{32\!\cdots\!43}a^{3}-\frac{16\!\cdots\!10}{32\!\cdots\!43}a^{2}+\frac{61\!\cdots\!78}{32\!\cdots\!43}a+\frac{80\!\cdots\!22}{32\!\cdots\!43}$, $\frac{78\!\cdots\!49}{32\!\cdots\!43}a^{20}-\frac{31\!\cdots\!86}{32\!\cdots\!43}a^{19}-\frac{56\!\cdots\!22}{32\!\cdots\!43}a^{18}+\frac{13\!\cdots\!50}{32\!\cdots\!43}a^{17}+\frac{31\!\cdots\!29}{32\!\cdots\!43}a^{16}-\frac{21\!\cdots\!47}{32\!\cdots\!43}a^{15}+\frac{43\!\cdots\!66}{32\!\cdots\!43}a^{14}+\frac{13\!\cdots\!28}{32\!\cdots\!43}a^{13}+\frac{38\!\cdots\!03}{32\!\cdots\!43}a^{12}-\frac{13\!\cdots\!26}{32\!\cdots\!43}a^{11}-\frac{36\!\cdots\!45}{32\!\cdots\!43}a^{10}-\frac{69\!\cdots\!51}{32\!\cdots\!43}a^{9}+\frac{64\!\cdots\!18}{32\!\cdots\!43}a^{8}+\frac{19\!\cdots\!71}{32\!\cdots\!43}a^{7}-\frac{26\!\cdots\!99}{32\!\cdots\!43}a^{6}-\frac{19\!\cdots\!30}{32\!\cdots\!43}a^{5}+\frac{14\!\cdots\!81}{32\!\cdots\!43}a^{4}-\frac{28\!\cdots\!50}{32\!\cdots\!43}a^{3}-\frac{37\!\cdots\!96}{32\!\cdots\!43}a^{2}+\frac{76\!\cdots\!23}{32\!\cdots\!43}a+\frac{11\!\cdots\!87}{32\!\cdots\!43}$, $\frac{57\!\cdots\!68}{32\!\cdots\!43}a^{20}-\frac{40\!\cdots\!68}{32\!\cdots\!43}a^{19}-\frac{42\!\cdots\!83}{32\!\cdots\!43}a^{18}-\frac{42\!\cdots\!74}{32\!\cdots\!43}a^{17}+\frac{28\!\cdots\!39}{32\!\cdots\!43}a^{16}-\frac{15\!\cdots\!89}{32\!\cdots\!43}a^{15}+\frac{26\!\cdots\!52}{32\!\cdots\!43}a^{14}+\frac{10\!\cdots\!63}{32\!\cdots\!43}a^{13}+\frac{59\!\cdots\!43}{32\!\cdots\!43}a^{12}-\frac{90\!\cdots\!95}{32\!\cdots\!43}a^{11}-\frac{30\!\cdots\!95}{32\!\cdots\!43}a^{10}-\frac{13\!\cdots\!23}{32\!\cdots\!43}a^{9}+\frac{46\!\cdots\!01}{32\!\cdots\!43}a^{8}+\frac{31\!\cdots\!58}{32\!\cdots\!43}a^{7}-\frac{16\!\cdots\!25}{32\!\cdots\!43}a^{6}-\frac{91\!\cdots\!15}{32\!\cdots\!43}a^{5}+\frac{16\!\cdots\!67}{32\!\cdots\!43}a^{4}-\frac{16\!\cdots\!77}{32\!\cdots\!43}a^{3}-\frac{98\!\cdots\!30}{32\!\cdots\!43}a^{2}+\frac{72\!\cdots\!77}{32\!\cdots\!43}a+\frac{29\!\cdots\!81}{32\!\cdots\!43}$, $\frac{13\!\cdots\!10}{32\!\cdots\!43}a^{20}-\frac{40\!\cdots\!60}{32\!\cdots\!43}a^{19}-\frac{98\!\cdots\!34}{32\!\cdots\!43}a^{18}+\frac{12\!\cdots\!20}{32\!\cdots\!43}a^{17}+\frac{53\!\cdots\!22}{32\!\cdots\!43}a^{16}-\frac{37\!\cdots\!97}{32\!\cdots\!43}a^{15}+\frac{71\!\cdots\!93}{32\!\cdots\!43}a^{14}+\frac{23\!\cdots\!38}{32\!\cdots\!43}a^{13}+\frac{93\!\cdots\!39}{32\!\cdots\!43}a^{12}-\frac{20\!\cdots\!05}{32\!\cdots\!43}a^{11}-\frac{65\!\cdots\!42}{32\!\cdots\!43}a^{10}-\frac{19\!\cdots\!33}{32\!\cdots\!43}a^{9}+\frac{10\!\cdots\!38}{32\!\cdots\!43}a^{8}+\frac{44\!\cdots\!41}{32\!\cdots\!43}a^{7}-\frac{39\!\cdots\!68}{32\!\cdots\!43}a^{6}-\frac{42\!\cdots\!34}{32\!\cdots\!43}a^{5}+\frac{25\!\cdots\!47}{32\!\cdots\!43}a^{4}-\frac{52\!\cdots\!49}{32\!\cdots\!43}a^{3}-\frac{41\!\cdots\!92}{32\!\cdots\!43}a^{2}+\frac{11\!\cdots\!81}{32\!\cdots\!43}a+\frac{20\!\cdots\!02}{32\!\cdots\!43}$, $\frac{22\!\cdots\!33}{32\!\cdots\!43}a^{20}-\frac{90\!\cdots\!40}{32\!\cdots\!43}a^{19}-\frac{16\!\cdots\!26}{32\!\cdots\!43}a^{18}+\frac{37\!\cdots\!31}{32\!\cdots\!43}a^{17}+\frac{94\!\cdots\!64}{32\!\cdots\!43}a^{16}-\frac{63\!\cdots\!21}{32\!\cdots\!43}a^{15}+\frac{12\!\cdots\!55}{32\!\cdots\!43}a^{14}+\frac{38\!\cdots\!70}{32\!\cdots\!43}a^{13}+\frac{11\!\cdots\!20}{32\!\cdots\!43}a^{12}-\frac{38\!\cdots\!27}{32\!\cdots\!43}a^{11}-\frac{10\!\cdots\!69}{32\!\cdots\!43}a^{10}-\frac{22\!\cdots\!40}{32\!\cdots\!43}a^{9}+\frac{19\!\cdots\!78}{32\!\cdots\!43}a^{8}+\frac{62\!\cdots\!77}{32\!\cdots\!43}a^{7}-\frac{78\!\cdots\!45}{32\!\cdots\!43}a^{6}-\frac{83\!\cdots\!34}{32\!\cdots\!43}a^{5}+\frac{36\!\cdots\!31}{32\!\cdots\!43}a^{4}-\frac{80\!\cdots\!55}{32\!\cdots\!43}a^{3}-\frac{71\!\cdots\!29}{32\!\cdots\!43}a^{2}+\frac{22\!\cdots\!85}{32\!\cdots\!43}a+\frac{43\!\cdots\!75}{32\!\cdots\!43}$, $\frac{22\!\cdots\!33}{32\!\cdots\!43}a^{20}-\frac{90\!\cdots\!40}{32\!\cdots\!43}a^{19}-\frac{16\!\cdots\!26}{32\!\cdots\!43}a^{18}+\frac{37\!\cdots\!31}{32\!\cdots\!43}a^{17}+\frac{94\!\cdots\!64}{32\!\cdots\!43}a^{16}-\frac{63\!\cdots\!21}{32\!\cdots\!43}a^{15}+\frac{12\!\cdots\!55}{32\!\cdots\!43}a^{14}+\frac{38\!\cdots\!70}{32\!\cdots\!43}a^{13}+\frac{11\!\cdots\!20}{32\!\cdots\!43}a^{12}-\frac{38\!\cdots\!27}{32\!\cdots\!43}a^{11}-\frac{10\!\cdots\!69}{32\!\cdots\!43}a^{10}-\frac{22\!\cdots\!40}{32\!\cdots\!43}a^{9}+\frac{19\!\cdots\!78}{32\!\cdots\!43}a^{8}+\frac{62\!\cdots\!77}{32\!\cdots\!43}a^{7}-\frac{78\!\cdots\!45}{32\!\cdots\!43}a^{6}-\frac{83\!\cdots\!34}{32\!\cdots\!43}a^{5}+\frac{36\!\cdots\!31}{32\!\cdots\!43}a^{4}-\frac{80\!\cdots\!55}{32\!\cdots\!43}a^{3}-\frac{71\!\cdots\!29}{32\!\cdots\!43}a^{2}+\frac{22\!\cdots\!85}{32\!\cdots\!43}a+\frac{40\!\cdots\!32}{32\!\cdots\!43}$, $\frac{19\!\cdots\!85}{32\!\cdots\!43}a^{20}-\frac{12\!\cdots\!38}{32\!\cdots\!43}a^{19}-\frac{13\!\cdots\!12}{32\!\cdots\!43}a^{18}+\frac{66\!\cdots\!81}{32\!\cdots\!43}a^{17}+\frac{62\!\cdots\!31}{32\!\cdots\!43}a^{16}-\frac{55\!\cdots\!27}{32\!\cdots\!43}a^{15}+\frac{12\!\cdots\!34}{32\!\cdots\!43}a^{14}+\frac{29\!\cdots\!86}{32\!\cdots\!43}a^{13}+\frac{25\!\cdots\!11}{32\!\cdots\!43}a^{12}-\frac{32\!\cdots\!80}{32\!\cdots\!43}a^{11}-\frac{84\!\cdots\!77}{32\!\cdots\!43}a^{10}+\frac{23\!\cdots\!77}{32\!\cdots\!43}a^{9}+\frac{15\!\cdots\!20}{32\!\cdots\!43}a^{8}+\frac{11\!\cdots\!54}{32\!\cdots\!43}a^{7}-\frac{68\!\cdots\!97}{32\!\cdots\!43}a^{6}+\frac{89\!\cdots\!29}{24\!\cdots\!11}a^{5}-\frac{63\!\cdots\!67}{32\!\cdots\!43}a^{4}-\frac{70\!\cdots\!79}{32\!\cdots\!43}a^{3}+\frac{19\!\cdots\!50}{32\!\cdots\!43}a^{2}+\frac{12\!\cdots\!31}{32\!\cdots\!43}a+\frac{13\!\cdots\!46}{32\!\cdots\!43}$, $\frac{89\!\cdots\!45}{32\!\cdots\!43}a^{20}+\frac{24\!\cdots\!80}{32\!\cdots\!43}a^{19}-\frac{81\!\cdots\!50}{32\!\cdots\!43}a^{18}-\frac{14\!\cdots\!34}{24\!\cdots\!11}a^{17}+\frac{11\!\cdots\!91}{32\!\cdots\!43}a^{16}-\frac{15\!\cdots\!53}{32\!\cdots\!43}a^{15}-\frac{30\!\cdots\!05}{32\!\cdots\!43}a^{14}+\frac{32\!\cdots\!47}{32\!\cdots\!43}a^{13}+\frac{48\!\cdots\!67}{32\!\cdots\!43}a^{12}-\frac{80\!\cdots\!83}{32\!\cdots\!43}a^{11}-\frac{89\!\cdots\!77}{32\!\cdots\!43}a^{10}-\frac{13\!\cdots\!13}{32\!\cdots\!43}a^{9}+\frac{69\!\cdots\!63}{32\!\cdots\!43}a^{8}+\frac{25\!\cdots\!46}{32\!\cdots\!43}a^{7}+\frac{11\!\cdots\!39}{32\!\cdots\!43}a^{6}-\frac{10\!\cdots\!97}{32\!\cdots\!43}a^{5}+\frac{14\!\cdots\!57}{32\!\cdots\!43}a^{4}-\frac{17\!\cdots\!90}{32\!\cdots\!43}a^{3}-\frac{11\!\cdots\!21}{32\!\cdots\!43}a^{2}+\frac{23\!\cdots\!99}{32\!\cdots\!43}a+\frac{23\!\cdots\!53}{32\!\cdots\!43}$
| 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: | \( 725840.889095 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 725840.889095 \cdot 1}{2\cdot\sqrt{31095511042786085990206459319}}\cr\approx \mathstrut & 0.251287790113 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 7*x^19 + 6*x^18 + 3*x^17 - 30*x^16 + 72*x^15 + 134*x^14 - 49*x^13 - 195*x^12 - 375*x^11 + 190*x^10 + 878*x^9 - 233*x^8 - 491*x^7 + 177*x^6 + 32*x^5 - 47*x^4 + 21*x^3 + 10*x^2 - 4*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 - x^20 - 7*x^19 + 6*x^18 + 3*x^17 - 30*x^16 + 72*x^15 + 134*x^14 - 49*x^13 - 195*x^12 - 375*x^11 + 190*x^10 + 878*x^9 - 233*x^8 - 491*x^7 + 177*x^6 + 32*x^5 - 47*x^4 + 21*x^3 + 10*x^2 - 4*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 - x^20 - 7*x^19 + 6*x^18 + 3*x^17 - 30*x^16 + 72*x^15 + 134*x^14 - 49*x^13 - 195*x^12 - 375*x^11 + 190*x^10 + 878*x^9 - 233*x^8 - 491*x^7 + 177*x^6 + 32*x^5 - 47*x^4 + 21*x^3 + 10*x^2 - 4*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 - x^20 - 7*x^19 + 6*x^18 + 3*x^17 - 30*x^16 + 72*x^15 + 134*x^14 - 49*x^13 - 195*x^12 - 375*x^11 + 190*x^10 + 878*x^9 - 233*x^8 - 491*x^7 + 177*x^6 + 32*x^5 - 47*x^4 + 21*x^3 + 10*x^2 - 4*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
$C_3\times D_7$ (as 21T3):
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 42 |
| The 15 conjugacy class representatives for $C_3\times D_7$ |
| Character table for $C_3\times D_7$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 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
| Galois closure: | deg 42 |
| 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 | $21$ | $21$ | $21$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.2.0.1}{2} }^{9}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $21$ | ${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.7.0.1}{7} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $21$ | ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.7.0.1}{7} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(71\)
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |