Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 10*x^20 - 103*x^19 + 1448*x^18 + 2152*x^17 - 83871*x^16 + 149173*x^15 + 2385250*x^14 - 9948420*x^13 - 29032971*x^12 + 247870758*x^11 - 109823779*x^10 - 2879373801*x^9 + 7247306792*x^8 + 9672661015*x^7 - 71078825238*x^6 + 89876215729*x^5 + 141470849513*x^4 - 604358702025*x^3 + 843239380035*x^2 - 569139139934*x + 156657136133)
gp: K = bnfinit(y^21 - 10*y^20 - 103*y^19 + 1448*y^18 + 2152*y^17 - 83871*y^16 + 149173*y^15 + 2385250*y^14 - 9948420*y^13 - 29032971*y^12 + 247870758*y^11 - 109823779*y^10 - 2879373801*y^9 + 7247306792*y^8 + 9672661015*y^7 - 71078825238*y^6 + 89876215729*y^5 + 141470849513*y^4 - 604358702025*y^3 + 843239380035*y^2 - 569139139934*y + 156657136133, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 10*x^20 - 103*x^19 + 1448*x^18 + 2152*x^17 - 83871*x^16 + 149173*x^15 + 2385250*x^14 - 9948420*x^13 - 29032971*x^12 + 247870758*x^11 - 109823779*x^10 - 2879373801*x^9 + 7247306792*x^8 + 9672661015*x^7 - 71078825238*x^6 + 89876215729*x^5 + 141470849513*x^4 - 604358702025*x^3 + 843239380035*x^2 - 569139139934*x + 156657136133);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 10*x^20 - 103*x^19 + 1448*x^18 + 2152*x^17 - 83871*x^16 + 149173*x^15 + 2385250*x^14 - 9948420*x^13 - 29032971*x^12 + 247870758*x^11 - 109823779*x^10 - 2879373801*x^9 + 7247306792*x^8 + 9672661015*x^7 - 71078825238*x^6 + 89876215729*x^5 + 141470849513*x^4 - 604358702025*x^3 + 843239380035*x^2 - 569139139934*x + 156657136133)
\( x^{21} - 10 x^{20} - 103 x^{19} + 1448 x^{18} + 2152 x^{17} - 83871 x^{16} + 149173 x^{15} + \cdots + 156657136133 \)
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: | $[5, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(45907693464999604842301273030242456313152526209\)
\(\medspace = 3^{2}\cdot 73^{2}\cdot 1249^{2}\cdot 2741^{6}\cdot 3877^{2}\cdot 9811^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(166.72\) | 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: | $3^{1/2}73^{2/3}1249^{2/3}2741^{1/2}3877^{2/3}9811^{2/3}\approx 20776771425.4863$ | ||
| Ramified primes: |
\(3\), \(73\), \(1249\), \(2741\), \(3877\), \(9811\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{22\!\cdots\!99}a^{20}-\frac{51\!\cdots\!29}{22\!\cdots\!99}a^{19}-\frac{70\!\cdots\!87}{22\!\cdots\!99}a^{18}-\frac{21\!\cdots\!14}{22\!\cdots\!99}a^{17}-\frac{10\!\cdots\!22}{22\!\cdots\!99}a^{16}+\frac{72\!\cdots\!95}{22\!\cdots\!99}a^{15}+\frac{19\!\cdots\!52}{22\!\cdots\!99}a^{14}+\frac{34\!\cdots\!54}{22\!\cdots\!99}a^{13}+\frac{88\!\cdots\!18}{22\!\cdots\!99}a^{12}-\frac{23\!\cdots\!39}{22\!\cdots\!99}a^{11}-\frac{39\!\cdots\!81}{22\!\cdots\!99}a^{10}+\frac{70\!\cdots\!80}{22\!\cdots\!99}a^{9}+\frac{52\!\cdots\!53}{22\!\cdots\!99}a^{8}+\frac{54\!\cdots\!85}{22\!\cdots\!99}a^{7}-\frac{83\!\cdots\!85}{22\!\cdots\!99}a^{6}+\frac{37\!\cdots\!28}{22\!\cdots\!99}a^{5}+\frac{10\!\cdots\!48}{22\!\cdots\!99}a^{4}-\frac{87\!\cdots\!91}{22\!\cdots\!99}a^{3}-\frac{32\!\cdots\!29}{22\!\cdots\!99}a^{2}+\frac{74\!\cdots\!69}{22\!\cdots\!99}a+\frac{15\!\cdots\!11}{22\!\cdots\!99}$
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
$C_{3}$, which has order $3$ (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: | $12$ | 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: |
$\frac{14\!\cdots\!48}{22\!\cdots\!99}a^{20}-\frac{11\!\cdots\!93}{22\!\cdots\!99}a^{19}-\frac{17\!\cdots\!81}{22\!\cdots\!99}a^{18}+\frac{16\!\cdots\!11}{22\!\cdots\!99}a^{17}+\frac{67\!\cdots\!15}{22\!\cdots\!99}a^{16}-\frac{10\!\cdots\!21}{22\!\cdots\!99}a^{15}-\frac{18\!\cdots\!86}{22\!\cdots\!99}a^{14}+\frac{33\!\cdots\!47}{22\!\cdots\!99}a^{13}-\frac{67\!\cdots\!74}{22\!\cdots\!99}a^{12}-\frac{57\!\cdots\!00}{22\!\cdots\!99}a^{11}+\frac{22\!\cdots\!30}{22\!\cdots\!99}a^{10}+\frac{36\!\cdots\!05}{22\!\cdots\!99}a^{9}-\frac{33\!\cdots\!34}{22\!\cdots\!99}a^{8}+\frac{28\!\cdots\!06}{22\!\cdots\!99}a^{7}+\frac{20\!\cdots\!06}{22\!\cdots\!99}a^{6}-\frac{55\!\cdots\!46}{22\!\cdots\!99}a^{5}+\frac{83\!\cdots\!05}{22\!\cdots\!99}a^{4}+\frac{21\!\cdots\!01}{22\!\cdots\!99}a^{3}-\frac{39\!\cdots\!31}{22\!\cdots\!99}a^{2}+\frac{30\!\cdots\!68}{22\!\cdots\!99}a-\frac{95\!\cdots\!44}{22\!\cdots\!99}$, $\frac{33\!\cdots\!13}{22\!\cdots\!99}a^{20}-\frac{26\!\cdots\!57}{22\!\cdots\!99}a^{19}-\frac{39\!\cdots\!37}{22\!\cdots\!99}a^{18}+\frac{39\!\cdots\!74}{22\!\cdots\!99}a^{17}+\frac{15\!\cdots\!46}{22\!\cdots\!99}a^{16}-\frac{24\!\cdots\!30}{22\!\cdots\!99}a^{15}-\frac{37\!\cdots\!94}{22\!\cdots\!99}a^{14}+\frac{78\!\cdots\!18}{22\!\cdots\!99}a^{13}-\frac{15\!\cdots\!59}{22\!\cdots\!99}a^{12}-\frac{13\!\cdots\!57}{22\!\cdots\!99}a^{11}+\frac{53\!\cdots\!36}{22\!\cdots\!99}a^{10}+\frac{81\!\cdots\!93}{22\!\cdots\!99}a^{9}-\frac{78\!\cdots\!21}{22\!\cdots\!99}a^{8}+\frac{69\!\cdots\!77}{22\!\cdots\!99}a^{7}+\frac{47\!\cdots\!16}{22\!\cdots\!99}a^{6}-\frac{13\!\cdots\!86}{22\!\cdots\!99}a^{5}+\frac{82\!\cdots\!22}{22\!\cdots\!99}a^{4}+\frac{49\!\cdots\!23}{22\!\cdots\!99}a^{3}-\frac{93\!\cdots\!68}{22\!\cdots\!99}a^{2}+\frac{75\!\cdots\!21}{22\!\cdots\!99}a-\frac{23\!\cdots\!55}{22\!\cdots\!99}$, $\frac{20\!\cdots\!58}{22\!\cdots\!99}a^{20}-\frac{14\!\cdots\!19}{22\!\cdots\!99}a^{19}-\frac{25\!\cdots\!76}{22\!\cdots\!99}a^{18}+\frac{22\!\cdots\!20}{22\!\cdots\!99}a^{17}+\frac{10\!\cdots\!00}{22\!\cdots\!99}a^{16}-\frac{13\!\cdots\!85}{22\!\cdots\!99}a^{15}-\frac{10\!\cdots\!05}{22\!\cdots\!99}a^{14}+\frac{45\!\cdots\!93}{22\!\cdots\!99}a^{13}-\frac{70\!\cdots\!00}{22\!\cdots\!99}a^{12}-\frac{79\!\cdots\!26}{22\!\cdots\!99}a^{11}+\frac{27\!\cdots\!11}{22\!\cdots\!99}a^{10}+\frac{57\!\cdots\!34}{22\!\cdots\!99}a^{9}-\frac{42\!\cdots\!93}{22\!\cdots\!99}a^{8}+\frac{25\!\cdots\!18}{22\!\cdots\!99}a^{7}+\frac{27\!\cdots\!71}{22\!\cdots\!99}a^{6}-\frac{65\!\cdots\!77}{22\!\cdots\!99}a^{5}-\frac{94\!\cdots\!53}{22\!\cdots\!99}a^{4}+\frac{26\!\cdots\!36}{22\!\cdots\!99}a^{3}-\frac{46\!\cdots\!79}{22\!\cdots\!99}a^{2}+\frac{35\!\cdots\!67}{22\!\cdots\!99}a-\frac{10\!\cdots\!44}{22\!\cdots\!99}$, $\frac{30\!\cdots\!83}{22\!\cdots\!99}a^{20}-\frac{24\!\cdots\!91}{22\!\cdots\!99}a^{19}-\frac{36\!\cdots\!20}{22\!\cdots\!99}a^{18}+\frac{36\!\cdots\!88}{22\!\cdots\!99}a^{17}+\frac{14\!\cdots\!87}{22\!\cdots\!99}a^{16}-\frac{22\!\cdots\!18}{22\!\cdots\!99}a^{15}-\frac{14\!\cdots\!06}{22\!\cdots\!99}a^{14}+\frac{73\!\cdots\!37}{22\!\cdots\!99}a^{13}-\frac{15\!\cdots\!58}{22\!\cdots\!99}a^{12}-\frac{12\!\cdots\!97}{22\!\cdots\!99}a^{11}+\frac{50\!\cdots\!35}{22\!\cdots\!99}a^{10}+\frac{73\!\cdots\!79}{22\!\cdots\!99}a^{9}-\frac{73\!\cdots\!48}{22\!\cdots\!99}a^{8}+\frac{68\!\cdots\!24}{22\!\cdots\!99}a^{7}+\frac{44\!\cdots\!31}{22\!\cdots\!99}a^{6}-\frac{12\!\cdots\!99}{22\!\cdots\!99}a^{5}+\frac{99\!\cdots\!84}{22\!\cdots\!99}a^{4}+\frac{46\!\cdots\!72}{22\!\cdots\!99}a^{3}-\frac{88\!\cdots\!38}{22\!\cdots\!99}a^{2}+\frac{71\!\cdots\!08}{22\!\cdots\!99}a-\frac{22\!\cdots\!74}{22\!\cdots\!99}$, $\frac{46\!\cdots\!44}{22\!\cdots\!99}a^{20}-\frac{35\!\cdots\!09}{22\!\cdots\!99}a^{19}-\frac{56\!\cdots\!74}{22\!\cdots\!99}a^{18}+\frac{54\!\cdots\!34}{22\!\cdots\!99}a^{17}+\frac{22\!\cdots\!38}{22\!\cdots\!99}a^{16}-\frac{33\!\cdots\!57}{22\!\cdots\!99}a^{15}-\frac{92\!\cdots\!25}{22\!\cdots\!99}a^{14}+\frac{10\!\cdots\!67}{22\!\cdots\!99}a^{13}-\frac{20\!\cdots\!10}{22\!\cdots\!99}a^{12}-\frac{18\!\cdots\!02}{22\!\cdots\!99}a^{11}+\frac{72\!\cdots\!88}{22\!\cdots\!99}a^{10}+\frac{11\!\cdots\!30}{22\!\cdots\!99}a^{9}-\frac{10\!\cdots\!89}{22\!\cdots\!99}a^{8}+\frac{88\!\cdots\!19}{22\!\cdots\!99}a^{7}+\frac{65\!\cdots\!21}{22\!\cdots\!99}a^{6}-\frac{17\!\cdots\!74}{22\!\cdots\!99}a^{5}+\frac{42\!\cdots\!14}{22\!\cdots\!99}a^{4}+\frac{66\!\cdots\!00}{22\!\cdots\!99}a^{3}-\frac{12\!\cdots\!50}{22\!\cdots\!99}a^{2}+\frac{99\!\cdots\!33}{22\!\cdots\!99}a-\frac{31\!\cdots\!36}{22\!\cdots\!99}$, $\frac{58\!\cdots\!00}{22\!\cdots\!99}a^{20}-\frac{35\!\cdots\!06}{22\!\cdots\!99}a^{19}-\frac{73\!\cdots\!29}{22\!\cdots\!99}a^{18}+\frac{55\!\cdots\!80}{22\!\cdots\!99}a^{17}+\frac{34\!\cdots\!69}{22\!\cdots\!99}a^{16}-\frac{34\!\cdots\!58}{22\!\cdots\!99}a^{15}-\frac{53\!\cdots\!06}{22\!\cdots\!99}a^{14}+\frac{11\!\cdots\!95}{22\!\cdots\!99}a^{13}-\frac{10\!\cdots\!72}{22\!\cdots\!99}a^{12}-\frac{20\!\cdots\!06}{22\!\cdots\!99}a^{11}+\frac{55\!\cdots\!47}{22\!\cdots\!99}a^{10}+\frac{16\!\cdots\!57}{22\!\cdots\!99}a^{9}-\frac{91\!\cdots\!82}{22\!\cdots\!99}a^{8}+\frac{21\!\cdots\!67}{22\!\cdots\!99}a^{7}+\frac{62\!\cdots\!95}{22\!\cdots\!99}a^{6}-\frac{12\!\cdots\!95}{22\!\cdots\!99}a^{5}-\frac{50\!\cdots\!46}{22\!\cdots\!99}a^{4}+\frac{55\!\cdots\!39}{22\!\cdots\!99}a^{3}-\frac{90\!\cdots\!50}{22\!\cdots\!99}a^{2}+\frac{66\!\cdots\!17}{22\!\cdots\!99}a-\frac{19\!\cdots\!16}{22\!\cdots\!99}$, $\frac{41\!\cdots\!30}{22\!\cdots\!99}a^{20}-\frac{69\!\cdots\!33}{22\!\cdots\!99}a^{19}-\frac{18\!\cdots\!03}{22\!\cdots\!99}a^{18}+\frac{71\!\cdots\!85}{22\!\cdots\!99}a^{17}-\frac{35\!\cdots\!19}{22\!\cdots\!99}a^{16}-\frac{21\!\cdots\!37}{22\!\cdots\!99}a^{15}+\frac{22\!\cdots\!91}{22\!\cdots\!99}a^{14}-\frac{12\!\cdots\!61}{22\!\cdots\!99}a^{13}-\frac{55\!\cdots\!71}{22\!\cdots\!99}a^{12}+\frac{19\!\cdots\!10}{22\!\cdots\!99}a^{11}+\frac{41\!\cdots\!47}{22\!\cdots\!99}a^{10}-\frac{39\!\cdots\!02}{22\!\cdots\!99}a^{9}+\frac{52\!\cdots\!43}{22\!\cdots\!99}a^{8}+\frac{25\!\cdots\!80}{22\!\cdots\!99}a^{7}-\frac{10\!\cdots\!68}{22\!\cdots\!99}a^{6}+\frac{80\!\cdots\!76}{22\!\cdots\!99}a^{5}+\frac{32\!\cdots\!50}{22\!\cdots\!99}a^{4}-\frac{10\!\cdots\!19}{22\!\cdots\!99}a^{3}+\frac{13\!\cdots\!89}{22\!\cdots\!99}a^{2}-\frac{93\!\cdots\!78}{22\!\cdots\!99}a+\frac{25\!\cdots\!97}{22\!\cdots\!99}$, $\frac{11\!\cdots\!04}{22\!\cdots\!99}a^{20}-\frac{39\!\cdots\!19}{22\!\cdots\!99}a^{19}-\frac{15\!\cdots\!73}{22\!\cdots\!99}a^{18}+\frac{67\!\cdots\!38}{22\!\cdots\!99}a^{17}+\frac{81\!\cdots\!88}{22\!\cdots\!99}a^{16}-\frac{46\!\cdots\!70}{22\!\cdots\!99}a^{15}-\frac{20\!\cdots\!45}{22\!\cdots\!99}a^{14}+\frac{16\!\cdots\!63}{22\!\cdots\!99}a^{13}+\frac{16\!\cdots\!79}{22\!\cdots\!99}a^{12}-\frac{34\!\cdots\!38}{22\!\cdots\!99}a^{11}+\frac{34\!\cdots\!32}{22\!\cdots\!99}a^{10}+\frac{35\!\cdots\!77}{22\!\cdots\!99}a^{9}-\frac{98\!\cdots\!15}{22\!\cdots\!99}a^{8}-\frac{11\!\cdots\!94}{22\!\cdots\!99}a^{7}+\frac{86\!\cdots\!12}{22\!\cdots\!99}a^{6}-\frac{97\!\cdots\!43}{22\!\cdots\!99}a^{5}-\frac{18\!\cdots\!10}{22\!\cdots\!99}a^{4}+\frac{66\!\cdots\!29}{22\!\cdots\!99}a^{3}-\frac{82\!\cdots\!17}{22\!\cdots\!99}a^{2}+\frac{49\!\cdots\!53}{22\!\cdots\!99}a-\frac{11\!\cdots\!21}{22\!\cdots\!99}$, $\frac{49\!\cdots\!62}{22\!\cdots\!99}a^{20}-\frac{21\!\cdots\!76}{22\!\cdots\!99}a^{19}-\frac{59\!\cdots\!32}{22\!\cdots\!99}a^{18}+\frac{32\!\cdots\!62}{22\!\cdots\!99}a^{17}+\frac{27\!\cdots\!13}{22\!\cdots\!99}a^{16}-\frac{19\!\cdots\!39}{22\!\cdots\!99}a^{15}-\frac{56\!\cdots\!42}{22\!\cdots\!99}a^{14}+\frac{59\!\cdots\!83}{22\!\cdots\!99}a^{13}+\frac{12\!\cdots\!33}{22\!\cdots\!99}a^{12}-\frac{10\!\cdots\!11}{22\!\cdots\!99}a^{11}+\frac{14\!\cdots\!29}{22\!\cdots\!99}a^{10}+\frac{82\!\cdots\!66}{22\!\cdots\!99}a^{9}-\frac{25\!\cdots\!97}{22\!\cdots\!99}a^{8}-\frac{16\!\cdots\!33}{22\!\cdots\!99}a^{7}+\frac{15\!\cdots\!07}{22\!\cdots\!99}a^{6}-\frac{16\!\cdots\!59}{22\!\cdots\!99}a^{5}-\frac{22\!\cdots\!94}{22\!\cdots\!99}a^{4}+\frac{59\!\cdots\!33}{22\!\cdots\!99}a^{3}-\frac{34\!\cdots\!44}{22\!\cdots\!99}a^{2}-\frac{90\!\cdots\!82}{22\!\cdots\!99}a+\frac{11\!\cdots\!37}{22\!\cdots\!99}$, $\frac{79\!\cdots\!91}{22\!\cdots\!99}a^{20}-\frac{40\!\cdots\!96}{22\!\cdots\!99}a^{19}-\frac{11\!\cdots\!60}{22\!\cdots\!99}a^{18}+\frac{66\!\cdots\!36}{22\!\cdots\!99}a^{17}+\frac{61\!\cdots\!03}{22\!\cdots\!99}a^{16}-\frac{46\!\cdots\!27}{22\!\cdots\!99}a^{15}-\frac{15\!\cdots\!81}{22\!\cdots\!99}a^{14}+\frac{17\!\cdots\!21}{22\!\cdots\!99}a^{13}+\frac{12\!\cdots\!91}{22\!\cdots\!99}a^{12}-\frac{37\!\cdots\!74}{22\!\cdots\!99}a^{11}+\frac{35\!\cdots\!37}{22\!\cdots\!99}a^{10}+\frac{46\!\cdots\!28}{22\!\cdots\!99}a^{9}-\frac{10\!\cdots\!19}{22\!\cdots\!99}a^{8}-\frac{28\!\cdots\!59}{22\!\cdots\!99}a^{7}+\frac{11\!\cdots\!66}{22\!\cdots\!99}a^{6}+\frac{10\!\cdots\!21}{22\!\cdots\!99}a^{5}-\frac{56\!\cdots\!20}{22\!\cdots\!99}a^{4}+\frac{66\!\cdots\!84}{22\!\cdots\!99}a^{3}+\frac{62\!\cdots\!31}{22\!\cdots\!99}a^{2}-\frac{17\!\cdots\!77}{22\!\cdots\!99}a+\frac{94\!\cdots\!51}{22\!\cdots\!99}$, $\frac{49\!\cdots\!00}{22\!\cdots\!99}a^{20}-\frac{56\!\cdots\!72}{22\!\cdots\!99}a^{19}-\frac{36\!\cdots\!01}{22\!\cdots\!99}a^{18}+\frac{71\!\cdots\!11}{22\!\cdots\!99}a^{17}-\frac{37\!\cdots\!01}{22\!\cdots\!99}a^{16}-\frac{33\!\cdots\!06}{22\!\cdots\!99}a^{15}+\frac{11\!\cdots\!35}{22\!\cdots\!99}a^{14}+\frac{66\!\cdots\!57}{22\!\cdots\!99}a^{13}-\frac{45\!\cdots\!69}{22\!\cdots\!99}a^{12}-\frac{14\!\cdots\!15}{22\!\cdots\!99}a^{11}+\frac{76\!\cdots\!82}{22\!\cdots\!99}a^{10}-\frac{15\!\cdots\!04}{22\!\cdots\!99}a^{9}-\frac{44\!\cdots\!81}{22\!\cdots\!99}a^{8}+\frac{22\!\cdots\!44}{22\!\cdots\!99}a^{7}-\frac{18\!\cdots\!64}{22\!\cdots\!99}a^{6}-\frac{80\!\cdots\!03}{22\!\cdots\!99}a^{5}+\frac{23\!\cdots\!27}{22\!\cdots\!99}a^{4}-\frac{23\!\cdots\!48}{22\!\cdots\!99}a^{3}+\frac{32\!\cdots\!35}{22\!\cdots\!99}a^{2}+\frac{10\!\cdots\!19}{22\!\cdots\!99}a-\frac{57\!\cdots\!61}{22\!\cdots\!99}$, $\frac{98\!\cdots\!13}{22\!\cdots\!99}a^{20}-\frac{59\!\cdots\!11}{22\!\cdots\!99}a^{19}-\frac{12\!\cdots\!21}{22\!\cdots\!99}a^{18}+\frac{93\!\cdots\!32}{22\!\cdots\!99}a^{17}+\frac{58\!\cdots\!22}{22\!\cdots\!99}a^{16}-\frac{59\!\cdots\!06}{22\!\cdots\!99}a^{15}-\frac{89\!\cdots\!48}{22\!\cdots\!99}a^{14}+\frac{20\!\cdots\!87}{22\!\cdots\!99}a^{13}-\frac{18\!\cdots\!88}{22\!\cdots\!99}a^{12}-\frac{36\!\cdots\!96}{22\!\cdots\!99}a^{11}+\frac{10\!\cdots\!13}{22\!\cdots\!99}a^{10}+\frac{29\!\cdots\!06}{22\!\cdots\!99}a^{9}-\frac{16\!\cdots\!92}{22\!\cdots\!99}a^{8}+\frac{48\!\cdots\!20}{22\!\cdots\!99}a^{7}+\frac{11\!\cdots\!21}{22\!\cdots\!99}a^{6}-\frac{24\!\cdots\!34}{22\!\cdots\!99}a^{5}-\frac{86\!\cdots\!01}{22\!\cdots\!99}a^{4}+\frac{10\!\cdots\!32}{22\!\cdots\!99}a^{3}-\frac{17\!\cdots\!57}{22\!\cdots\!99}a^{2}+\frac{13\!\cdots\!71}{22\!\cdots\!99}a-\frac{38\!\cdots\!04}{22\!\cdots\!99}$
| 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: | \( 830894950455000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{8}\cdot 830894950455000 \cdot 3}{2\cdot\sqrt{45907693464999604842301273030242456313152526209}}\cr\approx \mathstrut & 0.452151074614942 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 10*x^20 - 103*x^19 + 1448*x^18 + 2152*x^17 - 83871*x^16 + 149173*x^15 + 2385250*x^14 - 9948420*x^13 - 29032971*x^12 + 247870758*x^11 - 109823779*x^10 - 2879373801*x^9 + 7247306792*x^8 + 9672661015*x^7 - 71078825238*x^6 + 89876215729*x^5 + 141470849513*x^4 - 604358702025*x^3 + 843239380035*x^2 - 569139139934*x + 156657136133)
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 - 10*x^20 - 103*x^19 + 1448*x^18 + 2152*x^17 - 83871*x^16 + 149173*x^15 + 2385250*x^14 - 9948420*x^13 - 29032971*x^12 + 247870758*x^11 - 109823779*x^10 - 2879373801*x^9 + 7247306792*x^8 + 9672661015*x^7 - 71078825238*x^6 + 89876215729*x^5 + 141470849513*x^4 - 604358702025*x^3 + 843239380035*x^2 - 569139139934*x + 156657136133, 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 - 10*x^20 - 103*x^19 + 1448*x^18 + 2152*x^17 - 83871*x^16 + 149173*x^15 + 2385250*x^14 - 9948420*x^13 - 29032971*x^12 + 247870758*x^11 - 109823779*x^10 - 2879373801*x^9 + 7247306792*x^8 + 9672661015*x^7 - 71078825238*x^6 + 89876215729*x^5 + 141470849513*x^4 - 604358702025*x^3 + 843239380035*x^2 - 569139139934*x + 156657136133);
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 - 10*x^20 - 103*x^19 + 1448*x^18 + 2152*x^17 - 83871*x^16 + 149173*x^15 + 2385250*x^14 - 9948420*x^13 - 29032971*x^12 + 247870758*x^11 - 109823779*x^10 - 2879373801*x^9 + 7247306792*x^8 + 9672661015*x^7 - 71078825238*x^6 + 89876215729*x^5 + 141470849513*x^4 - 604358702025*x^3 + 843239380035*x^2 - 569139139934*x + 156657136133);
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
$C_3^7.C_2^6:\GL(3,2)$ (as 21T145):
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 23514624 |
| The 132 conjugacy class representatives for $C_3^7.C_2^6:\GL(3,2)$ |
| Character table for $C_3^7.C_2^6:\GL(3,2)$ |
Intermediate fields
| 7.3.7513081.1 |
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 | $21$ | R | $21$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $21$ | ${\href{/padicField/13.7.0.1}{7} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.7.0.1}{7} }^{3}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }^{3}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $21$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.8.0.1 | $x^{8} + 2 x^{5} + x^{4} + 2 x^{2} + 2 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(73\)
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.3.2.3 | $x^{3} + 146$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
|
\(1249\)
| $\Q_{1249}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1249}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1249}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1249}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1249}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1249}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1249}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1249}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $3$ | $1$ | $2$ | ||||
|
\(2741\)
| $\Q_{2741}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2741}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{2741}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{2741}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{2741}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{2741}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $8$ | $2$ | $4$ | $4$ | ||||
|
\(3877\)
| $\Q_{3877}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $3$ | $1$ | $2$ | ||||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
|
\(9811\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $3$ | $1$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |