Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 7*x^19 + 55*x^18 - 132*x^17 - 112*x^16 + 589*x^15 - 188*x^14 - 1078*x^13 + 1042*x^12 + 579*x^11 - 1297*x^10 + 411*x^9 + 431*x^8 - 127*x^7 - 455*x^6 - 126*x^5 + 888*x^4 - 460*x^3 - 146*x^2 + 127*x - 1)
gp: K = bnfinit(y^21 - 7*y^20 + 7*y^19 + 55*y^18 - 132*y^17 - 112*y^16 + 589*y^15 - 188*y^14 - 1078*y^13 + 1042*y^12 + 579*y^11 - 1297*y^10 + 411*y^9 + 431*y^8 - 127*y^7 - 455*y^6 - 126*y^5 + 888*y^4 - 460*y^3 - 146*y^2 + 127*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 7*x^20 + 7*x^19 + 55*x^18 - 132*x^17 - 112*x^16 + 589*x^15 - 188*x^14 - 1078*x^13 + 1042*x^12 + 579*x^11 - 1297*x^10 + 411*x^9 + 431*x^8 - 127*x^7 - 455*x^6 - 126*x^5 + 888*x^4 - 460*x^3 - 146*x^2 + 127*x - 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 7*x^20 + 7*x^19 + 55*x^18 - 132*x^17 - 112*x^16 + 589*x^15 - 188*x^14 - 1078*x^13 + 1042*x^12 + 579*x^11 - 1297*x^10 + 411*x^9 + 431*x^8 - 127*x^7 - 455*x^6 - 126*x^5 + 888*x^4 - 460*x^3 - 146*x^2 + 127*x - 1)
\( x^{21} - 7 x^{20} + 7 x^{19} + 55 x^{18} - 132 x^{17} - 112 x^{16} + 589 x^{15} - 188 x^{14} - 1078 x^{13} + \cdots - 1 \)
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: | $[9, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(13686230803976276974088591656489\)
\(\medspace = 17^{10}\cdot 19^{2}\cdot 61^{2}\cdot 131^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.39\) | 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: | $17^{5/6}19^{2/3}61^{2/3}131^{2/3}\approx 30172.450876657273$ | ||
| Ramified primes: |
\(17\), \(19\), \(61\), \(131\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{10\!\cdots\!91}a^{20}-\frac{44\!\cdots\!39}{10\!\cdots\!91}a^{19}-\frac{50\!\cdots\!45}{10\!\cdots\!91}a^{18}+\frac{11\!\cdots\!69}{10\!\cdots\!91}a^{17}+\frac{25\!\cdots\!00}{10\!\cdots\!91}a^{16}-\frac{27\!\cdots\!01}{10\!\cdots\!91}a^{15}+\frac{43\!\cdots\!47}{10\!\cdots\!91}a^{14}+\frac{32\!\cdots\!57}{10\!\cdots\!91}a^{13}-\frac{22\!\cdots\!53}{10\!\cdots\!91}a^{12}+\frac{20\!\cdots\!45}{10\!\cdots\!91}a^{11}-\frac{73\!\cdots\!31}{10\!\cdots\!91}a^{10}-\frac{36\!\cdots\!40}{10\!\cdots\!91}a^{9}+\frac{37\!\cdots\!47}{10\!\cdots\!91}a^{8}+\frac{93\!\cdots\!25}{10\!\cdots\!91}a^{7}+\frac{43\!\cdots\!30}{10\!\cdots\!91}a^{6}+\frac{41\!\cdots\!49}{10\!\cdots\!91}a^{5}-\frac{37\!\cdots\!00}{10\!\cdots\!91}a^{4}-\frac{17\!\cdots\!28}{10\!\cdots\!91}a^{3}-\frac{18\!\cdots\!46}{10\!\cdots\!91}a^{2}-\frac{39\!\cdots\!03}{10\!\cdots\!91}a-\frac{19\!\cdots\!94}{10\!\cdots\!91}$
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: |
$\frac{49\!\cdots\!01}{10\!\cdots\!91}a^{20}-\frac{30\!\cdots\!00}{10\!\cdots\!91}a^{19}+\frac{12\!\cdots\!29}{10\!\cdots\!91}a^{18}+\frac{26\!\cdots\!77}{10\!\cdots\!91}a^{17}-\frac{42\!\cdots\!59}{10\!\cdots\!91}a^{16}-\frac{78\!\cdots\!54}{10\!\cdots\!91}a^{15}+\frac{20\!\cdots\!74}{10\!\cdots\!91}a^{14}+\frac{52\!\cdots\!70}{10\!\cdots\!91}a^{13}-\frac{40\!\cdots\!60}{10\!\cdots\!91}a^{12}+\frac{16\!\cdots\!56}{10\!\cdots\!91}a^{11}+\frac{30\!\cdots\!02}{10\!\cdots\!91}a^{10}-\frac{30\!\cdots\!92}{10\!\cdots\!91}a^{9}-\frac{39\!\cdots\!66}{10\!\cdots\!91}a^{8}+\frac{15\!\cdots\!66}{10\!\cdots\!91}a^{7}+\frac{60\!\cdots\!93}{10\!\cdots\!91}a^{6}-\frac{15\!\cdots\!38}{10\!\cdots\!91}a^{5}-\frac{14\!\cdots\!37}{10\!\cdots\!91}a^{4}+\frac{23\!\cdots\!33}{10\!\cdots\!91}a^{3}-\frac{39\!\cdots\!55}{10\!\cdots\!91}a^{2}-\frac{56\!\cdots\!12}{10\!\cdots\!91}a+\frac{19\!\cdots\!81}{10\!\cdots\!91}$, $\frac{67\!\cdots\!81}{10\!\cdots\!91}a^{20}-\frac{42\!\cdots\!38}{10\!\cdots\!91}a^{19}+\frac{21\!\cdots\!95}{10\!\cdots\!91}a^{18}+\frac{36\!\cdots\!09}{10\!\cdots\!91}a^{17}-\frac{62\!\cdots\!45}{10\!\cdots\!91}a^{16}-\frac{10\!\cdots\!08}{10\!\cdots\!91}a^{15}+\frac{29\!\cdots\!98}{10\!\cdots\!91}a^{14}+\frac{32\!\cdots\!14}{10\!\cdots\!91}a^{13}-\frac{57\!\cdots\!18}{10\!\cdots\!91}a^{12}+\frac{31\!\cdots\!34}{10\!\cdots\!91}a^{11}+\frac{38\!\cdots\!48}{10\!\cdots\!91}a^{10}-\frac{48\!\cdots\!78}{10\!\cdots\!91}a^{9}+\frac{67\!\cdots\!92}{10\!\cdots\!91}a^{8}+\frac{16\!\cdots\!60}{10\!\cdots\!91}a^{7}+\frac{62\!\cdots\!19}{10\!\cdots\!91}a^{6}-\frac{21\!\cdots\!36}{10\!\cdots\!91}a^{5}-\frac{18\!\cdots\!58}{10\!\cdots\!91}a^{4}+\frac{38\!\cdots\!17}{10\!\cdots\!91}a^{3}-\frac{10\!\cdots\!50}{10\!\cdots\!91}a^{2}-\frac{49\!\cdots\!69}{10\!\cdots\!91}a+\frac{92\!\cdots\!38}{10\!\cdots\!91}$, $\frac{80\!\cdots\!58}{10\!\cdots\!91}a^{20}-\frac{48\!\cdots\!88}{10\!\cdots\!91}a^{19}+\frac{11\!\cdots\!40}{10\!\cdots\!91}a^{18}+\frac{44\!\cdots\!15}{10\!\cdots\!91}a^{17}-\frac{63\!\cdots\!42}{10\!\cdots\!91}a^{16}-\frac{14\!\cdots\!58}{10\!\cdots\!91}a^{15}+\frac{32\!\cdots\!49}{10\!\cdots\!91}a^{14}+\frac{15\!\cdots\!14}{10\!\cdots\!91}a^{13}-\frac{68\!\cdots\!31}{10\!\cdots\!91}a^{12}+\frac{15\!\cdots\!50}{10\!\cdots\!91}a^{11}+\frac{58\!\cdots\!06}{10\!\cdots\!91}a^{10}-\frac{42\!\cdots\!19}{10\!\cdots\!91}a^{9}-\frac{99\!\cdots\!03}{10\!\cdots\!91}a^{8}+\frac{22\!\cdots\!06}{10\!\cdots\!91}a^{7}+\frac{13\!\cdots\!51}{10\!\cdots\!91}a^{6}-\frac{23\!\cdots\!45}{10\!\cdots\!91}a^{5}-\frac{30\!\cdots\!56}{10\!\cdots\!91}a^{4}+\frac{37\!\cdots\!49}{10\!\cdots\!91}a^{3}+\frac{18\!\cdots\!86}{10\!\cdots\!91}a^{2}-\frac{99\!\cdots\!98}{10\!\cdots\!91}a+\frac{46\!\cdots\!69}{10\!\cdots\!91}$, $\frac{46\!\cdots\!21}{10\!\cdots\!91}a^{20}-\frac{29\!\cdots\!62}{10\!\cdots\!91}a^{19}+\frac{14\!\cdots\!65}{10\!\cdots\!91}a^{18}+\frac{26\!\cdots\!18}{10\!\cdots\!91}a^{17}-\frac{44\!\cdots\!51}{10\!\cdots\!91}a^{16}-\frac{80\!\cdots\!90}{10\!\cdots\!91}a^{15}+\frac{22\!\cdots\!67}{10\!\cdots\!91}a^{14}+\frac{58\!\cdots\!44}{10\!\cdots\!91}a^{13}-\frac{46\!\cdots\!15}{10\!\cdots\!91}a^{12}+\frac{16\!\cdots\!24}{10\!\cdots\!91}a^{11}+\frac{39\!\cdots\!64}{10\!\cdots\!91}a^{10}-\frac{32\!\cdots\!67}{10\!\cdots\!91}a^{9}-\frac{50\!\cdots\!20}{10\!\cdots\!91}a^{8}+\frac{16\!\cdots\!90}{10\!\cdots\!91}a^{7}+\frac{56\!\cdots\!66}{10\!\cdots\!91}a^{6}-\frac{17\!\cdots\!73}{10\!\cdots\!91}a^{5}-\frac{17\!\cdots\!19}{10\!\cdots\!91}a^{4}+\frac{29\!\cdots\!92}{10\!\cdots\!91}a^{3}+\frac{71\!\cdots\!51}{10\!\cdots\!91}a^{2}-\frac{75\!\cdots\!14}{10\!\cdots\!91}a-\frac{40\!\cdots\!26}{10\!\cdots\!91}$, $\frac{16\!\cdots\!78}{10\!\cdots\!91}a^{20}-\frac{84\!\cdots\!94}{10\!\cdots\!91}a^{19}-\frac{51\!\cdots\!76}{10\!\cdots\!91}a^{18}+\frac{86\!\cdots\!13}{10\!\cdots\!91}a^{17}-\frac{52\!\cdots\!30}{10\!\cdots\!91}a^{16}-\frac{35\!\cdots\!68}{10\!\cdots\!91}a^{15}+\frac{35\!\cdots\!93}{10\!\cdots\!91}a^{14}+\frac{68\!\cdots\!38}{10\!\cdots\!91}a^{13}-\frac{82\!\cdots\!96}{10\!\cdots\!91}a^{12}-\frac{61\!\cdots\!70}{10\!\cdots\!91}a^{11}+\frac{78\!\cdots\!71}{10\!\cdots\!91}a^{10}+\frac{18\!\cdots\!90}{10\!\cdots\!91}a^{9}-\frac{37\!\cdots\!07}{10\!\cdots\!91}a^{8}-\frac{18\!\cdots\!68}{10\!\cdots\!91}a^{7}+\frac{68\!\cdots\!99}{10\!\cdots\!91}a^{6}-\frac{13\!\cdots\!92}{10\!\cdots\!91}a^{5}-\frac{90\!\cdots\!27}{10\!\cdots\!91}a^{4}+\frac{69\!\cdots\!14}{10\!\cdots\!91}a^{3}+\frac{29\!\cdots\!31}{10\!\cdots\!91}a^{2}+\frac{22\!\cdots\!45}{10\!\cdots\!91}a-\frac{14\!\cdots\!96}{10\!\cdots\!91}$, $\frac{17\!\cdots\!59}{10\!\cdots\!91}a^{20}-\frac{10\!\cdots\!32}{10\!\cdots\!91}a^{19}+\frac{21\!\cdots\!51}{10\!\cdots\!91}a^{18}+\frac{93\!\cdots\!84}{10\!\cdots\!91}a^{17}-\frac{12\!\cdots\!61}{10\!\cdots\!91}a^{16}-\frac{29\!\cdots\!00}{10\!\cdots\!91}a^{15}+\frac{64\!\cdots\!89}{10\!\cdots\!91}a^{14}+\frac{24\!\cdots\!32}{10\!\cdots\!91}a^{13}-\frac{13\!\cdots\!20}{10\!\cdots\!91}a^{12}+\frac{51\!\cdots\!96}{10\!\cdots\!91}a^{11}+\frac{10\!\cdots\!71}{10\!\cdots\!91}a^{10}-\frac{11\!\cdots\!42}{10\!\cdots\!91}a^{9}-\frac{12\!\cdots\!76}{10\!\cdots\!91}a^{8}+\frac{63\!\cdots\!84}{10\!\cdots\!91}a^{7}+\frac{12\!\cdots\!86}{10\!\cdots\!91}a^{6}-\frac{43\!\cdots\!80}{10\!\cdots\!91}a^{5}-\frac{37\!\cdots\!10}{10\!\cdots\!91}a^{4}+\frac{77\!\cdots\!43}{10\!\cdots\!91}a^{3}-\frac{72\!\cdots\!05}{10\!\cdots\!91}a^{2}-\frac{36\!\cdots\!38}{10\!\cdots\!91}a+\frac{12\!\cdots\!00}{10\!\cdots\!91}$, $\frac{11\!\cdots\!07}{10\!\cdots\!91}a^{20}-\frac{94\!\cdots\!88}{10\!\cdots\!91}a^{19}+\frac{13\!\cdots\!39}{10\!\cdots\!91}a^{18}+\frac{77\!\cdots\!08}{10\!\cdots\!91}a^{17}-\frac{23\!\cdots\!08}{10\!\cdots\!91}a^{16}-\frac{15\!\cdots\!34}{10\!\cdots\!91}a^{15}+\frac{11\!\cdots\!99}{10\!\cdots\!91}a^{14}-\frac{34\!\cdots\!15}{10\!\cdots\!91}a^{13}-\frac{23\!\cdots\!62}{10\!\cdots\!91}a^{12}+\frac{18\!\cdots\!75}{10\!\cdots\!91}a^{11}+\frac{19\!\cdots\!99}{10\!\cdots\!91}a^{10}-\frac{25\!\cdots\!41}{10\!\cdots\!91}a^{9}+\frac{48\!\cdots\!46}{10\!\cdots\!91}a^{8}+\frac{11\!\cdots\!01}{10\!\cdots\!91}a^{7}-\frac{33\!\cdots\!17}{10\!\cdots\!91}a^{6}-\frac{10\!\cdots\!81}{10\!\cdots\!91}a^{5}-\frac{28\!\cdots\!29}{10\!\cdots\!91}a^{4}+\frac{21\!\cdots\!97}{10\!\cdots\!91}a^{3}-\frac{36\!\cdots\!79}{10\!\cdots\!91}a^{2}-\frac{64\!\cdots\!56}{10\!\cdots\!91}a+\frac{40\!\cdots\!26}{10\!\cdots\!91}$, $\frac{49\!\cdots\!25}{10\!\cdots\!91}a^{20}-\frac{30\!\cdots\!28}{10\!\cdots\!91}a^{19}+\frac{65\!\cdots\!90}{10\!\cdots\!91}a^{18}+\frac{27\!\cdots\!78}{10\!\cdots\!91}a^{17}-\frac{38\!\cdots\!79}{10\!\cdots\!91}a^{16}-\frac{91\!\cdots\!77}{10\!\cdots\!91}a^{15}+\frac{19\!\cdots\!51}{10\!\cdots\!91}a^{14}+\frac{10\!\cdots\!32}{10\!\cdots\!91}a^{13}-\frac{40\!\cdots\!61}{10\!\cdots\!91}a^{12}+\frac{61\!\cdots\!75}{10\!\cdots\!91}a^{11}+\frac{32\!\cdots\!50}{10\!\cdots\!91}a^{10}-\frac{18\!\cdots\!22}{10\!\cdots\!91}a^{9}-\frac{52\!\cdots\!48}{10\!\cdots\!91}a^{8}+\frac{75\!\cdots\!49}{10\!\cdots\!91}a^{7}+\frac{13\!\cdots\!66}{10\!\cdots\!91}a^{6}-\frac{13\!\cdots\!58}{10\!\cdots\!91}a^{5}-\frac{22\!\cdots\!23}{10\!\cdots\!91}a^{4}+\frac{20\!\cdots\!09}{10\!\cdots\!91}a^{3}+\frac{20\!\cdots\!80}{10\!\cdots\!91}a^{2}+\frac{53\!\cdots\!78}{10\!\cdots\!91}a-\frac{33\!\cdots\!96}{10\!\cdots\!91}$, $\frac{12\!\cdots\!84}{10\!\cdots\!91}a^{20}-\frac{59\!\cdots\!95}{10\!\cdots\!91}a^{19}-\frac{72\!\cdots\!08}{10\!\cdots\!91}a^{18}+\frac{71\!\cdots\!34}{10\!\cdots\!91}a^{17}-\frac{91\!\cdots\!02}{10\!\cdots\!91}a^{16}-\frac{36\!\cdots\!89}{10\!\cdots\!91}a^{15}+\frac{20\!\cdots\!06}{10\!\cdots\!91}a^{14}+\frac{10\!\cdots\!10}{10\!\cdots\!91}a^{13}-\frac{79\!\cdots\!11}{10\!\cdots\!91}a^{12}-\frac{14\!\cdots\!10}{10\!\cdots\!91}a^{11}+\frac{15\!\cdots\!70}{10\!\cdots\!91}a^{10}+\frac{87\!\cdots\!76}{10\!\cdots\!91}a^{9}-\frac{15\!\cdots\!58}{10\!\cdots\!91}a^{8}+\frac{26\!\cdots\!47}{10\!\cdots\!91}a^{7}+\frac{10\!\cdots\!87}{10\!\cdots\!91}a^{6}-\frac{38\!\cdots\!95}{10\!\cdots\!91}a^{5}-\frac{97\!\cdots\!50}{10\!\cdots\!91}a^{4}-\frac{13\!\cdots\!41}{10\!\cdots\!91}a^{3}+\frac{96\!\cdots\!14}{10\!\cdots\!91}a^{2}-\frac{12\!\cdots\!23}{10\!\cdots\!91}a-\frac{30\!\cdots\!23}{10\!\cdots\!91}$, $\frac{64\!\cdots\!35}{10\!\cdots\!91}a^{20}-\frac{44\!\cdots\!22}{10\!\cdots\!91}a^{19}+\frac{37\!\cdots\!84}{10\!\cdots\!91}a^{18}+\frac{36\!\cdots\!17}{10\!\cdots\!91}a^{17}-\frac{78\!\cdots\!41}{10\!\cdots\!91}a^{16}-\frac{90\!\cdots\!77}{10\!\cdots\!91}a^{15}+\frac{35\!\cdots\!51}{10\!\cdots\!91}a^{14}-\frac{28\!\cdots\!33}{10\!\cdots\!91}a^{13}-\frac{69\!\cdots\!58}{10\!\cdots\!91}a^{12}+\frac{45\!\cdots\!92}{10\!\cdots\!91}a^{11}+\frac{48\!\cdots\!31}{10\!\cdots\!91}a^{10}-\frac{63\!\cdots\!65}{10\!\cdots\!91}a^{9}+\frac{61\!\cdots\!59}{10\!\cdots\!91}a^{8}+\frac{27\!\cdots\!98}{10\!\cdots\!91}a^{7}+\frac{14\!\cdots\!45}{10\!\cdots\!91}a^{6}-\frac{31\!\cdots\!83}{10\!\cdots\!91}a^{5}-\frac{15\!\cdots\!96}{10\!\cdots\!91}a^{4}+\frac{48\!\cdots\!63}{10\!\cdots\!91}a^{3}-\frac{10\!\cdots\!42}{10\!\cdots\!91}a^{2}-\frac{11\!\cdots\!34}{10\!\cdots\!91}a+\frac{41\!\cdots\!21}{10\!\cdots\!91}$, $\frac{47\!\cdots\!58}{98\!\cdots\!81}a^{20}-\frac{28\!\cdots\!78}{98\!\cdots\!81}a^{19}+\frac{22\!\cdots\!10}{98\!\cdots\!81}a^{18}+\frac{27\!\cdots\!48}{98\!\cdots\!81}a^{17}-\frac{36\!\cdots\!37}{98\!\cdots\!81}a^{16}-\frac{10\!\cdots\!70}{98\!\cdots\!81}a^{15}+\frac{20\!\cdots\!62}{98\!\cdots\!81}a^{14}+\frac{13\!\cdots\!98}{98\!\cdots\!81}a^{13}-\frac{45\!\cdots\!74}{98\!\cdots\!81}a^{12}+\frac{50\!\cdots\!32}{98\!\cdots\!81}a^{11}+\frac{40\!\cdots\!32}{98\!\cdots\!81}a^{10}-\frac{25\!\cdots\!74}{98\!\cdots\!81}a^{9}-\frac{50\!\cdots\!38}{98\!\cdots\!81}a^{8}+\frac{14\!\cdots\!12}{98\!\cdots\!81}a^{7}+\frac{79\!\cdots\!60}{98\!\cdots\!81}a^{6}-\frac{12\!\cdots\!35}{98\!\cdots\!81}a^{5}-\frac{25\!\cdots\!04}{98\!\cdots\!81}a^{4}+\frac{23\!\cdots\!86}{98\!\cdots\!81}a^{3}+\frac{18\!\cdots\!42}{98\!\cdots\!81}a^{2}-\frac{27\!\cdots\!68}{98\!\cdots\!81}a+\frac{15\!\cdots\!79}{98\!\cdots\!81}$, $\frac{49\!\cdots\!37}{10\!\cdots\!91}a^{20}-\frac{28\!\cdots\!86}{10\!\cdots\!91}a^{19}-\frac{28\!\cdots\!23}{10\!\cdots\!91}a^{18}+\frac{28\!\cdots\!97}{10\!\cdots\!91}a^{17}-\frac{32\!\cdots\!61}{10\!\cdots\!91}a^{16}-\frac{10\!\cdots\!94}{10\!\cdots\!91}a^{15}+\frac{18\!\cdots\!91}{10\!\cdots\!91}a^{14}+\frac{14\!\cdots\!00}{10\!\cdots\!91}a^{13}-\frac{43\!\cdots\!17}{10\!\cdots\!91}a^{12}+\frac{42\!\cdots\!42}{10\!\cdots\!91}a^{11}+\frac{41\!\cdots\!27}{10\!\cdots\!91}a^{10}-\frac{27\!\cdots\!30}{10\!\cdots\!91}a^{9}-\frac{83\!\cdots\!26}{10\!\cdots\!91}a^{8}+\frac{16\!\cdots\!31}{10\!\cdots\!91}a^{7}+\frac{87\!\cdots\!85}{10\!\cdots\!91}a^{6}-\frac{12\!\cdots\!73}{10\!\cdots\!91}a^{5}-\frac{23\!\cdots\!76}{10\!\cdots\!91}a^{4}+\frac{26\!\cdots\!85}{10\!\cdots\!91}a^{3}+\frac{28\!\cdots\!67}{10\!\cdots\!91}a^{2}-\frac{73\!\cdots\!37}{10\!\cdots\!91}a-\frac{68\!\cdots\!84}{10\!\cdots\!91}$, $\frac{11\!\cdots\!77}{10\!\cdots\!91}a^{20}+\frac{84\!\cdots\!80}{10\!\cdots\!91}a^{19}-\frac{48\!\cdots\!55}{10\!\cdots\!91}a^{18}-\frac{56\!\cdots\!33}{10\!\cdots\!91}a^{17}+\frac{71\!\cdots\!51}{10\!\cdots\!91}a^{16}-\frac{51\!\cdots\!44}{10\!\cdots\!91}a^{15}-\frac{34\!\cdots\!11}{10\!\cdots\!91}a^{14}+\frac{50\!\cdots\!71}{10\!\cdots\!91}a^{13}+\frac{64\!\cdots\!64}{10\!\cdots\!91}a^{12}-\frac{14\!\cdots\!17}{10\!\cdots\!91}a^{11}-\frac{54\!\cdots\!05}{10\!\cdots\!91}a^{10}+\frac{16\!\cdots\!73}{10\!\cdots\!91}a^{9}-\frac{10\!\cdots\!78}{10\!\cdots\!91}a^{8}-\frac{92\!\cdots\!24}{10\!\cdots\!91}a^{7}+\frac{43\!\cdots\!26}{10\!\cdots\!91}a^{6}+\frac{56\!\cdots\!20}{10\!\cdots\!91}a^{5}+\frac{12\!\cdots\!27}{10\!\cdots\!91}a^{4}-\frac{11\!\cdots\!55}{10\!\cdots\!91}a^{3}+\frac{89\!\cdots\!91}{10\!\cdots\!91}a^{2}+\frac{25\!\cdots\!23}{10\!\cdots\!91}a-\frac{30\!\cdots\!15}{10\!\cdots\!91}$, $\frac{22\!\cdots\!20}{10\!\cdots\!91}a^{20}-\frac{14\!\cdots\!54}{10\!\cdots\!91}a^{19}+\frac{43\!\cdots\!73}{10\!\cdots\!91}a^{18}+\frac{12\!\cdots\!23}{10\!\cdots\!91}a^{17}-\frac{19\!\cdots\!51}{10\!\cdots\!91}a^{16}-\frac{41\!\cdots\!54}{10\!\cdots\!91}a^{15}+\frac{98\!\cdots\!94}{10\!\cdots\!91}a^{14}+\frac{42\!\cdots\!14}{10\!\cdots\!91}a^{13}-\frac{20\!\cdots\!18}{10\!\cdots\!91}a^{12}+\frac{52\!\cdots\!85}{10\!\cdots\!91}a^{11}+\frac{18\!\cdots\!82}{10\!\cdots\!91}a^{10}-\frac{13\!\cdots\!30}{10\!\cdots\!91}a^{9}-\frac{29\!\cdots\!76}{10\!\cdots\!91}a^{8}+\frac{71\!\cdots\!79}{10\!\cdots\!91}a^{7}+\frac{39\!\cdots\!52}{10\!\cdots\!91}a^{6}-\frac{74\!\cdots\!83}{10\!\cdots\!91}a^{5}-\frac{92\!\cdots\!32}{10\!\cdots\!91}a^{4}+\frac{12\!\cdots\!69}{10\!\cdots\!91}a^{3}+\frac{85\!\cdots\!58}{10\!\cdots\!91}a^{2}-\frac{28\!\cdots\!20}{10\!\cdots\!91}a-\frac{16\!\cdots\!36}{10\!\cdots\!91}$
| 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: | \( 39718417.9458 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{6}\cdot 39718417.9458 \cdot 1}{2\cdot\sqrt{13686230803976276974088591656489}}\cr\approx \mathstrut & 0.169109960109 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 7*x^19 + 55*x^18 - 132*x^17 - 112*x^16 + 589*x^15 - 188*x^14 - 1078*x^13 + 1042*x^12 + 579*x^11 - 1297*x^10 + 411*x^9 + 431*x^8 - 127*x^7 - 455*x^6 - 126*x^5 + 888*x^4 - 460*x^3 - 146*x^2 + 127*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 - 7*x^20 + 7*x^19 + 55*x^18 - 132*x^17 - 112*x^16 + 589*x^15 - 188*x^14 - 1078*x^13 + 1042*x^12 + 579*x^11 - 1297*x^10 + 411*x^9 + 431*x^8 - 127*x^7 - 455*x^6 - 126*x^5 + 888*x^4 - 460*x^3 - 146*x^2 + 127*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 - 7*x^20 + 7*x^19 + 55*x^18 - 132*x^17 - 112*x^16 + 589*x^15 - 188*x^14 - 1078*x^13 + 1042*x^12 + 579*x^11 - 1297*x^10 + 411*x^9 + 431*x^8 - 127*x^7 - 455*x^6 - 126*x^5 + 888*x^4 - 460*x^3 - 146*x^2 + 127*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 - 7*x^20 + 7*x^19 + 55*x^18 - 132*x^17 - 112*x^16 + 589*x^15 - 188*x^14 - 1078*x^13 + 1042*x^12 + 579*x^11 - 1297*x^10 + 411*x^9 + 431*x^8 - 127*x^7 - 455*x^6 - 126*x^5 + 888*x^4 - 460*x^3 - 146*x^2 + 127*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
$C_3^7.A_7$ (as 21T132):
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 5511240 |
| The 246 conjugacy class representatives for $C_3^7.A_7$ |
| Character table for $C_3^7.A_7$ |
Intermediate fields
| 7.3.4959529.1 |
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 |
| 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 | $21$ | $21$ | $15{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | $21$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.7.0.1}{7} }^{3}$ | R | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.3.0.1}{3} }$ | $21$ | $21$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.5.0.1}{5} }^{3}{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.5.0.1}{5} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | $21$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.9.0.1 | $x^{9} + 7 x^{2} + 8 x + 14$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |
| 17.12.10.2 | $x^{12} - 3060 x^{6} - 197676$ | $6$ | $2$ | $10$ | $C_6\times S_3$ | $[\ ]_{6}^{6}$ | |
|
\(19\)
| 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 19.3.2.3 | $x^{3} + 38$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 19.15.0.1 | $x^{15} + x^{7} + 10 x^{6} + 11 x^{5} + 13 x^{4} + 15 x^{3} + 14 x^{2} + 17$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
|
\(61\)
| 61.3.2.3 | $x^{3} + 244$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 61.6.0.1 | $x^{6} + 49 x^{3} + 3 x^{2} + 29 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 61.12.0.1 | $x^{12} + 2 x^{8} + 42 x^{7} + 33 x^{6} + 8 x^{5} + 38 x^{4} + 14 x^{3} + x^{2} + 15 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(131\)
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 131.6.0.1 | $x^{6} + 2 x^{4} + 66 x^{3} + 4 x^{2} + 22 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 131.9.6.1 | $x^{9} + 9 x^{7} + 780 x^{6} + 27 x^{5} + 1143 x^{4} - 253446 x^{3} + 7020 x^{2} - 311220 x + 17579537$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |