Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 100*x^19 + 456*x^18 + 3776*x^17 - 20240*x^16 - 65058*x^15 + 451646*x^14 + 423688*x^13 - 5452225*x^12 + 1655193*x^11 + 35371490*x^10 - 39386623*x^9 - 111354014*x^8 + 208156183*x^7 + 105332901*x^6 - 422345627*x^5 + 144428798*x^4 + 234723315*x^3 - 152780306*x^2 - 25641148*x + 26494201)
gp: K = bnfinit(y^21 - 4*y^20 - 100*y^19 + 456*y^18 + 3776*y^17 - 20240*y^16 - 65058*y^15 + 451646*y^14 + 423688*y^13 - 5452225*y^12 + 1655193*y^11 + 35371490*y^10 - 39386623*y^9 - 111354014*y^8 + 208156183*y^7 + 105332901*y^6 - 422345627*y^5 + 144428798*y^4 + 234723315*y^3 - 152780306*y^2 - 25641148*y + 26494201, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 4*x^20 - 100*x^19 + 456*x^18 + 3776*x^17 - 20240*x^16 - 65058*x^15 + 451646*x^14 + 423688*x^13 - 5452225*x^12 + 1655193*x^11 + 35371490*x^10 - 39386623*x^9 - 111354014*x^8 + 208156183*x^7 + 105332901*x^6 - 422345627*x^5 + 144428798*x^4 + 234723315*x^3 - 152780306*x^2 - 25641148*x + 26494201);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 4*x^20 - 100*x^19 + 456*x^18 + 3776*x^17 - 20240*x^16 - 65058*x^15 + 451646*x^14 + 423688*x^13 - 5452225*x^12 + 1655193*x^11 + 35371490*x^10 - 39386623*x^9 - 111354014*x^8 + 208156183*x^7 + 105332901*x^6 - 422345627*x^5 + 144428798*x^4 + 234723315*x^3 - 152780306*x^2 - 25641148*x + 26494201)
\( x^{21} - 4 x^{20} - 100 x^{19} + 456 x^{18} + 3776 x^{17} - 20240 x^{16} - 65058 x^{15} + \cdots + 26494201 \)
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: | $[21, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1425682560385717640661443870033677687676469889\)
\(\medspace = 7^{14}\cdot 71^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(141.32\) | 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: | $7^{2/3}71^{6/7}\approx 141.31607405847367$ | ||
| 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: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $21$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(497=7\cdot 71\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{497}(1,·)$, $\chi_{497}(387,·)$, $\chi_{497}(261,·)$, $\chi_{497}(72,·)$, $\chi_{497}(463,·)$, $\chi_{497}(400,·)$, $\chi_{497}(403,·)$, $\chi_{497}(214,·)$, $\chi_{497}(471,·)$, $\chi_{497}(30,·)$, $\chi_{497}(32,·)$, $\chi_{497}(162,·)$, $\chi_{497}(37,·)$, $\chi_{497}(233,·)$, $\chi_{497}(172,·)$, $\chi_{497}(456,·)$, $\chi_{497}(179,·)$, $\chi_{497}(116,·)$, $\chi_{497}(375,·)$, $\chi_{497}(316,·)$, $\chi_{497}(190,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}+\frac{1}{5}a^{11}-\frac{2}{5}a^{10}-\frac{1}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{13}+\frac{2}{5}a^{11}+\frac{1}{5}a^{10}+\frac{2}{5}a^{9}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{14}-\frac{1}{5}a^{11}+\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{15}+\frac{2}{5}a^{11}-\frac{2}{5}a^{7}-\frac{2}{5}a^{5}-\frac{1}{5}a^{3}+\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{16}-\frac{2}{5}a^{11}-\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{17}+\frac{1}{5}a^{11}-\frac{2}{5}a^{10}-\frac{1}{5}a^{9}+\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{425}a^{18}+\frac{1}{17}a^{17}+\frac{7}{425}a^{16}-\frac{2}{425}a^{15}-\frac{38}{425}a^{14}+\frac{32}{425}a^{13}-\frac{14}{425}a^{12}+\frac{42}{425}a^{11}+\frac{196}{425}a^{10}-\frac{183}{425}a^{9}+\frac{161}{425}a^{8}+\frac{197}{425}a^{7}-\frac{4}{85}a^{6}+\frac{149}{425}a^{5}-\frac{4}{25}a^{4}+\frac{27}{425}a^{3}-\frac{189}{425}a^{2}-\frac{56}{425}a-\frac{44}{425}$, $\frac{1}{425}a^{19}-\frac{23}{425}a^{17}-\frac{7}{425}a^{16}+\frac{12}{425}a^{15}-\frac{38}{425}a^{14}+\frac{36}{425}a^{13}-\frac{33}{425}a^{12}-\frac{4}{425}a^{11}+\frac{11}{25}a^{10}-\frac{109}{425}a^{9}-\frac{3}{425}a^{8}-\frac{4}{17}a^{7}-\frac{31}{425}a^{6}+\frac{117}{425}a^{5}-\frac{143}{425}a^{4}-\frac{184}{425}a^{3}-\frac{176}{425}a^{2}-\frac{4}{425}a+\frac{16}{85}$, $\frac{1}{35\!\cdots\!75}a^{20}+\frac{11\!\cdots\!93}{35\!\cdots\!75}a^{19}-\frac{17\!\cdots\!03}{35\!\cdots\!75}a^{18}-\frac{28\!\cdots\!66}{35\!\cdots\!75}a^{17}+\frac{23\!\cdots\!31}{35\!\cdots\!75}a^{16}+\frac{31\!\cdots\!78}{35\!\cdots\!75}a^{15}-\frac{21\!\cdots\!23}{35\!\cdots\!75}a^{14}+\frac{53\!\cdots\!66}{70\!\cdots\!95}a^{13}-\frac{28\!\cdots\!43}{35\!\cdots\!75}a^{12}+\frac{13\!\cdots\!61}{70\!\cdots\!95}a^{11}-\frac{22\!\cdots\!73}{35\!\cdots\!75}a^{10}+\frac{27\!\cdots\!08}{70\!\cdots\!95}a^{9}-\frac{38\!\cdots\!52}{20\!\cdots\!75}a^{8}-\frac{13\!\cdots\!71}{35\!\cdots\!75}a^{7}+\frac{10\!\cdots\!09}{35\!\cdots\!75}a^{6}-\frac{11\!\cdots\!82}{35\!\cdots\!75}a^{5}-\frac{13\!\cdots\!03}{35\!\cdots\!75}a^{4}+\frac{11\!\cdots\!92}{35\!\cdots\!75}a^{3}-\frac{34\!\cdots\!47}{35\!\cdots\!75}a^{2}+\frac{55\!\cdots\!28}{35\!\cdots\!75}a-\frac{10\!\cdots\!01}{70\!\cdots\!95}$
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: | $20$ | 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: |
$\frac{85\!\cdots\!26}{83\!\cdots\!07}a^{20}-\frac{26\!\cdots\!18}{83\!\cdots\!07}a^{19}-\frac{87\!\cdots\!00}{83\!\cdots\!07}a^{18}+\frac{30\!\cdots\!54}{83\!\cdots\!07}a^{17}+\frac{35\!\cdots\!87}{83\!\cdots\!07}a^{16}-\frac{13\!\cdots\!86}{83\!\cdots\!07}a^{15}-\frac{68\!\cdots\!36}{83\!\cdots\!07}a^{14}+\frac{32\!\cdots\!62}{83\!\cdots\!07}a^{13}+\frac{66\!\cdots\!55}{83\!\cdots\!07}a^{12}-\frac{40\!\cdots\!38}{83\!\cdots\!07}a^{11}-\frac{23\!\cdots\!45}{83\!\cdots\!07}a^{10}+\frac{27\!\cdots\!77}{83\!\cdots\!07}a^{9}-\frac{74\!\cdots\!50}{83\!\cdots\!07}a^{8}-\frac{10\!\cdots\!30}{83\!\cdots\!07}a^{7}+\frac{82\!\cdots\!96}{83\!\cdots\!07}a^{6}+\frac{16\!\cdots\!17}{83\!\cdots\!07}a^{5}-\frac{20\!\cdots\!20}{83\!\cdots\!07}a^{4}-\frac{67\!\cdots\!73}{83\!\cdots\!07}a^{3}+\frac{13\!\cdots\!14}{83\!\cdots\!07}a^{2}-\frac{27\!\cdots\!71}{83\!\cdots\!07}a-\frac{24\!\cdots\!57}{83\!\cdots\!07}$, $\frac{34\!\cdots\!10}{83\!\cdots\!07}a^{20}-\frac{10\!\cdots\!42}{83\!\cdots\!07}a^{19}-\frac{35\!\cdots\!34}{83\!\cdots\!07}a^{18}+\frac{12\!\cdots\!47}{83\!\cdots\!07}a^{17}+\frac{14\!\cdots\!80}{83\!\cdots\!07}a^{16}-\frac{58\!\cdots\!95}{83\!\cdots\!07}a^{15}-\frac{27\!\cdots\!25}{83\!\cdots\!07}a^{14}+\frac{13\!\cdots\!68}{83\!\cdots\!07}a^{13}+\frac{26\!\cdots\!47}{83\!\cdots\!07}a^{12}-\frac{16\!\cdots\!16}{83\!\cdots\!07}a^{11}-\frac{85\!\cdots\!70}{83\!\cdots\!07}a^{10}+\frac{11\!\cdots\!07}{83\!\cdots\!07}a^{9}-\frac{37\!\cdots\!47}{83\!\cdots\!07}a^{8}-\frac{41\!\cdots\!34}{83\!\cdots\!07}a^{7}+\frac{36\!\cdots\!29}{83\!\cdots\!07}a^{6}+\frac{67\!\cdots\!40}{83\!\cdots\!07}a^{5}-\frac{88\!\cdots\!49}{83\!\cdots\!07}a^{4}-\frac{26\!\cdots\!74}{83\!\cdots\!07}a^{3}+\frac{59\!\cdots\!58}{83\!\cdots\!07}a^{2}-\frac{20\!\cdots\!64}{83\!\cdots\!07}a-\frac{10\!\cdots\!91}{83\!\cdots\!07}$, $\frac{22\!\cdots\!67}{35\!\cdots\!75}a^{20}-\frac{68\!\cdots\!08}{35\!\cdots\!75}a^{19}-\frac{22\!\cdots\!93}{35\!\cdots\!75}a^{18}+\frac{16\!\cdots\!78}{70\!\cdots\!95}a^{17}+\frac{90\!\cdots\!11}{35\!\cdots\!75}a^{16}-\frac{36\!\cdots\!03}{35\!\cdots\!75}a^{15}-\frac{17\!\cdots\!73}{35\!\cdots\!75}a^{14}+\frac{83\!\cdots\!12}{35\!\cdots\!75}a^{13}+\frac{16\!\cdots\!69}{35\!\cdots\!75}a^{12}-\frac{10\!\cdots\!48}{35\!\cdots\!75}a^{11}-\frac{58\!\cdots\!36}{35\!\cdots\!75}a^{10}+\frac{72\!\cdots\!67}{35\!\cdots\!75}a^{9}-\frac{21\!\cdots\!53}{35\!\cdots\!75}a^{8}-\frac{26\!\cdots\!86}{35\!\cdots\!75}a^{7}+\frac{22\!\cdots\!52}{35\!\cdots\!75}a^{6}+\frac{17\!\cdots\!40}{14\!\cdots\!19}a^{5}-\frac{54\!\cdots\!58}{35\!\cdots\!75}a^{4}-\frac{17\!\cdots\!19}{35\!\cdots\!75}a^{3}+\frac{36\!\cdots\!38}{35\!\cdots\!75}a^{2}-\frac{87\!\cdots\!21}{35\!\cdots\!75}a-\frac{64\!\cdots\!92}{35\!\cdots\!75}$, $\frac{47\!\cdots\!93}{35\!\cdots\!75}a^{20}-\frac{14\!\cdots\!02}{35\!\cdots\!75}a^{19}-\frac{49\!\cdots\!97}{35\!\cdots\!75}a^{18}+\frac{33\!\cdots\!76}{70\!\cdots\!95}a^{17}+\frac{19\!\cdots\!94}{35\!\cdots\!75}a^{16}-\frac{77\!\cdots\!27}{35\!\cdots\!75}a^{15}-\frac{38\!\cdots\!17}{35\!\cdots\!75}a^{14}+\frac{17\!\cdots\!18}{35\!\cdots\!75}a^{13}+\frac{37\!\cdots\!31}{35\!\cdots\!75}a^{12}-\frac{22\!\cdots\!62}{35\!\cdots\!75}a^{11}-\frac{14\!\cdots\!24}{35\!\cdots\!75}a^{10}+\frac{15\!\cdots\!73}{35\!\cdots\!75}a^{9}-\frac{35\!\cdots\!57}{35\!\cdots\!75}a^{8}-\frac{56\!\cdots\!44}{35\!\cdots\!75}a^{7}+\frac{43\!\cdots\!58}{35\!\cdots\!75}a^{6}+\frac{18\!\cdots\!18}{70\!\cdots\!95}a^{5}-\frac{11\!\cdots\!67}{35\!\cdots\!75}a^{4}-\frac{39\!\cdots\!91}{35\!\cdots\!75}a^{3}+\frac{73\!\cdots\!72}{35\!\cdots\!75}a^{2}-\frac{64\!\cdots\!24}{35\!\cdots\!75}a-\frac{12\!\cdots\!18}{35\!\cdots\!75}$, $\frac{58\!\cdots\!62}{35\!\cdots\!75}a^{20}-\frac{16\!\cdots\!98}{35\!\cdots\!75}a^{19}-\frac{60\!\cdots\!58}{35\!\cdots\!75}a^{18}+\frac{78\!\cdots\!59}{14\!\cdots\!19}a^{17}+\frac{24\!\cdots\!76}{35\!\cdots\!75}a^{16}-\frac{89\!\cdots\!43}{35\!\cdots\!75}a^{15}-\frac{49\!\cdots\!43}{35\!\cdots\!75}a^{14}+\frac{20\!\cdots\!42}{35\!\cdots\!75}a^{13}+\frac{50\!\cdots\!54}{35\!\cdots\!75}a^{12}-\frac{26\!\cdots\!78}{35\!\cdots\!75}a^{11}-\frac{22\!\cdots\!01}{35\!\cdots\!75}a^{10}+\frac{18\!\cdots\!92}{35\!\cdots\!75}a^{9}-\frac{58\!\cdots\!58}{35\!\cdots\!75}a^{8}-\frac{68\!\cdots\!26}{35\!\cdots\!75}a^{7}+\frac{38\!\cdots\!37}{35\!\cdots\!75}a^{6}+\frac{23\!\cdots\!01}{70\!\cdots\!95}a^{5}-\frac{10\!\cdots\!03}{35\!\cdots\!75}a^{4}-\frac{58\!\cdots\!89}{35\!\cdots\!75}a^{3}+\frac{70\!\cdots\!58}{35\!\cdots\!75}a^{2}+\frac{50\!\cdots\!54}{35\!\cdots\!75}a-\frac{11\!\cdots\!72}{35\!\cdots\!75}$, $\frac{18\!\cdots\!17}{35\!\cdots\!75}a^{20}-\frac{57\!\cdots\!58}{35\!\cdots\!75}a^{19}-\frac{19\!\cdots\!93}{35\!\cdots\!75}a^{18}+\frac{13\!\cdots\!88}{70\!\cdots\!95}a^{17}+\frac{75\!\cdots\!36}{35\!\cdots\!75}a^{16}-\frac{30\!\cdots\!53}{35\!\cdots\!75}a^{15}-\frac{14\!\cdots\!73}{35\!\cdots\!75}a^{14}+\frac{70\!\cdots\!62}{35\!\cdots\!75}a^{13}+\frac{14\!\cdots\!94}{35\!\cdots\!75}a^{12}-\frac{88\!\cdots\!98}{35\!\cdots\!75}a^{11}-\frac{48\!\cdots\!11}{35\!\cdots\!75}a^{10}+\frac{61\!\cdots\!42}{35\!\cdots\!75}a^{9}-\frac{17\!\cdots\!03}{35\!\cdots\!75}a^{8}-\frac{22\!\cdots\!36}{35\!\cdots\!75}a^{7}+\frac{18\!\cdots\!52}{35\!\cdots\!75}a^{6}+\frac{14\!\cdots\!51}{14\!\cdots\!19}a^{5}-\frac{45\!\cdots\!58}{35\!\cdots\!75}a^{4}-\frac{14\!\cdots\!94}{35\!\cdots\!75}a^{3}+\frac{30\!\cdots\!88}{35\!\cdots\!75}a^{2}-\frac{75\!\cdots\!71}{35\!\cdots\!75}a-\frac{54\!\cdots\!67}{35\!\cdots\!75}$, $\frac{24\!\cdots\!54}{70\!\cdots\!95}a^{20}-\frac{15\!\cdots\!50}{14\!\cdots\!19}a^{19}-\frac{25\!\cdots\!36}{70\!\cdots\!95}a^{18}+\frac{90\!\cdots\!44}{70\!\cdots\!95}a^{17}+\frac{10\!\cdots\!32}{70\!\cdots\!95}a^{16}-\frac{40\!\cdots\!84}{70\!\cdots\!95}a^{15}-\frac{19\!\cdots\!86}{70\!\cdots\!95}a^{14}+\frac{94\!\cdots\!98}{70\!\cdots\!95}a^{13}+\frac{18\!\cdots\!89}{70\!\cdots\!95}a^{12}-\frac{23\!\cdots\!54}{14\!\cdots\!19}a^{11}-\frac{64\!\cdots\!93}{70\!\cdots\!95}a^{10}+\frac{16\!\cdots\!61}{14\!\cdots\!19}a^{9}-\frac{48\!\cdots\!64}{14\!\cdots\!19}a^{8}-\frac{29\!\cdots\!02}{70\!\cdots\!95}a^{7}+\frac{24\!\cdots\!74}{70\!\cdots\!95}a^{6}+\frac{48\!\cdots\!33}{70\!\cdots\!95}a^{5}-\frac{61\!\cdots\!48}{70\!\cdots\!95}a^{4}-\frac{19\!\cdots\!41}{70\!\cdots\!95}a^{3}+\frac{81\!\cdots\!67}{14\!\cdots\!19}a^{2}-\frac{21\!\cdots\!70}{14\!\cdots\!19}a-\frac{73\!\cdots\!84}{70\!\cdots\!95}$, $\frac{85\!\cdots\!42}{35\!\cdots\!75}a^{20}-\frac{26\!\cdots\!78}{35\!\cdots\!75}a^{19}-\frac{87\!\cdots\!08}{35\!\cdots\!75}a^{18}+\frac{61\!\cdots\!92}{70\!\cdots\!95}a^{17}+\frac{35\!\cdots\!81}{35\!\cdots\!75}a^{16}-\frac{14\!\cdots\!78}{35\!\cdots\!75}a^{15}-\frac{68\!\cdots\!58}{35\!\cdots\!75}a^{14}+\frac{32\!\cdots\!82}{35\!\cdots\!75}a^{13}+\frac{66\!\cdots\!44}{35\!\cdots\!75}a^{12}-\frac{40\!\cdots\!78}{35\!\cdots\!75}a^{11}-\frac{23\!\cdots\!76}{35\!\cdots\!75}a^{10}+\frac{27\!\cdots\!32}{35\!\cdots\!75}a^{9}-\frac{76\!\cdots\!58}{35\!\cdots\!75}a^{8}-\frac{10\!\cdots\!06}{35\!\cdots\!75}a^{7}+\frac{83\!\cdots\!12}{35\!\cdots\!75}a^{6}+\frac{33\!\cdots\!86}{70\!\cdots\!95}a^{5}-\frac{20\!\cdots\!83}{35\!\cdots\!75}a^{4}-\frac{66\!\cdots\!24}{35\!\cdots\!75}a^{3}+\frac{13\!\cdots\!78}{35\!\cdots\!75}a^{2}-\frac{29\!\cdots\!41}{35\!\cdots\!75}a-\frac{24\!\cdots\!57}{35\!\cdots\!75}$, $\frac{11\!\cdots\!79}{35\!\cdots\!75}a^{20}-\frac{36\!\cdots\!37}{35\!\cdots\!75}a^{19}-\frac{12\!\cdots\!31}{35\!\cdots\!75}a^{18}+\frac{43\!\cdots\!13}{35\!\cdots\!75}a^{17}+\frac{48\!\cdots\!59}{35\!\cdots\!75}a^{16}-\frac{19\!\cdots\!13}{35\!\cdots\!75}a^{15}-\frac{95\!\cdots\!38}{35\!\cdots\!75}a^{14}+\frac{26\!\cdots\!24}{20\!\cdots\!75}a^{13}+\frac{91\!\cdots\!86}{35\!\cdots\!75}a^{12}-\frac{56\!\cdots\!72}{35\!\cdots\!75}a^{11}-\frac{31\!\cdots\!24}{35\!\cdots\!75}a^{10}+\frac{39\!\cdots\!58}{35\!\cdots\!75}a^{9}-\frac{11\!\cdots\!53}{35\!\cdots\!75}a^{8}-\frac{14\!\cdots\!47}{35\!\cdots\!75}a^{7}+\frac{23\!\cdots\!22}{70\!\cdots\!95}a^{6}+\frac{23\!\cdots\!68}{35\!\cdots\!75}a^{5}-\frac{29\!\cdots\!13}{35\!\cdots\!75}a^{4}-\frac{93\!\cdots\!69}{35\!\cdots\!75}a^{3}+\frac{19\!\cdots\!82}{35\!\cdots\!75}a^{2}-\frac{44\!\cdots\!33}{35\!\cdots\!75}a-\frac{34\!\cdots\!39}{35\!\cdots\!75}$, $\frac{12\!\cdots\!71}{35\!\cdots\!75}a^{20}-\frac{23\!\cdots\!09}{20\!\cdots\!75}a^{19}-\frac{13\!\cdots\!42}{35\!\cdots\!75}a^{18}+\frac{47\!\cdots\!87}{35\!\cdots\!75}a^{17}+\frac{10\!\cdots\!92}{70\!\cdots\!95}a^{16}-\frac{21\!\cdots\!41}{35\!\cdots\!75}a^{15}-\frac{10\!\cdots\!48}{35\!\cdots\!75}a^{14}+\frac{49\!\cdots\!71}{35\!\cdots\!75}a^{13}+\frac{98\!\cdots\!96}{35\!\cdots\!75}a^{12}-\frac{61\!\cdots\!14}{35\!\cdots\!75}a^{11}-\frac{33\!\cdots\!54}{35\!\cdots\!75}a^{10}+\frac{42\!\cdots\!41}{35\!\cdots\!75}a^{9}-\frac{25\!\cdots\!19}{70\!\cdots\!95}a^{8}-\frac{15\!\cdots\!49}{35\!\cdots\!75}a^{7}+\frac{26\!\cdots\!37}{70\!\cdots\!95}a^{6}+\frac{50\!\cdots\!31}{70\!\cdots\!95}a^{5}-\frac{32\!\cdots\!73}{35\!\cdots\!75}a^{4}-\frac{99\!\cdots\!37}{35\!\cdots\!75}a^{3}+\frac{86\!\cdots\!76}{14\!\cdots\!19}a^{2}-\frac{59\!\cdots\!54}{35\!\cdots\!75}a-\frac{38\!\cdots\!79}{35\!\cdots\!75}$, $\frac{33\!\cdots\!03}{35\!\cdots\!75}a^{20}-\frac{60\!\cdots\!34}{20\!\cdots\!75}a^{19}-\frac{34\!\cdots\!62}{35\!\cdots\!75}a^{18}+\frac{12\!\cdots\!98}{35\!\cdots\!75}a^{17}+\frac{13\!\cdots\!96}{35\!\cdots\!75}a^{16}-\frac{54\!\cdots\!34}{35\!\cdots\!75}a^{15}-\frac{27\!\cdots\!09}{35\!\cdots\!75}a^{14}+\frac{12\!\cdots\!07}{35\!\cdots\!75}a^{13}+\frac{26\!\cdots\!29}{35\!\cdots\!75}a^{12}-\frac{15\!\cdots\!43}{35\!\cdots\!75}a^{11}-\frac{94\!\cdots\!11}{35\!\cdots\!75}a^{10}+\frac{64\!\cdots\!56}{20\!\cdots\!75}a^{9}-\frac{28\!\cdots\!99}{35\!\cdots\!75}a^{8}-\frac{39\!\cdots\!54}{35\!\cdots\!75}a^{7}+\frac{18\!\cdots\!72}{20\!\cdots\!75}a^{6}+\frac{65\!\cdots\!88}{35\!\cdots\!75}a^{5}-\frac{80\!\cdots\!69}{35\!\cdots\!75}a^{4}-\frac{26\!\cdots\!87}{35\!\cdots\!75}a^{3}+\frac{53\!\cdots\!73}{35\!\cdots\!75}a^{2}-\frac{21\!\cdots\!48}{70\!\cdots\!95}a-\frac{95\!\cdots\!88}{35\!\cdots\!75}$, $\frac{66\!\cdots\!79}{70\!\cdots\!95}a^{20}-\frac{10\!\cdots\!93}{35\!\cdots\!75}a^{19}-\frac{34\!\cdots\!58}{35\!\cdots\!75}a^{18}+\frac{12\!\cdots\!69}{35\!\cdots\!75}a^{17}+\frac{27\!\cdots\!27}{70\!\cdots\!95}a^{16}-\frac{11\!\cdots\!97}{70\!\cdots\!95}a^{15}-\frac{26\!\cdots\!52}{35\!\cdots\!75}a^{14}+\frac{12\!\cdots\!36}{35\!\cdots\!75}a^{13}+\frac{25\!\cdots\!41}{35\!\cdots\!75}a^{12}-\frac{15\!\cdots\!34}{35\!\cdots\!75}a^{11}-\frac{84\!\cdots\!84}{35\!\cdots\!75}a^{10}+\frac{11\!\cdots\!46}{35\!\cdots\!75}a^{9}-\frac{34\!\cdots\!19}{35\!\cdots\!75}a^{8}-\frac{40\!\cdots\!86}{35\!\cdots\!75}a^{7}+\frac{34\!\cdots\!83}{35\!\cdots\!75}a^{6}+\frac{65\!\cdots\!67}{35\!\cdots\!75}a^{5}-\frac{83\!\cdots\!72}{35\!\cdots\!75}a^{4}-\frac{25\!\cdots\!44}{35\!\cdots\!75}a^{3}+\frac{11\!\cdots\!91}{70\!\cdots\!95}a^{2}-\frac{33\!\cdots\!17}{70\!\cdots\!95}a-\frac{10\!\cdots\!33}{35\!\cdots\!75}$, $\frac{18\!\cdots\!27}{35\!\cdots\!75}a^{20}-\frac{58\!\cdots\!06}{35\!\cdots\!75}a^{19}-\frac{19\!\cdots\!31}{35\!\cdots\!75}a^{18}+\frac{40\!\cdots\!57}{20\!\cdots\!75}a^{17}+\frac{77\!\cdots\!11}{35\!\cdots\!75}a^{16}-\frac{31\!\cdots\!73}{35\!\cdots\!75}a^{15}-\frac{30\!\cdots\!61}{70\!\cdots\!95}a^{14}+\frac{71\!\cdots\!43}{35\!\cdots\!75}a^{13}+\frac{29\!\cdots\!87}{70\!\cdots\!95}a^{12}-\frac{90\!\cdots\!92}{35\!\cdots\!75}a^{11}-\frac{10\!\cdots\!41}{70\!\cdots\!95}a^{10}+\frac{36\!\cdots\!84}{20\!\cdots\!75}a^{9}-\frac{18\!\cdots\!82}{35\!\cdots\!75}a^{8}-\frac{22\!\cdots\!82}{35\!\cdots\!75}a^{7}+\frac{75\!\cdots\!88}{14\!\cdots\!19}a^{6}+\frac{37\!\cdots\!57}{35\!\cdots\!75}a^{5}-\frac{54\!\cdots\!27}{41\!\cdots\!35}a^{4}-\frac{14\!\cdots\!43}{35\!\cdots\!75}a^{3}+\frac{31\!\cdots\!13}{35\!\cdots\!75}a^{2}-\frac{76\!\cdots\!46}{35\!\cdots\!75}a-\frac{22\!\cdots\!82}{14\!\cdots\!19}$, $\frac{80\!\cdots\!44}{35\!\cdots\!75}a^{20}-\frac{25\!\cdots\!68}{35\!\cdots\!75}a^{19}-\frac{82\!\cdots\!87}{35\!\cdots\!75}a^{18}+\frac{29\!\cdots\!11}{35\!\cdots\!75}a^{17}+\frac{32\!\cdots\!94}{35\!\cdots\!75}a^{16}-\frac{13\!\cdots\!93}{35\!\cdots\!75}a^{15}-\frac{63\!\cdots\!57}{35\!\cdots\!75}a^{14}+\frac{61\!\cdots\!29}{70\!\cdots\!95}a^{13}+\frac{60\!\cdots\!28}{35\!\cdots\!75}a^{12}-\frac{77\!\cdots\!87}{70\!\cdots\!95}a^{11}-\frac{20\!\cdots\!37}{35\!\cdots\!75}a^{10}+\frac{53\!\cdots\!32}{70\!\cdots\!95}a^{9}-\frac{82\!\cdots\!96}{35\!\cdots\!75}a^{8}-\frac{97\!\cdots\!54}{35\!\cdots\!75}a^{7}+\frac{82\!\cdots\!06}{35\!\cdots\!75}a^{6}+\frac{15\!\cdots\!32}{35\!\cdots\!75}a^{5}-\frac{20\!\cdots\!77}{35\!\cdots\!75}a^{4}-\frac{64\!\cdots\!82}{35\!\cdots\!75}a^{3}+\frac{13\!\cdots\!22}{35\!\cdots\!75}a^{2}-\frac{29\!\cdots\!73}{35\!\cdots\!75}a-\frac{47\!\cdots\!92}{70\!\cdots\!95}$, $\frac{16\!\cdots\!09}{35\!\cdots\!75}a^{20}-\frac{50\!\cdots\!99}{35\!\cdots\!75}a^{19}-\frac{16\!\cdots\!78}{35\!\cdots\!75}a^{18}+\frac{58\!\cdots\!24}{35\!\cdots\!75}a^{17}+\frac{65\!\cdots\!49}{35\!\cdots\!75}a^{16}-\frac{26\!\cdots\!48}{35\!\cdots\!75}a^{15}-\frac{12\!\cdots\!31}{35\!\cdots\!75}a^{14}+\frac{61\!\cdots\!57}{35\!\cdots\!75}a^{13}+\frac{24\!\cdots\!52}{70\!\cdots\!95}a^{12}-\frac{76\!\cdots\!23}{35\!\cdots\!75}a^{11}-\frac{80\!\cdots\!49}{70\!\cdots\!95}a^{10}+\frac{53\!\cdots\!27}{35\!\cdots\!75}a^{9}-\frac{99\!\cdots\!07}{20\!\cdots\!75}a^{8}-\frac{19\!\cdots\!21}{35\!\cdots\!75}a^{7}+\frac{16\!\cdots\!72}{35\!\cdots\!75}a^{6}+\frac{31\!\cdots\!96}{35\!\cdots\!75}a^{5}-\frac{40\!\cdots\!16}{35\!\cdots\!75}a^{4}-\frac{48\!\cdots\!38}{14\!\cdots\!19}a^{3}+\frac{26\!\cdots\!92}{35\!\cdots\!75}a^{2}-\frac{81\!\cdots\!88}{35\!\cdots\!75}a-\frac{48\!\cdots\!81}{35\!\cdots\!75}$, $\frac{78\!\cdots\!63}{35\!\cdots\!75}a^{20}-\frac{24\!\cdots\!11}{35\!\cdots\!75}a^{19}-\frac{80\!\cdots\!18}{35\!\cdots\!75}a^{18}+\frac{29\!\cdots\!57}{35\!\cdots\!75}a^{17}+\frac{64\!\cdots\!43}{70\!\cdots\!95}a^{16}-\frac{13\!\cdots\!58}{35\!\cdots\!75}a^{15}-\frac{62\!\cdots\!37}{35\!\cdots\!75}a^{14}+\frac{30\!\cdots\!17}{35\!\cdots\!75}a^{13}+\frac{59\!\cdots\!02}{35\!\cdots\!75}a^{12}-\frac{38\!\cdots\!53}{35\!\cdots\!75}a^{11}-\frac{19\!\cdots\!48}{35\!\cdots\!75}a^{10}+\frac{26\!\cdots\!82}{35\!\cdots\!75}a^{9}-\frac{84\!\cdots\!31}{35\!\cdots\!75}a^{8}-\frac{95\!\cdots\!01}{35\!\cdots\!75}a^{7}+\frac{82\!\cdots\!62}{35\!\cdots\!75}a^{6}+\frac{15\!\cdots\!23}{35\!\cdots\!75}a^{5}-\frac{19\!\cdots\!67}{35\!\cdots\!75}a^{4}-\frac{61\!\cdots\!17}{35\!\cdots\!75}a^{3}+\frac{26\!\cdots\!22}{70\!\cdots\!95}a^{2}-\frac{36\!\cdots\!02}{35\!\cdots\!75}a-\frac{23\!\cdots\!34}{35\!\cdots\!75}$, $\frac{55\!\cdots\!39}{35\!\cdots\!75}a^{20}-\frac{68\!\cdots\!52}{14\!\cdots\!19}a^{19}-\frac{56\!\cdots\!14}{35\!\cdots\!75}a^{18}+\frac{19\!\cdots\!47}{35\!\cdots\!75}a^{17}+\frac{22\!\cdots\!94}{35\!\cdots\!75}a^{16}-\frac{90\!\cdots\!48}{35\!\cdots\!75}a^{15}-\frac{88\!\cdots\!32}{70\!\cdots\!95}a^{14}+\frac{20\!\cdots\!44}{35\!\cdots\!75}a^{13}+\frac{42\!\cdots\!92}{35\!\cdots\!75}a^{12}-\frac{26\!\cdots\!46}{35\!\cdots\!75}a^{11}-\frac{15\!\cdots\!48}{35\!\cdots\!75}a^{10}+\frac{17\!\cdots\!34}{35\!\cdots\!75}a^{9}-\frac{48\!\cdots\!17}{35\!\cdots\!75}a^{8}-\frac{65\!\cdots\!13}{35\!\cdots\!75}a^{7}+\frac{52\!\cdots\!83}{35\!\cdots\!75}a^{6}+\frac{21\!\cdots\!57}{70\!\cdots\!95}a^{5}-\frac{25\!\cdots\!02}{70\!\cdots\!95}a^{4}-\frac{43\!\cdots\!83}{35\!\cdots\!75}a^{3}+\frac{86\!\cdots\!32}{35\!\cdots\!75}a^{2}-\frac{11\!\cdots\!18}{35\!\cdots\!75}a-\frac{15\!\cdots\!92}{35\!\cdots\!75}$, $\frac{39\!\cdots\!56}{35\!\cdots\!75}a^{20}-\frac{49\!\cdots\!72}{14\!\cdots\!19}a^{19}-\frac{81\!\cdots\!79}{70\!\cdots\!95}a^{18}+\frac{14\!\cdots\!18}{35\!\cdots\!75}a^{17}+\frac{16\!\cdots\!03}{35\!\cdots\!75}a^{16}-\frac{66\!\cdots\!94}{35\!\cdots\!75}a^{15}-\frac{31\!\cdots\!23}{35\!\cdots\!75}a^{14}+\frac{15\!\cdots\!33}{35\!\cdots\!75}a^{13}+\frac{30\!\cdots\!79}{35\!\cdots\!75}a^{12}-\frac{19\!\cdots\!87}{35\!\cdots\!75}a^{11}-\frac{10\!\cdots\!36}{35\!\cdots\!75}a^{10}+\frac{77\!\cdots\!74}{20\!\cdots\!75}a^{9}-\frac{24\!\cdots\!16}{20\!\cdots\!75}a^{8}-\frac{96\!\cdots\!19}{70\!\cdots\!95}a^{7}+\frac{41\!\cdots\!12}{35\!\cdots\!75}a^{6}+\frac{77\!\cdots\!94}{35\!\cdots\!75}a^{5}-\frac{10\!\cdots\!38}{35\!\cdots\!75}a^{4}-\frac{60\!\cdots\!39}{70\!\cdots\!95}a^{3}+\frac{67\!\cdots\!39}{35\!\cdots\!75}a^{2}-\frac{20\!\cdots\!28}{35\!\cdots\!75}a-\frac{12\!\cdots\!87}{35\!\cdots\!75}$, $\frac{61\!\cdots\!92}{35\!\cdots\!75}a^{20}-\frac{19\!\cdots\!48}{35\!\cdots\!75}a^{19}-\frac{63\!\cdots\!91}{35\!\cdots\!75}a^{18}+\frac{44\!\cdots\!09}{70\!\cdots\!95}a^{17}+\frac{29\!\cdots\!57}{41\!\cdots\!35}a^{16}-\frac{10\!\cdots\!22}{35\!\cdots\!75}a^{15}-\frac{49\!\cdots\!09}{35\!\cdots\!75}a^{14}+\frac{23\!\cdots\!46}{35\!\cdots\!75}a^{13}+\frac{46\!\cdots\!56}{35\!\cdots\!75}a^{12}-\frac{29\!\cdots\!49}{35\!\cdots\!75}a^{11}-\frac{16\!\cdots\!44}{35\!\cdots\!75}a^{10}+\frac{20\!\cdots\!96}{35\!\cdots\!75}a^{9}-\frac{59\!\cdots\!06}{35\!\cdots\!75}a^{8}-\frac{74\!\cdots\!77}{35\!\cdots\!75}a^{7}+\frac{61\!\cdots\!87}{35\!\cdots\!75}a^{6}+\frac{12\!\cdots\!98}{35\!\cdots\!75}a^{5}-\frac{15\!\cdots\!64}{35\!\cdots\!75}a^{4}-\frac{98\!\cdots\!58}{70\!\cdots\!95}a^{3}+\frac{40\!\cdots\!24}{14\!\cdots\!19}a^{2}-\frac{18\!\cdots\!88}{35\!\cdots\!75}a-\frac{35\!\cdots\!31}{70\!\cdots\!95}$, $\frac{39\!\cdots\!17}{35\!\cdots\!75}a^{20}-\frac{12\!\cdots\!12}{35\!\cdots\!75}a^{19}-\frac{40\!\cdots\!56}{35\!\cdots\!75}a^{18}+\frac{14\!\cdots\!77}{35\!\cdots\!75}a^{17}+\frac{16\!\cdots\!08}{35\!\cdots\!75}a^{16}-\frac{26\!\cdots\!26}{14\!\cdots\!19}a^{15}-\frac{31\!\cdots\!82}{35\!\cdots\!75}a^{14}+\frac{15\!\cdots\!27}{35\!\cdots\!75}a^{13}+\frac{30\!\cdots\!83}{35\!\cdots\!75}a^{12}-\frac{11\!\cdots\!19}{20\!\cdots\!75}a^{11}-\frac{10\!\cdots\!37}{35\!\cdots\!75}a^{10}+\frac{13\!\cdots\!22}{35\!\cdots\!75}a^{9}-\frac{39\!\cdots\!54}{35\!\cdots\!75}a^{8}-\frac{27\!\cdots\!41}{20\!\cdots\!75}a^{7}+\frac{39\!\cdots\!61}{35\!\cdots\!75}a^{6}+\frac{15\!\cdots\!14}{70\!\cdots\!95}a^{5}-\frac{98\!\cdots\!17}{35\!\cdots\!75}a^{4}-\frac{30\!\cdots\!59}{35\!\cdots\!75}a^{3}+\frac{65\!\cdots\!09}{35\!\cdots\!75}a^{2}-\frac{18\!\cdots\!22}{35\!\cdots\!75}a-\frac{23\!\cdots\!26}{70\!\cdots\!95}$
| 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: | \( 4244799371179891.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{21}\cdot(2\pi)^{0}\cdot 4244799371179891.5 \cdot 1}{2\cdot\sqrt{1425682560385717640661443870033677687676469889}}\cr\approx \mathstrut & 0.117881502593591 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 100*x^19 + 456*x^18 + 3776*x^17 - 20240*x^16 - 65058*x^15 + 451646*x^14 + 423688*x^13 - 5452225*x^12 + 1655193*x^11 + 35371490*x^10 - 39386623*x^9 - 111354014*x^8 + 208156183*x^7 + 105332901*x^6 - 422345627*x^5 + 144428798*x^4 + 234723315*x^3 - 152780306*x^2 - 25641148*x + 26494201)
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^20 - 100*x^19 + 456*x^18 + 3776*x^17 - 20240*x^16 - 65058*x^15 + 451646*x^14 + 423688*x^13 - 5452225*x^12 + 1655193*x^11 + 35371490*x^10 - 39386623*x^9 - 111354014*x^8 + 208156183*x^7 + 105332901*x^6 - 422345627*x^5 + 144428798*x^4 + 234723315*x^3 - 152780306*x^2 - 25641148*x + 26494201, 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^20 - 100*x^19 + 456*x^18 + 3776*x^17 - 20240*x^16 - 65058*x^15 + 451646*x^14 + 423688*x^13 - 5452225*x^12 + 1655193*x^11 + 35371490*x^10 - 39386623*x^9 - 111354014*x^8 + 208156183*x^7 + 105332901*x^6 - 422345627*x^5 + 144428798*x^4 + 234723315*x^3 - 152780306*x^2 - 25641148*x + 26494201);
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^20 - 100*x^19 + 456*x^18 + 3776*x^17 - 20240*x^16 - 65058*x^15 + 451646*x^14 + 423688*x^13 - 5452225*x^12 + 1655193*x^11 + 35371490*x^10 - 39386623*x^9 - 111354014*x^8 + 208156183*x^7 + 105332901*x^6 - 422345627*x^5 + 144428798*x^4 + 234723315*x^3 - 152780306*x^2 - 25641148*x + 26494201);
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
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 cyclic group of order 21 |
| The 21 conjugacy class representatives for $C_{21}$ |
| Character table for $C_{21}$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 7.7.128100283921.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $21$ | $21$ | ${\href{/padicField/5.3.0.1}{3} }^{7}$ | R | $21$ | ${\href{/padicField/13.7.0.1}{7} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{7}$ | $21$ | $21$ | ${\href{/padicField/29.7.0.1}{7} }^{3}$ | $21$ | $21$ | ${\href{/padicField/41.7.0.1}{7} }^{3}$ | ${\href{/padicField/43.7.0.1}{7} }^{3}$ | $21$ | $21$ | $21$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| Deg $21$ | $3$ | $7$ | $14$ | |||
|
\(71\)
| 71.7.6.1 | $x^{7} + 71$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
| 71.7.6.1 | $x^{7} + 71$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ | |
| 71.7.6.1 | $x^{7} + 71$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |