Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 + 3*x^19 + 4*x^18 + 8*x^17 - 12*x^16 + 9*x^15 + 113*x^14 - 287*x^13 + 404*x^12 - 526*x^11 + 1106*x^10 - 2438*x^9 + 2381*x^8 - 2385*x^7 + 3637*x^6 - 6756*x^5 + 4734*x^4 - 3615*x^3 + 1513*x^2 - 2017*x - 1319)
gp: K = bnfinit(y^21 - y^20 + 3*y^19 + 4*y^18 + 8*y^17 - 12*y^16 + 9*y^15 + 113*y^14 - 287*y^13 + 404*y^12 - 526*y^11 + 1106*y^10 - 2438*y^9 + 2381*y^8 - 2385*y^7 + 3637*y^6 - 6756*y^5 + 4734*y^4 - 3615*y^3 + 1513*y^2 - 2017*y - 1319, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - x^20 + 3*x^19 + 4*x^18 + 8*x^17 - 12*x^16 + 9*x^15 + 113*x^14 - 287*x^13 + 404*x^12 - 526*x^11 + 1106*x^10 - 2438*x^9 + 2381*x^8 - 2385*x^7 + 3637*x^6 - 6756*x^5 + 4734*x^4 - 3615*x^3 + 1513*x^2 - 2017*x - 1319);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - x^20 + 3*x^19 + 4*x^18 + 8*x^17 - 12*x^16 + 9*x^15 + 113*x^14 - 287*x^13 + 404*x^12 - 526*x^11 + 1106*x^10 - 2438*x^9 + 2381*x^8 - 2385*x^7 + 3637*x^6 - 6756*x^5 + 4734*x^4 - 3615*x^3 + 1513*x^2 - 2017*x - 1319)
\( x^{21} - x^{20} + 3 x^{19} + 4 x^{18} + 8 x^{17} - 12 x^{16} + 9 x^{15} + 113 x^{14} - 287 x^{13} + \cdots - 1319 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[3, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-8971378199137136745057872231\)
\(\medspace = -\,7^{2}\cdot 13^{2}\cdot 71^{9}\cdot 4861^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.43\) | 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: | $7^{2/3}13^{2/3}71^{1/2}4861^{2/3}\approx 48918.580904922295$ | ||
| Ramified primes: |
\(7\), \(13\), \(71\), \(4861\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-71}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{7}a^{19}-\frac{2}{7}a^{18}-\frac{3}{7}a^{17}+\frac{2}{7}a^{16}+\frac{2}{7}a^{15}-\frac{2}{7}a^{14}+\frac{2}{7}a^{13}+\frac{1}{7}a^{12}-\frac{3}{7}a^{11}+\frac{2}{7}a^{9}-\frac{2}{7}a^{8}-\frac{2}{7}a^{7}-\frac{2}{7}a^{6}-\frac{1}{7}a^{5}+\frac{2}{7}a^{2}+\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{38\!\cdots\!11}a^{20}-\frac{22\!\cdots\!14}{38\!\cdots\!11}a^{19}-\frac{12\!\cdots\!40}{38\!\cdots\!11}a^{18}+\frac{14\!\cdots\!86}{38\!\cdots\!11}a^{17}+\frac{77\!\cdots\!94}{38\!\cdots\!11}a^{16}+\frac{53\!\cdots\!28}{38\!\cdots\!11}a^{15}+\frac{35\!\cdots\!05}{38\!\cdots\!11}a^{14}+\frac{13\!\cdots\!80}{38\!\cdots\!11}a^{13}+\frac{13\!\cdots\!70}{38\!\cdots\!11}a^{12}-\frac{19\!\cdots\!71}{38\!\cdots\!11}a^{11}-\frac{73\!\cdots\!73}{38\!\cdots\!11}a^{10}-\frac{64\!\cdots\!59}{38\!\cdots\!11}a^{9}+\frac{73\!\cdots\!55}{38\!\cdots\!11}a^{8}+\frac{14\!\cdots\!13}{38\!\cdots\!11}a^{7}-\frac{42\!\cdots\!92}{38\!\cdots\!11}a^{6}+\frac{16\!\cdots\!11}{38\!\cdots\!11}a^{5}-\frac{25\!\cdots\!92}{54\!\cdots\!73}a^{4}+\frac{47\!\cdots\!79}{38\!\cdots\!11}a^{3}+\frac{19\!\cdots\!03}{38\!\cdots\!11}a^{2}+\frac{91\!\cdots\!42}{38\!\cdots\!11}a-\frac{15\!\cdots\!23}{38\!\cdots\!11}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{63\!\cdots\!01}{38\!\cdots\!11}a^{20}-\frac{40\!\cdots\!07}{38\!\cdots\!11}a^{19}+\frac{13\!\cdots\!40}{38\!\cdots\!11}a^{18}+\frac{21\!\cdots\!94}{54\!\cdots\!73}a^{17}+\frac{57\!\cdots\!08}{38\!\cdots\!11}a^{16}-\frac{11\!\cdots\!26}{38\!\cdots\!11}a^{15}-\frac{96\!\cdots\!99}{38\!\cdots\!11}a^{14}+\frac{90\!\cdots\!64}{54\!\cdots\!73}a^{13}-\frac{19\!\cdots\!37}{54\!\cdots\!73}a^{12}+\frac{18\!\cdots\!67}{38\!\cdots\!11}a^{11}-\frac{33\!\cdots\!90}{38\!\cdots\!11}a^{10}+\frac{90\!\cdots\!47}{38\!\cdots\!11}a^{9}-\frac{15\!\cdots\!01}{38\!\cdots\!11}a^{8}+\frac{11\!\cdots\!87}{38\!\cdots\!11}a^{7}-\frac{25\!\cdots\!63}{54\!\cdots\!73}a^{6}+\frac{36\!\cdots\!37}{38\!\cdots\!11}a^{5}-\frac{50\!\cdots\!23}{54\!\cdots\!73}a^{4}+\frac{14\!\cdots\!77}{38\!\cdots\!11}a^{3}-\frac{43\!\cdots\!78}{38\!\cdots\!11}a^{2}+\frac{32\!\cdots\!24}{38\!\cdots\!11}a-\frac{11\!\cdots\!41}{38\!\cdots\!11}$, $\frac{21\!\cdots\!62}{38\!\cdots\!11}a^{20}-\frac{26\!\cdots\!77}{38\!\cdots\!11}a^{19}+\frac{57\!\cdots\!24}{38\!\cdots\!11}a^{18}+\frac{93\!\cdots\!58}{54\!\cdots\!73}a^{17}+\frac{12\!\cdots\!79}{38\!\cdots\!11}a^{16}-\frac{39\!\cdots\!12}{38\!\cdots\!11}a^{15}+\frac{55\!\cdots\!45}{38\!\cdots\!11}a^{14}+\frac{34\!\cdots\!42}{54\!\cdots\!73}a^{13}-\frac{95\!\cdots\!41}{54\!\cdots\!73}a^{12}+\frac{87\!\cdots\!75}{38\!\cdots\!11}a^{11}-\frac{11\!\cdots\!21}{38\!\cdots\!11}a^{10}+\frac{25\!\cdots\!68}{38\!\cdots\!11}a^{9}-\frac{56\!\cdots\!62}{38\!\cdots\!11}a^{8}+\frac{55\!\cdots\!99}{38\!\cdots\!11}a^{7}-\frac{66\!\cdots\!40}{54\!\cdots\!73}a^{6}+\frac{89\!\cdots\!40}{38\!\cdots\!11}a^{5}-\frac{21\!\cdots\!67}{54\!\cdots\!73}a^{4}+\frac{11\!\cdots\!84}{38\!\cdots\!11}a^{3}-\frac{64\!\cdots\!44}{38\!\cdots\!11}a^{2}+\frac{53\!\cdots\!85}{38\!\cdots\!11}a-\frac{11\!\cdots\!34}{38\!\cdots\!11}$, $\frac{34\!\cdots\!73}{38\!\cdots\!11}a^{20}-\frac{38\!\cdots\!21}{38\!\cdots\!11}a^{19}+\frac{31\!\cdots\!48}{38\!\cdots\!11}a^{18}+\frac{45\!\cdots\!26}{54\!\cdots\!73}a^{17}+\frac{88\!\cdots\!46}{38\!\cdots\!11}a^{16}+\frac{14\!\cdots\!06}{38\!\cdots\!11}a^{15}+\frac{32\!\cdots\!84}{38\!\cdots\!11}a^{14}+\frac{48\!\cdots\!99}{54\!\cdots\!73}a^{13}-\frac{13\!\cdots\!84}{54\!\cdots\!73}a^{12}+\frac{28\!\cdots\!80}{38\!\cdots\!11}a^{11}-\frac{41\!\cdots\!20}{38\!\cdots\!11}a^{10}+\frac{30\!\cdots\!45}{38\!\cdots\!11}a^{9}-\frac{90\!\cdots\!10}{38\!\cdots\!11}a^{8}+\frac{15\!\cdots\!62}{38\!\cdots\!11}a^{7}-\frac{45\!\cdots\!35}{54\!\cdots\!73}a^{6}+\frac{32\!\cdots\!87}{38\!\cdots\!11}a^{5}-\frac{38\!\cdots\!94}{54\!\cdots\!73}a^{4}+\frac{47\!\cdots\!30}{38\!\cdots\!11}a^{3}-\frac{68\!\cdots\!31}{38\!\cdots\!11}a^{2}-\frac{12\!\cdots\!92}{38\!\cdots\!11}a-\frac{98\!\cdots\!48}{38\!\cdots\!11}$, $\frac{51\!\cdots\!14}{54\!\cdots\!73}a^{20}+\frac{92\!\cdots\!84}{38\!\cdots\!11}a^{19}-\frac{36\!\cdots\!99}{38\!\cdots\!11}a^{18}+\frac{40\!\cdots\!12}{38\!\cdots\!11}a^{17}+\frac{78\!\cdots\!47}{38\!\cdots\!11}a^{16}+\frac{39\!\cdots\!21}{38\!\cdots\!11}a^{15}-\frac{16\!\cdots\!41}{38\!\cdots\!11}a^{14}+\frac{45\!\cdots\!55}{38\!\cdots\!11}a^{13}+\frac{69\!\cdots\!76}{38\!\cdots\!11}a^{12}-\frac{18\!\cdots\!38}{38\!\cdots\!11}a^{11}+\frac{38\!\cdots\!90}{54\!\cdots\!73}a^{10}-\frac{11\!\cdots\!54}{38\!\cdots\!11}a^{9}+\frac{38\!\cdots\!63}{38\!\cdots\!11}a^{8}-\frac{20\!\cdots\!76}{38\!\cdots\!11}a^{7}+\frac{88\!\cdots\!54}{38\!\cdots\!11}a^{6}-\frac{31\!\cdots\!18}{38\!\cdots\!11}a^{5}+\frac{30\!\cdots\!17}{54\!\cdots\!73}a^{4}-\frac{87\!\cdots\!30}{54\!\cdots\!73}a^{3}+\frac{20\!\cdots\!82}{38\!\cdots\!11}a^{2}-\frac{37\!\cdots\!70}{38\!\cdots\!11}a+\frac{32\!\cdots\!94}{38\!\cdots\!11}$, $\frac{87\!\cdots\!04}{54\!\cdots\!73}a^{20}-\frac{56\!\cdots\!13}{38\!\cdots\!11}a^{19}+\frac{66\!\cdots\!20}{38\!\cdots\!11}a^{18}-\frac{31\!\cdots\!06}{38\!\cdots\!11}a^{17}+\frac{47\!\cdots\!90}{38\!\cdots\!11}a^{16}-\frac{24\!\cdots\!11}{38\!\cdots\!11}a^{15}-\frac{32\!\cdots\!44}{38\!\cdots\!11}a^{14}+\frac{20\!\cdots\!33}{38\!\cdots\!11}a^{13}-\frac{19\!\cdots\!23}{38\!\cdots\!11}a^{12}+\frac{10\!\cdots\!26}{38\!\cdots\!11}a^{11}-\frac{52\!\cdots\!74}{54\!\cdots\!73}a^{10}+\frac{88\!\cdots\!68}{38\!\cdots\!11}a^{9}-\frac{14\!\cdots\!70}{38\!\cdots\!11}a^{8}+\frac{17\!\cdots\!19}{38\!\cdots\!11}a^{7}-\frac{14\!\cdots\!25}{38\!\cdots\!11}a^{6}+\frac{52\!\cdots\!93}{38\!\cdots\!11}a^{5}-\frac{31\!\cdots\!35}{54\!\cdots\!73}a^{4}+\frac{69\!\cdots\!13}{54\!\cdots\!73}a^{3}-\frac{32\!\cdots\!29}{38\!\cdots\!11}a^{2}+\frac{79\!\cdots\!70}{38\!\cdots\!11}a+\frac{40\!\cdots\!88}{38\!\cdots\!11}$, $\frac{15\!\cdots\!17}{38\!\cdots\!11}a^{20}-\frac{57\!\cdots\!92}{38\!\cdots\!11}a^{19}+\frac{16\!\cdots\!99}{38\!\cdots\!11}a^{18}-\frac{14\!\cdots\!74}{54\!\cdots\!73}a^{17}+\frac{12\!\cdots\!18}{38\!\cdots\!11}a^{16}-\frac{17\!\cdots\!19}{38\!\cdots\!11}a^{15}+\frac{12\!\cdots\!25}{38\!\cdots\!11}a^{14}+\frac{10\!\cdots\!51}{54\!\cdots\!73}a^{13}-\frac{13\!\cdots\!79}{54\!\cdots\!73}a^{12}+\frac{27\!\cdots\!34}{38\!\cdots\!11}a^{11}-\frac{41\!\cdots\!25}{38\!\cdots\!11}a^{10}+\frac{61\!\cdots\!40}{38\!\cdots\!11}a^{9}-\frac{10\!\cdots\!51}{38\!\cdots\!11}a^{8}+\frac{20\!\cdots\!73}{38\!\cdots\!11}a^{7}-\frac{36\!\cdots\!63}{54\!\cdots\!73}a^{6}+\frac{21\!\cdots\!06}{38\!\cdots\!11}a^{5}-\frac{48\!\cdots\!23}{54\!\cdots\!73}a^{4}+\frac{52\!\cdots\!44}{38\!\cdots\!11}a^{3}-\frac{58\!\cdots\!79}{38\!\cdots\!11}a^{2}+\frac{21\!\cdots\!15}{38\!\cdots\!11}a-\frac{16\!\cdots\!40}{38\!\cdots\!11}$, $\frac{46\!\cdots\!02}{38\!\cdots\!11}a^{20}-\frac{20\!\cdots\!56}{54\!\cdots\!73}a^{19}+\frac{33\!\cdots\!73}{54\!\cdots\!73}a^{18}-\frac{14\!\cdots\!87}{38\!\cdots\!11}a^{17}-\frac{34\!\cdots\!10}{38\!\cdots\!11}a^{16}-\frac{17\!\cdots\!35}{38\!\cdots\!11}a^{15}+\frac{28\!\cdots\!46}{38\!\cdots\!11}a^{14}+\frac{92\!\cdots\!50}{38\!\cdots\!11}a^{13}-\frac{30\!\cdots\!29}{38\!\cdots\!11}a^{12}+\frac{42\!\cdots\!91}{38\!\cdots\!11}a^{11}-\frac{66\!\cdots\!58}{38\!\cdots\!11}a^{10}+\frac{84\!\cdots\!45}{38\!\cdots\!11}a^{9}-\frac{18\!\cdots\!28}{38\!\cdots\!11}a^{8}+\frac{36\!\cdots\!13}{38\!\cdots\!11}a^{7}-\frac{26\!\cdots\!01}{38\!\cdots\!11}a^{6}+\frac{35\!\cdots\!81}{38\!\cdots\!11}a^{5}-\frac{49\!\cdots\!44}{54\!\cdots\!73}a^{4}+\frac{10\!\cdots\!90}{38\!\cdots\!11}a^{3}-\frac{28\!\cdots\!73}{38\!\cdots\!11}a^{2}+\frac{38\!\cdots\!49}{38\!\cdots\!11}a-\frac{45\!\cdots\!08}{38\!\cdots\!11}$, $\frac{43\!\cdots\!27}{38\!\cdots\!11}a^{20}-\frac{38\!\cdots\!72}{38\!\cdots\!11}a^{19}+\frac{23\!\cdots\!05}{38\!\cdots\!11}a^{18}-\frac{65\!\cdots\!94}{38\!\cdots\!11}a^{17}-\frac{24\!\cdots\!52}{54\!\cdots\!73}a^{16}-\frac{40\!\cdots\!43}{38\!\cdots\!11}a^{15}+\frac{24\!\cdots\!44}{38\!\cdots\!11}a^{14}+\frac{44\!\cdots\!31}{38\!\cdots\!11}a^{13}-\frac{50\!\cdots\!15}{38\!\cdots\!11}a^{12}+\frac{91\!\cdots\!31}{38\!\cdots\!11}a^{11}-\frac{99\!\cdots\!81}{38\!\cdots\!11}a^{10}+\frac{13\!\cdots\!52}{38\!\cdots\!11}a^{9}-\frac{47\!\cdots\!31}{54\!\cdots\!73}a^{8}+\frac{92\!\cdots\!37}{54\!\cdots\!73}a^{7}-\frac{37\!\cdots\!66}{38\!\cdots\!11}a^{6}+\frac{50\!\cdots\!89}{38\!\cdots\!11}a^{5}-\frac{13\!\cdots\!44}{54\!\cdots\!73}a^{4}+\frac{14\!\cdots\!64}{38\!\cdots\!11}a^{3}-\frac{72\!\cdots\!27}{54\!\cdots\!73}a^{2}+\frac{53\!\cdots\!63}{38\!\cdots\!11}a+\frac{51\!\cdots\!02}{38\!\cdots\!11}$, $\frac{29\!\cdots\!68}{38\!\cdots\!11}a^{20}-\frac{49\!\cdots\!00}{54\!\cdots\!73}a^{19}+\frac{12\!\cdots\!93}{54\!\cdots\!73}a^{18}+\frac{88\!\cdots\!41}{38\!\cdots\!11}a^{17}+\frac{19\!\cdots\!22}{38\!\cdots\!11}a^{16}-\frac{44\!\cdots\!77}{38\!\cdots\!11}a^{15}+\frac{22\!\cdots\!99}{38\!\cdots\!11}a^{14}+\frac{29\!\cdots\!40}{38\!\cdots\!11}a^{13}-\frac{90\!\cdots\!30}{38\!\cdots\!11}a^{12}+\frac{13\!\cdots\!62}{38\!\cdots\!11}a^{11}-\frac{18\!\cdots\!74}{38\!\cdots\!11}a^{10}+\frac{35\!\cdots\!05}{38\!\cdots\!11}a^{9}-\frac{77\!\cdots\!81}{38\!\cdots\!11}a^{8}+\frac{87\!\cdots\!60}{38\!\cdots\!11}a^{7}-\frac{93\!\cdots\!34}{38\!\cdots\!11}a^{6}+\frac{13\!\cdots\!65}{38\!\cdots\!11}a^{5}-\frac{31\!\cdots\!53}{54\!\cdots\!73}a^{4}+\frac{20\!\cdots\!03}{38\!\cdots\!11}a^{3}-\frac{17\!\cdots\!43}{38\!\cdots\!11}a^{2}+\frac{12\!\cdots\!32}{38\!\cdots\!11}a-\frac{83\!\cdots\!44}{38\!\cdots\!11}$, $\frac{14\!\cdots\!91}{38\!\cdots\!11}a^{20}-\frac{19\!\cdots\!73}{38\!\cdots\!11}a^{19}+\frac{19\!\cdots\!92}{38\!\cdots\!11}a^{18}+\frac{16\!\cdots\!56}{38\!\cdots\!11}a^{17}-\frac{57\!\cdots\!64}{54\!\cdots\!73}a^{16}-\frac{46\!\cdots\!46}{38\!\cdots\!11}a^{15}-\frac{39\!\cdots\!93}{38\!\cdots\!11}a^{14}+\frac{12\!\cdots\!46}{38\!\cdots\!11}a^{13}-\frac{47\!\cdots\!09}{38\!\cdots\!11}a^{12}+\frac{48\!\cdots\!82}{38\!\cdots\!11}a^{11}-\frac{60\!\cdots\!77}{38\!\cdots\!11}a^{10}+\frac{16\!\cdots\!21}{38\!\cdots\!11}a^{9}-\frac{43\!\cdots\!30}{54\!\cdots\!73}a^{8}+\frac{41\!\cdots\!44}{54\!\cdots\!73}a^{7}-\frac{15\!\cdots\!58}{38\!\cdots\!11}a^{6}+\frac{64\!\cdots\!28}{38\!\cdots\!11}a^{5}-\frac{80\!\cdots\!42}{54\!\cdots\!73}a^{4}+\frac{52\!\cdots\!61}{38\!\cdots\!11}a^{3}-\frac{21\!\cdots\!06}{54\!\cdots\!73}a^{2}+\frac{34\!\cdots\!63}{38\!\cdots\!11}a+\frac{33\!\cdots\!98}{38\!\cdots\!11}$, $\frac{14\!\cdots\!91}{38\!\cdots\!11}a^{20}-\frac{19\!\cdots\!73}{38\!\cdots\!11}a^{19}+\frac{19\!\cdots\!92}{38\!\cdots\!11}a^{18}+\frac{16\!\cdots\!56}{38\!\cdots\!11}a^{17}-\frac{57\!\cdots\!64}{54\!\cdots\!73}a^{16}-\frac{46\!\cdots\!46}{38\!\cdots\!11}a^{15}-\frac{39\!\cdots\!93}{38\!\cdots\!11}a^{14}+\frac{12\!\cdots\!46}{38\!\cdots\!11}a^{13}-\frac{47\!\cdots\!09}{38\!\cdots\!11}a^{12}+\frac{48\!\cdots\!82}{38\!\cdots\!11}a^{11}-\frac{60\!\cdots\!77}{38\!\cdots\!11}a^{10}+\frac{16\!\cdots\!21}{38\!\cdots\!11}a^{9}-\frac{43\!\cdots\!30}{54\!\cdots\!73}a^{8}+\frac{41\!\cdots\!44}{54\!\cdots\!73}a^{7}-\frac{15\!\cdots\!58}{38\!\cdots\!11}a^{6}+\frac{64\!\cdots\!28}{38\!\cdots\!11}a^{5}-\frac{80\!\cdots\!42}{54\!\cdots\!73}a^{4}+\frac{52\!\cdots\!61}{38\!\cdots\!11}a^{3}-\frac{21\!\cdots\!06}{54\!\cdots\!73}a^{2}+\frac{34\!\cdots\!63}{38\!\cdots\!11}a+\frac{71\!\cdots\!09}{38\!\cdots\!11}$
| 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: | \( 277898.768894 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 277898.768894 \cdot 1}{2\cdot\sqrt{8971378199137136745057872231}}\cr\approx \mathstrut & 0.179116604085 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 + 3*x^19 + 4*x^18 + 8*x^17 - 12*x^16 + 9*x^15 + 113*x^14 - 287*x^13 + 404*x^12 - 526*x^11 + 1106*x^10 - 2438*x^9 + 2381*x^8 - 2385*x^7 + 3637*x^6 - 6756*x^5 + 4734*x^4 - 3615*x^3 + 1513*x^2 - 2017*x - 1319)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^21 - x^20 + 3*x^19 + 4*x^18 + 8*x^17 - 12*x^16 + 9*x^15 + 113*x^14 - 287*x^13 + 404*x^12 - 526*x^11 + 1106*x^10 - 2438*x^9 + 2381*x^8 - 2385*x^7 + 3637*x^6 - 6756*x^5 + 4734*x^4 - 3615*x^3 + 1513*x^2 - 2017*x - 1319, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^21 - x^20 + 3*x^19 + 4*x^18 + 8*x^17 - 12*x^16 + 9*x^15 + 113*x^14 - 287*x^13 + 404*x^12 - 526*x^11 + 1106*x^10 - 2438*x^9 + 2381*x^8 - 2385*x^7 + 3637*x^6 - 6756*x^5 + 4734*x^4 - 3615*x^3 + 1513*x^2 - 2017*x - 1319);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - x^20 + 3*x^19 + 4*x^18 + 8*x^17 - 12*x^16 + 9*x^15 + 113*x^14 - 287*x^13 + 404*x^12 - 526*x^11 + 1106*x^10 - 2438*x^9 + 2381*x^8 - 2385*x^7 + 3637*x^6 - 6756*x^5 + 4734*x^4 - 3615*x^3 + 1513*x^2 - 2017*x - 1319);
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
$C_3^7:D_7$ (as 21T76):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 30618 |
| The 288 conjugacy class representatives for $C_3^7:D_7$ are not computed |
| Character table for $C_3^7:D_7$ is not computed |
Intermediate fields
| 7.1.357911.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 21 siblings: | data not computed |
| 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 | $21$ | ${\href{/padicField/3.7.0.1}{7} }^{3}$ | $21$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.3.0.1}{3} }$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | $21$ | ${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.7.0.1}{7} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $21$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{6}$ | $21$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(13\)
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.3.2.1 | $x^{3} + 26$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(71\)
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.3.0.1 | $x^{3} + 4 x + 64$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 71.6.3.2 | $x^{6} + 221 x^{4} + 128 x^{3} + 15139 x^{2} - 26752 x + 322815$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(4861\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $3$ | $3$ | $1$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |