Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^19 + 23*x^17 - 4*x^16 - 54*x^15 + 19*x^14 + 94*x^13 - 41*x^12 - 115*x^11 + 59*x^10 + 30*x^9 - 81*x^8 + 20*x^7 + 46*x^6 - 18*x^5 - 5*x^4 + 13*x^3 - 3*x^2 + 1)
gp: K = bnfinit(y^21 - 4*y^19 + 23*y^17 - 4*y^16 - 54*y^15 + 19*y^14 + 94*y^13 - 41*y^12 - 115*y^11 + 59*y^10 + 30*y^9 - 81*y^8 + 20*y^7 + 46*y^6 - 18*y^5 - 5*y^4 + 13*y^3 - 3*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 4*x^19 + 23*x^17 - 4*x^16 - 54*x^15 + 19*x^14 + 94*x^13 - 41*x^12 - 115*x^11 + 59*x^10 + 30*x^9 - 81*x^8 + 20*x^7 + 46*x^6 - 18*x^5 - 5*x^4 + 13*x^3 - 3*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 4*x^19 + 23*x^17 - 4*x^16 - 54*x^15 + 19*x^14 + 94*x^13 - 41*x^12 - 115*x^11 + 59*x^10 + 30*x^9 - 81*x^8 + 20*x^7 + 46*x^6 - 18*x^5 - 5*x^4 + 13*x^3 - 3*x^2 + 1)
\( x^{21} - 4 x^{19} + 23 x^{17} - 4 x^{16} - 54 x^{15} + 19 x^{14} + 94 x^{13} - 41 x^{12} - 115 x^{11} + \cdots + 1 \)
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: | $[5, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(85594961606810871789418849\)
\(\medspace = 23^{8}\cdot 239^{3}\cdot 431^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.17\) | 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: | $23^{1/2}239^{1/2}431^{1/2}\approx 1539.2228558594106$ | ||
| Ramified primes: |
\(23\), \(239\), \(431\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{103009}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{23}a^{19}+\frac{5}{23}a^{18}+\frac{9}{23}a^{17}+\frac{8}{23}a^{16}+\frac{1}{23}a^{15}-\frac{3}{23}a^{14}+\frac{11}{23}a^{13}-\frac{5}{23}a^{12}+\frac{6}{23}a^{11}+\frac{3}{23}a^{10}-\frac{11}{23}a^{9}-\frac{9}{23}a^{8}+\frac{2}{23}a^{7}-\frac{9}{23}a^{6}-\frac{3}{23}a^{5}+\frac{1}{23}a^{4}+\frac{6}{23}a^{2}-\frac{3}{23}a+\frac{2}{23}$, $\frac{1}{15\!\cdots\!23}a^{20}+\frac{129923724112386}{15\!\cdots\!23}a^{19}+\frac{188530018391087}{670662867513001}a^{18}+\frac{45\!\cdots\!51}{15\!\cdots\!23}a^{17}+\frac{59\!\cdots\!08}{15\!\cdots\!23}a^{16}+\frac{76\!\cdots\!84}{15\!\cdots\!23}a^{15}-\frac{31\!\cdots\!23}{15\!\cdots\!23}a^{14}+\frac{42\!\cdots\!16}{15\!\cdots\!23}a^{13}+\frac{36285041965030}{14\!\cdots\!93}a^{12}-\frac{26\!\cdots\!11}{15\!\cdots\!23}a^{11}+\frac{17\!\cdots\!47}{15\!\cdots\!23}a^{10}+\frac{13\!\cdots\!95}{15\!\cdots\!23}a^{9}+\frac{52\!\cdots\!44}{15\!\cdots\!23}a^{8}-\frac{32\!\cdots\!93}{15\!\cdots\!23}a^{7}-\frac{505211536564710}{14\!\cdots\!93}a^{6}+\frac{67\!\cdots\!83}{15\!\cdots\!23}a^{5}-\frac{10\!\cdots\!90}{15\!\cdots\!23}a^{4}-\frac{75\!\cdots\!47}{15\!\cdots\!23}a^{3}-\frac{29058979650835}{670662867513001}a^{2}+\frac{101136254658787}{15\!\cdots\!23}a+\frac{31\!\cdots\!09}{15\!\cdots\!23}$
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: | $12$ | 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: |
$\frac{95\!\cdots\!26}{15\!\cdots\!23}a^{20}+\frac{28\!\cdots\!24}{15\!\cdots\!23}a^{19}-\frac{37\!\cdots\!66}{15\!\cdots\!23}a^{18}-\frac{11\!\cdots\!47}{15\!\cdots\!23}a^{17}+\frac{21\!\cdots\!31}{15\!\cdots\!23}a^{16}+\frac{29\!\cdots\!38}{15\!\cdots\!23}a^{15}-\frac{51\!\cdots\!61}{15\!\cdots\!23}a^{14}+\frac{12\!\cdots\!72}{15\!\cdots\!23}a^{13}+\frac{83\!\cdots\!65}{14\!\cdots\!93}a^{12}-\frac{77\!\cdots\!78}{15\!\cdots\!23}a^{11}-\frac{11\!\cdots\!78}{15\!\cdots\!23}a^{10}+\frac{13\!\cdots\!78}{15\!\cdots\!23}a^{9}+\frac{39\!\cdots\!60}{15\!\cdots\!23}a^{8}-\frac{54\!\cdots\!47}{15\!\cdots\!23}a^{7}-\frac{113759778437755}{14\!\cdots\!93}a^{6}+\frac{38\!\cdots\!43}{15\!\cdots\!23}a^{5}+\frac{33\!\cdots\!63}{15\!\cdots\!23}a^{4}-\frac{32\!\cdots\!46}{15\!\cdots\!23}a^{3}+\frac{51\!\cdots\!48}{15\!\cdots\!23}a^{2}-\frac{27\!\cdots\!00}{15\!\cdots\!23}a+\frac{11\!\cdots\!40}{15\!\cdots\!23}$, $\frac{10\!\cdots\!12}{15\!\cdots\!23}a^{20}-\frac{71\!\cdots\!12}{15\!\cdots\!23}a^{19}-\frac{46\!\cdots\!21}{15\!\cdots\!23}a^{18}+\frac{29\!\cdots\!09}{15\!\cdots\!23}a^{17}+\frac{27\!\cdots\!93}{15\!\cdots\!23}a^{16}-\frac{16\!\cdots\!15}{15\!\cdots\!23}a^{15}-\frac{44\!\cdots\!31}{15\!\cdots\!23}a^{14}+\frac{41\!\cdots\!22}{15\!\cdots\!23}a^{13}+\frac{20\!\cdots\!74}{14\!\cdots\!93}a^{12}-\frac{74\!\cdots\!81}{15\!\cdots\!23}a^{11}+\frac{44\!\cdots\!69}{15\!\cdots\!23}a^{10}+\frac{95\!\cdots\!21}{15\!\cdots\!23}a^{9}-\frac{18\!\cdots\!47}{15\!\cdots\!23}a^{8}-\frac{39\!\cdots\!20}{15\!\cdots\!23}a^{7}+\frac{34\!\cdots\!55}{14\!\cdots\!93}a^{6}-\frac{31\!\cdots\!13}{15\!\cdots\!23}a^{5}-\frac{30\!\cdots\!53}{15\!\cdots\!23}a^{4}+\frac{98\!\cdots\!45}{15\!\cdots\!23}a^{3}+\frac{74\!\cdots\!83}{15\!\cdots\!23}a^{2}-\frac{32\!\cdots\!85}{15\!\cdots\!23}a+\frac{13\!\cdots\!02}{15\!\cdots\!23}$, $\frac{16\!\cdots\!50}{15\!\cdots\!23}a^{20}-\frac{42\!\cdots\!58}{15\!\cdots\!23}a^{19}-\frac{62\!\cdots\!14}{15\!\cdots\!23}a^{18}+\frac{16\!\cdots\!19}{15\!\cdots\!23}a^{17}+\frac{38\!\cdots\!64}{15\!\cdots\!23}a^{16}-\frac{10\!\cdots\!38}{15\!\cdots\!23}a^{15}-\frac{71\!\cdots\!10}{15\!\cdots\!23}a^{14}+\frac{24\!\cdots\!02}{15\!\cdots\!23}a^{13}+\frac{88\!\cdots\!55}{14\!\cdots\!93}a^{12}-\frac{44\!\cdots\!27}{15\!\cdots\!23}a^{11}-\frac{75\!\cdots\!45}{15\!\cdots\!23}a^{10}+\frac{55\!\cdots\!10}{15\!\cdots\!23}a^{9}-\frac{78\!\cdots\!23}{15\!\cdots\!23}a^{8}-\frac{24\!\cdots\!50}{15\!\cdots\!23}a^{7}+\frac{19\!\cdots\!96}{14\!\cdots\!93}a^{6}+\frac{29\!\cdots\!36}{15\!\cdots\!23}a^{5}-\frac{15\!\cdots\!29}{15\!\cdots\!23}a^{4}+\frac{112688629191889}{15\!\cdots\!23}a^{3}+\frac{41\!\cdots\!85}{15\!\cdots\!23}a^{2}-\frac{13\!\cdots\!22}{15\!\cdots\!23}a+\frac{49\!\cdots\!54}{15\!\cdots\!23}$, $\frac{565856428375638}{15\!\cdots\!23}a^{20}+\frac{28\!\cdots\!54}{15\!\cdots\!23}a^{19}-\frac{16\!\cdots\!93}{15\!\cdots\!23}a^{18}-\frac{12\!\cdots\!90}{15\!\cdots\!23}a^{17}+\frac{10\!\cdots\!71}{15\!\cdots\!23}a^{16}+\frac{67\!\cdots\!77}{15\!\cdots\!23}a^{15}-\frac{27\!\cdots\!79}{15\!\cdots\!23}a^{14}-\frac{17\!\cdots\!20}{15\!\cdots\!23}a^{13}+\frac{67\!\cdots\!81}{14\!\cdots\!93}a^{12}+\frac{30\!\cdots\!54}{15\!\cdots\!23}a^{11}-\frac{12\!\cdots\!14}{15\!\cdots\!23}a^{10}-\frac{39\!\cdots\!11}{15\!\cdots\!23}a^{9}+\frac{10\!\cdots\!24}{15\!\cdots\!23}a^{8}+\frac{15\!\cdots\!70}{15\!\cdots\!23}a^{7}-\frac{15\!\cdots\!59}{14\!\cdots\!93}a^{6}+\frac{60\!\cdots\!49}{15\!\cdots\!23}a^{5}+\frac{14\!\cdots\!24}{15\!\cdots\!23}a^{4}-\frac{97\!\cdots\!56}{15\!\cdots\!23}a^{3}-\frac{33\!\cdots\!98}{15\!\cdots\!23}a^{2}+\frac{18\!\cdots\!63}{15\!\cdots\!23}a-\frac{84\!\cdots\!48}{15\!\cdots\!23}$, $\frac{322837316403313}{15\!\cdots\!23}a^{20}-\frac{82348049617269}{670662867513001}a^{19}-\frac{28\!\cdots\!22}{15\!\cdots\!23}a^{18}+\frac{72\!\cdots\!99}{15\!\cdots\!23}a^{17}+\frac{13\!\cdots\!56}{15\!\cdots\!23}a^{16}-\frac{44\!\cdots\!24}{15\!\cdots\!23}a^{15}-\frac{45\!\cdots\!27}{15\!\cdots\!23}a^{14}+\frac{10\!\cdots\!67}{15\!\cdots\!23}a^{13}+\frac{76\!\cdots\!71}{14\!\cdots\!93}a^{12}-\frac{21\!\cdots\!88}{15\!\cdots\!23}a^{11}-\frac{12\!\cdots\!65}{15\!\cdots\!23}a^{10}+\frac{13\!\cdots\!18}{670662867513001}a^{9}+\frac{11\!\cdots\!25}{15\!\cdots\!23}a^{8}-\frac{19\!\cdots\!51}{15\!\cdots\!23}a^{7}+\frac{50\!\cdots\!08}{14\!\cdots\!93}a^{6}+\frac{16\!\cdots\!72}{15\!\cdots\!23}a^{5}-\frac{76\!\cdots\!72}{15\!\cdots\!23}a^{4}-\frac{93\!\cdots\!80}{15\!\cdots\!23}a^{3}+\frac{29\!\cdots\!15}{15\!\cdots\!23}a^{2}+\frac{78\!\cdots\!15}{15\!\cdots\!23}a-\frac{13\!\cdots\!90}{15\!\cdots\!23}$, $\frac{43\!\cdots\!36}{15\!\cdots\!23}a^{20}+\frac{74\!\cdots\!36}{15\!\cdots\!23}a^{19}-\frac{17\!\cdots\!48}{15\!\cdots\!23}a^{18}-\frac{30\!\cdots\!17}{15\!\cdots\!23}a^{17}+\frac{99\!\cdots\!38}{15\!\cdots\!23}a^{16}+\frac{15\!\cdots\!81}{15\!\cdots\!23}a^{15}-\frac{26\!\cdots\!17}{15\!\cdots\!23}a^{14}-\frac{33\!\cdots\!51}{15\!\cdots\!23}a^{13}+\frac{49\!\cdots\!48}{14\!\cdots\!93}a^{12}+\frac{55\!\cdots\!05}{15\!\cdots\!23}a^{11}-\frac{78\!\cdots\!99}{15\!\cdots\!23}a^{10}-\frac{67\!\cdots\!43}{15\!\cdots\!23}a^{9}+\frac{52\!\cdots\!71}{15\!\cdots\!23}a^{8}-\frac{19\!\cdots\!16}{15\!\cdots\!23}a^{7}-\frac{39\!\cdots\!38}{14\!\cdots\!93}a^{6}+\frac{27\!\cdots\!40}{15\!\cdots\!23}a^{5}+\frac{27\!\cdots\!13}{15\!\cdots\!23}a^{4}-\frac{84\!\cdots\!30}{15\!\cdots\!23}a^{3}-\frac{96\!\cdots\!32}{15\!\cdots\!23}a^{2}+\frac{32\!\cdots\!11}{15\!\cdots\!23}a-\frac{10\!\cdots\!78}{15\!\cdots\!23}$, $\frac{234681528936476}{670662867513001}a^{20}+\frac{367238815819824}{15\!\cdots\!23}a^{19}-\frac{22\!\cdots\!69}{15\!\cdots\!23}a^{18}-\frac{13\!\cdots\!08}{15\!\cdots\!23}a^{17}+\frac{12\!\cdots\!31}{15\!\cdots\!23}a^{16}-\frac{12\!\cdots\!34}{15\!\cdots\!23}a^{15}-\frac{31\!\cdots\!48}{15\!\cdots\!23}a^{14}+\frac{85\!\cdots\!71}{15\!\cdots\!23}a^{13}+\frac{51\!\cdots\!22}{14\!\cdots\!93}a^{12}-\frac{18\!\cdots\!76}{15\!\cdots\!23}a^{11}-\frac{73\!\cdots\!30}{15\!\cdots\!23}a^{10}+\frac{28\!\cdots\!78}{15\!\cdots\!23}a^{9}+\frac{33\!\cdots\!24}{15\!\cdots\!23}a^{8}-\frac{41\!\cdots\!36}{15\!\cdots\!23}a^{7}-\frac{45\!\cdots\!83}{14\!\cdots\!93}a^{6}+\frac{24\!\cdots\!27}{15\!\cdots\!23}a^{5}-\frac{28\!\cdots\!40}{15\!\cdots\!23}a^{4}-\frac{32\!\cdots\!95}{670662867513001}a^{3}+\frac{64\!\cdots\!51}{15\!\cdots\!23}a^{2}+\frac{248168651017626}{15\!\cdots\!23}a+\frac{32\!\cdots\!24}{15\!\cdots\!23}$, $\frac{44\!\cdots\!75}{15\!\cdots\!23}a^{20}+\frac{850021807636161}{15\!\cdots\!23}a^{19}-\frac{17\!\cdots\!18}{15\!\cdots\!23}a^{18}-\frac{26\!\cdots\!11}{15\!\cdots\!23}a^{17}+\frac{10\!\cdots\!76}{15\!\cdots\!23}a^{16}-\frac{63865994710413}{670662867513001}a^{15}-\frac{24\!\cdots\!48}{15\!\cdots\!23}a^{14}+\frac{56\!\cdots\!08}{15\!\cdots\!23}a^{13}+\frac{38\!\cdots\!89}{14\!\cdots\!93}a^{12}-\frac{14\!\cdots\!13}{15\!\cdots\!23}a^{11}-\frac{53\!\cdots\!53}{15\!\cdots\!23}a^{10}+\frac{25\!\cdots\!09}{15\!\cdots\!23}a^{9}+\frac{17\!\cdots\!08}{15\!\cdots\!23}a^{8}-\frac{46\!\cdots\!63}{15\!\cdots\!23}a^{7}+\frac{19\!\cdots\!14}{14\!\cdots\!93}a^{6}+\frac{30\!\cdots\!21}{15\!\cdots\!23}a^{5}-\frac{70\!\cdots\!33}{15\!\cdots\!23}a^{4}-\frac{59\!\cdots\!70}{15\!\cdots\!23}a^{3}+\frac{81\!\cdots\!61}{15\!\cdots\!23}a^{2}+\frac{31\!\cdots\!89}{15\!\cdots\!23}a-\frac{20\!\cdots\!71}{15\!\cdots\!23}$, $\frac{38\!\cdots\!00}{15\!\cdots\!23}a^{20}-\frac{559675775891108}{15\!\cdots\!23}a^{19}-\frac{17\!\cdots\!68}{15\!\cdots\!23}a^{18}+\frac{420978145573290}{15\!\cdots\!23}a^{17}+\frac{98\!\cdots\!55}{15\!\cdots\!23}a^{16}-\frac{20\!\cdots\!08}{15\!\cdots\!23}a^{15}-\frac{26\!\cdots\!69}{15\!\cdots\!23}a^{14}+\frac{68\!\cdots\!75}{15\!\cdots\!23}a^{13}+\frac{43\!\cdots\!62}{14\!\cdots\!93}a^{12}-\frac{13\!\cdots\!66}{15\!\cdots\!23}a^{11}-\frac{66\!\cdots\!42}{15\!\cdots\!23}a^{10}+\frac{17\!\cdots\!57}{15\!\cdots\!23}a^{9}+\frac{39\!\cdots\!84}{15\!\cdots\!23}a^{8}-\frac{16\!\cdots\!69}{15\!\cdots\!23}a^{7}-\frac{793648210492288}{14\!\cdots\!93}a^{6}+\frac{79\!\cdots\!00}{670662867513001}a^{5}-\frac{17\!\cdots\!61}{15\!\cdots\!23}a^{4}-\frac{76\!\cdots\!30}{15\!\cdots\!23}a^{3}-\frac{48\!\cdots\!09}{15\!\cdots\!23}a^{2}-\frac{37\!\cdots\!59}{15\!\cdots\!23}a+\frac{78\!\cdots\!01}{15\!\cdots\!23}$, $\frac{29\!\cdots\!92}{15\!\cdots\!23}a^{20}-\frac{66\!\cdots\!24}{15\!\cdots\!23}a^{19}-\frac{13\!\cdots\!63}{15\!\cdots\!23}a^{18}+\frac{24\!\cdots\!23}{15\!\cdots\!23}a^{17}+\frac{75\!\cdots\!57}{15\!\cdots\!23}a^{16}-\frac{15\!\cdots\!58}{15\!\cdots\!23}a^{15}-\frac{17\!\cdots\!72}{15\!\cdots\!23}a^{14}+\frac{37\!\cdots\!67}{15\!\cdots\!23}a^{13}+\frac{22\!\cdots\!57}{14\!\cdots\!93}a^{12}-\frac{67\!\cdots\!84}{15\!\cdots\!23}a^{11}-\frac{25\!\cdots\!35}{15\!\cdots\!23}a^{10}+\frac{82\!\cdots\!55}{15\!\cdots\!23}a^{9}-\frac{56\!\cdots\!57}{15\!\cdots\!23}a^{8}-\frac{30\!\cdots\!70}{15\!\cdots\!23}a^{7}+\frac{46\!\cdots\!30}{14\!\cdots\!93}a^{6}+\frac{69\!\cdots\!15}{15\!\cdots\!23}a^{5}-\frac{28\!\cdots\!50}{15\!\cdots\!23}a^{4}-\frac{82\!\cdots\!32}{15\!\cdots\!23}a^{3}+\frac{25\!\cdots\!52}{15\!\cdots\!23}a^{2}-\frac{65\!\cdots\!31}{15\!\cdots\!23}a+\frac{15\!\cdots\!09}{15\!\cdots\!23}$, $\frac{99\!\cdots\!23}{15\!\cdots\!23}a^{20}-\frac{12\!\cdots\!80}{15\!\cdots\!23}a^{19}-\frac{38\!\cdots\!89}{15\!\cdots\!23}a^{18}+\frac{56\!\cdots\!35}{15\!\cdots\!23}a^{17}+\frac{97\!\cdots\!09}{670662867513001}a^{16}-\frac{70\!\cdots\!62}{15\!\cdots\!23}a^{15}-\frac{50\!\cdots\!02}{15\!\cdots\!23}a^{14}+\frac{26\!\cdots\!04}{15\!\cdots\!23}a^{13}+\frac{77\!\cdots\!00}{14\!\cdots\!93}a^{12}-\frac{53\!\cdots\!40}{15\!\cdots\!23}a^{11}-\frac{10\!\cdots\!00}{15\!\cdots\!23}a^{10}+\frac{74\!\cdots\!99}{15\!\cdots\!23}a^{9}+\frac{15\!\cdots\!97}{15\!\cdots\!23}a^{8}-\frac{85\!\cdots\!36}{15\!\cdots\!23}a^{7}+\frac{21\!\cdots\!15}{14\!\cdots\!93}a^{6}+\frac{35\!\cdots\!82}{15\!\cdots\!23}a^{5}-\frac{14\!\cdots\!84}{15\!\cdots\!23}a^{4}+\frac{17\!\cdots\!77}{15\!\cdots\!23}a^{3}+\frac{10\!\cdots\!17}{15\!\cdots\!23}a^{2}-\frac{30\!\cdots\!58}{15\!\cdots\!23}a+\frac{10\!\cdots\!82}{15\!\cdots\!23}$, $\frac{67\!\cdots\!39}{15\!\cdots\!23}a^{20}-\frac{12\!\cdots\!64}{15\!\cdots\!23}a^{19}-\frac{29\!\cdots\!68}{15\!\cdots\!23}a^{18}+\frac{56\!\cdots\!73}{15\!\cdots\!23}a^{17}+\frac{16\!\cdots\!46}{15\!\cdots\!23}a^{16}-\frac{57\!\cdots\!42}{15\!\cdots\!23}a^{15}-\frac{40\!\cdots\!11}{15\!\cdots\!23}a^{14}+\frac{21\!\cdots\!79}{15\!\cdots\!23}a^{13}+\frac{65\!\cdots\!81}{14\!\cdots\!93}a^{12}-\frac{44\!\cdots\!00}{15\!\cdots\!23}a^{11}-\frac{92\!\cdots\!90}{15\!\cdots\!23}a^{10}+\frac{64\!\cdots\!81}{15\!\cdots\!23}a^{9}+\frac{38\!\cdots\!26}{15\!\cdots\!23}a^{8}-\frac{70\!\cdots\!90}{15\!\cdots\!23}a^{7}+\frac{11\!\cdots\!71}{14\!\cdots\!93}a^{6}+\frac{39\!\cdots\!21}{15\!\cdots\!23}a^{5}-\frac{76\!\cdots\!42}{670662867513001}a^{4}-\frac{95\!\cdots\!04}{15\!\cdots\!23}a^{3}+\frac{11\!\cdots\!52}{15\!\cdots\!23}a^{2}-\frac{18\!\cdots\!88}{15\!\cdots\!23}a-\frac{132761901113025}{670662867513001}$
| 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: | \( 45009.9851463 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{8}\cdot 45009.9851463 \cdot 1}{2\cdot\sqrt{85594961606810871789418849}}\cr\approx \mathstrut & 0.189079174872 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^19 + 23*x^17 - 4*x^16 - 54*x^15 + 19*x^14 + 94*x^13 - 41*x^12 - 115*x^11 + 59*x^10 + 30*x^9 - 81*x^8 + 20*x^7 + 46*x^6 - 18*x^5 - 5*x^4 + 13*x^3 - 3*x^2 + 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 - 4*x^19 + 23*x^17 - 4*x^16 - 54*x^15 + 19*x^14 + 94*x^13 - 41*x^12 - 115*x^11 + 59*x^10 + 30*x^9 - 81*x^8 + 20*x^7 + 46*x^6 - 18*x^5 - 5*x^4 + 13*x^3 - 3*x^2 + 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 - 4*x^19 + 23*x^17 - 4*x^16 - 54*x^15 + 19*x^14 + 94*x^13 - 41*x^12 - 115*x^11 + 59*x^10 + 30*x^9 - 81*x^8 + 20*x^7 + 46*x^6 - 18*x^5 - 5*x^4 + 13*x^3 - 3*x^2 + 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 - 4*x^19 + 23*x^17 - 4*x^16 - 54*x^15 + 19*x^14 + 94*x^13 - 41*x^12 - 115*x^11 + 59*x^10 + 30*x^9 - 81*x^8 + 20*x^7 + 46*x^6 - 18*x^5 - 5*x^4 + 13*x^3 - 3*x^2 + 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 $S_3\times S_7$ |
Intermediate fields
| 3.1.23.1, 7.5.2369207.1 |
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 | $21$ | $21$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | $21$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | R | $15{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{10}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.3.0.1 | $x^{3} + 2 x + 18$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(239\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $6$ | $2$ | $3$ | $3$ | ||||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(431\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |