Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 + x^19 - 5*x^18 + 8*x^17 - 5*x^16 + 11*x^15 - 40*x^14 + 33*x^13 + 2*x^12 + 62*x^11 - 87*x^10 - 12*x^9 - 43*x^8 + 76*x^7 + 27*x^6 - 15*x^5 - 7*x^4 - 10*x^3 + 2*x^2 + x + 1)
gp: K = bnfinit(y^21 + y^19 - 5*y^18 + 8*y^17 - 5*y^16 + 11*y^15 - 40*y^14 + 33*y^13 + 2*y^12 + 62*y^11 - 87*y^10 - 12*y^9 - 43*y^8 + 76*y^7 + 27*y^6 - 15*y^5 - 7*y^4 - 10*y^3 + 2*y^2 + y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 + x^19 - 5*x^18 + 8*x^17 - 5*x^16 + 11*x^15 - 40*x^14 + 33*x^13 + 2*x^12 + 62*x^11 - 87*x^10 - 12*x^9 - 43*x^8 + 76*x^7 + 27*x^6 - 15*x^5 - 7*x^4 - 10*x^3 + 2*x^2 + x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 + x^19 - 5*x^18 + 8*x^17 - 5*x^16 + 11*x^15 - 40*x^14 + 33*x^13 + 2*x^12 + 62*x^11 - 87*x^10 - 12*x^9 - 43*x^8 + 76*x^7 + 27*x^6 - 15*x^5 - 7*x^4 - 10*x^3 + 2*x^2 + x + 1)
\( x^{21} + x^{19} - 5 x^{18} + 8 x^{17} - 5 x^{16} + 11 x^{15} - 40 x^{14} + 33 x^{13} + 2 x^{12} + \cdots + 1 \)
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: | $[1, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(23714597990105054274863104\)
\(\medspace = 2^{14}\cdot 11^{8}\cdot 41^{3}\cdot 461^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.16\) | 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: | $2^{2/3}11^{1/2}41^{1/2}461^{1/2}\approx 723.8113641062746$ | ||
| Ramified primes: |
\(2\), \(11\), \(41\), \(461\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{18901}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{11}a^{19}-\frac{4}{11}a^{18}+\frac{2}{11}a^{17}+\frac{3}{11}a^{16}-\frac{1}{11}a^{15}-\frac{2}{11}a^{14}+\frac{1}{11}a^{13}-\frac{3}{11}a^{12}-\frac{3}{11}a^{11}+\frac{4}{11}a^{10}+\frac{3}{11}a^{9}-\frac{5}{11}a^{8}-\frac{4}{11}a^{7}+\frac{4}{11}a^{6}-\frac{1}{11}a^{5}+\frac{4}{11}a^{4}-\frac{5}{11}a^{3}-\frac{3}{11}a^{2}+\frac{3}{11}$, $\frac{1}{13\!\cdots\!59}a^{20}+\frac{12286583537248}{13\!\cdots\!59}a^{19}+\frac{191668500441346}{13\!\cdots\!59}a^{18}-\frac{345610731565114}{13\!\cdots\!59}a^{17}+\frac{422414217258764}{13\!\cdots\!59}a^{16}+\frac{488917770915768}{13\!\cdots\!59}a^{15}-\frac{581584913027979}{13\!\cdots\!59}a^{14}-\frac{351668375273832}{13\!\cdots\!59}a^{13}+\frac{161636401398155}{13\!\cdots\!59}a^{12}+\frac{172805438621979}{13\!\cdots\!59}a^{11}+\frac{235695191095}{120799188155869}a^{10}-\frac{411605789161335}{13\!\cdots\!59}a^{9}+\frac{82949821253362}{13\!\cdots\!59}a^{8}+\frac{300167517426487}{13\!\cdots\!59}a^{7}+\frac{94700910671880}{13\!\cdots\!59}a^{6}-\frac{260498236608982}{13\!\cdots\!59}a^{5}-\frac{89433799389327}{13\!\cdots\!59}a^{4}-\frac{545203816347170}{13\!\cdots\!59}a^{3}-\frac{199115360932769}{13\!\cdots\!59}a^{2}+\frac{65972783522363}{13\!\cdots\!59}a+\frac{156756481586128}{13\!\cdots\!59}$
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: | $10$ | 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: |
$\frac{755310368979667}{13\!\cdots\!59}a^{20}+\frac{53748058787158}{13\!\cdots\!59}a^{19}+\frac{826826649780458}{13\!\cdots\!59}a^{18}-\frac{36\!\cdots\!74}{13\!\cdots\!59}a^{17}+\frac{59\!\cdots\!51}{13\!\cdots\!59}a^{16}-\frac{35\!\cdots\!28}{13\!\cdots\!59}a^{15}+\frac{85\!\cdots\!50}{13\!\cdots\!59}a^{14}-\frac{29\!\cdots\!74}{13\!\cdots\!59}a^{13}+\frac{23\!\cdots\!82}{13\!\cdots\!59}a^{12}+\frac{13\!\cdots\!18}{13\!\cdots\!59}a^{11}+\frac{43\!\cdots\!02}{120799188155869}a^{10}-\frac{62\!\cdots\!54}{13\!\cdots\!59}a^{9}-\frac{95\!\cdots\!36}{13\!\cdots\!59}a^{8}-\frac{34\!\cdots\!96}{13\!\cdots\!59}a^{7}+\frac{54\!\cdots\!28}{13\!\cdots\!59}a^{6}+\frac{20\!\cdots\!04}{13\!\cdots\!59}a^{5}-\frac{10\!\cdots\!29}{13\!\cdots\!59}a^{4}-\frac{47\!\cdots\!32}{13\!\cdots\!59}a^{3}-\frac{73\!\cdots\!78}{13\!\cdots\!59}a^{2}+\frac{13\!\cdots\!30}{13\!\cdots\!59}a+\frac{684626140719557}{13\!\cdots\!59}$, $\frac{297059834235095}{13\!\cdots\!59}a^{20}+\frac{360000003954201}{13\!\cdots\!59}a^{19}+\frac{183746038812464}{13\!\cdots\!59}a^{18}-\frac{12\!\cdots\!03}{13\!\cdots\!59}a^{17}+\frac{364694720935607}{13\!\cdots\!59}a^{16}+\frac{16\!\cdots\!69}{13\!\cdots\!59}a^{15}+\frac{751787780176827}{13\!\cdots\!59}a^{14}-\frac{80\!\cdots\!44}{13\!\cdots\!59}a^{13}-\frac{56\!\cdots\!77}{13\!\cdots\!59}a^{12}+\frac{15\!\cdots\!54}{13\!\cdots\!59}a^{11}+\frac{16\!\cdots\!51}{120799188155869}a^{10}-\frac{50\!\cdots\!16}{13\!\cdots\!59}a^{9}-\frac{40\!\cdots\!10}{13\!\cdots\!59}a^{8}-\frac{14\!\cdots\!62}{13\!\cdots\!59}a^{7}+\frac{94\!\cdots\!06}{13\!\cdots\!59}a^{6}+\frac{37\!\cdots\!62}{13\!\cdots\!59}a^{5}+\frac{51\!\cdots\!33}{13\!\cdots\!59}a^{4}-\frac{87\!\cdots\!53}{13\!\cdots\!59}a^{3}-\frac{14\!\cdots\!69}{13\!\cdots\!59}a^{2}-\frac{31\!\cdots\!84}{13\!\cdots\!59}a-\frac{150840250017882}{13\!\cdots\!59}$, $\frac{442527599481139}{13\!\cdots\!59}a^{20}+\frac{527034342486649}{13\!\cdots\!59}a^{19}+\frac{515538102398970}{13\!\cdots\!59}a^{18}-\frac{16\!\cdots\!71}{13\!\cdots\!59}a^{17}+\frac{10\!\cdots\!60}{13\!\cdots\!59}a^{16}+\frac{16\!\cdots\!21}{13\!\cdots\!59}a^{15}+\frac{25\!\cdots\!09}{13\!\cdots\!59}a^{14}-\frac{12\!\cdots\!32}{13\!\cdots\!59}a^{13}-\frac{55\!\cdots\!70}{13\!\cdots\!59}a^{12}+\frac{15\!\cdots\!02}{13\!\cdots\!59}a^{11}+\frac{26\!\cdots\!39}{120799188155869}a^{10}-\frac{65\!\cdots\!38}{13\!\cdots\!59}a^{9}-\frac{44\!\cdots\!26}{13\!\cdots\!59}a^{8}-\frac{27\!\cdots\!18}{13\!\cdots\!59}a^{7}+\frac{91\!\cdots\!72}{13\!\cdots\!59}a^{6}+\frac{42\!\cdots\!42}{13\!\cdots\!59}a^{5}+\frac{73\!\cdots\!14}{13\!\cdots\!59}a^{4}-\frac{76\!\cdots\!19}{13\!\cdots\!59}a^{3}-\frac{20\!\cdots\!27}{13\!\cdots\!59}a^{2}-\frac{10\!\cdots\!36}{13\!\cdots\!59}a-\frac{318522420098555}{13\!\cdots\!59}$, $\frac{53748058787158}{13\!\cdots\!59}a^{20}+\frac{71516280800791}{13\!\cdots\!59}a^{19}+\frac{112984138599261}{13\!\cdots\!59}a^{18}-\frac{115213485757685}{13\!\cdots\!59}a^{17}+\frac{244023795718907}{13\!\cdots\!59}a^{16}+\frac{194102467565813}{13\!\cdots\!59}a^{15}+\frac{540564902171506}{13\!\cdots\!59}a^{14}-\frac{12\!\cdots\!29}{13\!\cdots\!59}a^{13}-\frac{145822893715116}{13\!\cdots\!59}a^{12}+\frac{992712973733868}{13\!\cdots\!59}a^{11}+\frac{317882427562025}{120799188155869}a^{10}-\frac{505607399786732}{13\!\cdots\!59}a^{9}-\frac{18\!\cdots\!15}{13\!\cdots\!59}a^{8}-\frac{28\!\cdots\!64}{13\!\cdots\!59}a^{7}+\frac{8597974636695}{13\!\cdots\!59}a^{6}+\frac{10\!\cdots\!76}{13\!\cdots\!59}a^{5}+\frac{517549734835537}{13\!\cdots\!59}a^{4}+\frac{206427656617492}{13\!\cdots\!59}a^{3}-\frac{135837380624104}{13\!\cdots\!59}a^{2}-\frac{70684228260110}{13\!\cdots\!59}a-\frac{755310368979667}{13\!\cdots\!59}$, $\frac{315749009474460}{13\!\cdots\!59}a^{20}+\frac{483218921332671}{13\!\cdots\!59}a^{19}+\frac{444323145635316}{13\!\cdots\!59}a^{18}-\frac{982626167866439}{13\!\cdots\!59}a^{17}+\frac{294969879507654}{13\!\cdots\!59}a^{16}+\frac{17\!\cdots\!04}{13\!\cdots\!59}a^{15}+\frac{15\!\cdots\!57}{13\!\cdots\!59}a^{14}-\frac{73\!\cdots\!64}{13\!\cdots\!59}a^{13}-\frac{77\!\cdots\!94}{13\!\cdots\!59}a^{12}+\frac{12\!\cdots\!31}{13\!\cdots\!59}a^{11}+\frac{20\!\cdots\!62}{13\!\cdots\!59}a^{10}+\frac{46\!\cdots\!36}{13\!\cdots\!59}a^{9}-\frac{37\!\cdots\!02}{13\!\cdots\!59}a^{8}-\frac{21\!\cdots\!23}{13\!\cdots\!59}a^{7}-\frac{54\!\cdots\!00}{13\!\cdots\!59}a^{6}+\frac{34\!\cdots\!13}{13\!\cdots\!59}a^{5}+\frac{85\!\cdots\!61}{13\!\cdots\!59}a^{4}+\frac{16\!\cdots\!18}{13\!\cdots\!59}a^{3}-\frac{182234544528305}{120799188155869}a^{2}-\frac{22\!\cdots\!86}{13\!\cdots\!59}a-\frac{68182380271880}{120799188155869}$, $\frac{275454944961918}{13\!\cdots\!59}a^{20}-\frac{132304543061547}{13\!\cdots\!59}a^{19}+\frac{237745114453712}{13\!\cdots\!59}a^{18}-\frac{15\!\cdots\!14}{13\!\cdots\!59}a^{17}+\frac{28\!\cdots\!11}{13\!\cdots\!59}a^{16}-\frac{22\!\cdots\!21}{13\!\cdots\!59}a^{15}+\frac{36\!\cdots\!13}{13\!\cdots\!59}a^{14}-\frac{12\!\cdots\!50}{13\!\cdots\!59}a^{13}+\frac{13\!\cdots\!30}{13\!\cdots\!59}a^{12}-\frac{25\!\cdots\!62}{13\!\cdots\!59}a^{11}+\frac{17\!\cdots\!63}{13\!\cdots\!59}a^{10}-\frac{33\!\cdots\!71}{13\!\cdots\!59}a^{9}+\frac{42\!\cdots\!62}{13\!\cdots\!59}a^{8}-\frac{888178175816530}{120799188155869}a^{7}+\frac{32\!\cdots\!89}{13\!\cdots\!59}a^{6}+\frac{18\!\cdots\!84}{13\!\cdots\!59}a^{5}-\frac{10\!\cdots\!17}{13\!\cdots\!59}a^{4}-\frac{71\!\cdots\!52}{13\!\cdots\!59}a^{3}-\frac{19\!\cdots\!19}{13\!\cdots\!59}a^{2}+\frac{23\!\cdots\!57}{13\!\cdots\!59}a+\frac{13\!\cdots\!62}{13\!\cdots\!59}$, $\frac{683604998634152}{13\!\cdots\!59}a^{20}-\frac{56110509955374}{13\!\cdots\!59}a^{19}+\frac{10\!\cdots\!60}{13\!\cdots\!59}a^{18}-\frac{33\!\cdots\!37}{13\!\cdots\!59}a^{17}+\frac{61\!\cdots\!63}{13\!\cdots\!59}a^{16}-\frac{471480857370338}{120799188155869}a^{15}+\frac{96\!\cdots\!74}{13\!\cdots\!59}a^{14}-\frac{28\!\cdots\!65}{13\!\cdots\!59}a^{13}+\frac{27\!\cdots\!35}{13\!\cdots\!59}a^{12}-\frac{11\!\cdots\!44}{13\!\cdots\!59}a^{11}+\frac{46\!\cdots\!56}{13\!\cdots\!59}a^{10}-\frac{53\!\cdots\!88}{120799188155869}a^{9}+\frac{16\!\cdots\!34}{13\!\cdots\!59}a^{8}-\frac{44\!\cdots\!42}{13\!\cdots\!59}a^{7}+\frac{42\!\cdots\!65}{13\!\cdots\!59}a^{6}-\frac{27\!\cdots\!18}{13\!\cdots\!59}a^{5}-\frac{765353918379492}{13\!\cdots\!59}a^{4}+\frac{97\!\cdots\!05}{13\!\cdots\!59}a^{3}-\frac{63\!\cdots\!48}{13\!\cdots\!59}a^{2}+\frac{39\!\cdots\!96}{13\!\cdots\!59}a-\frac{977766694681552}{13\!\cdots\!59}$, $\frac{380475925410286}{13\!\cdots\!59}a^{20}+\frac{220496744457618}{13\!\cdots\!59}a^{19}+\frac{696306711748832}{13\!\cdots\!59}a^{18}-\frac{15\!\cdots\!67}{13\!\cdots\!59}a^{17}+\frac{24\!\cdots\!07}{13\!\cdots\!59}a^{16}-\frac{14\!\cdots\!97}{13\!\cdots\!59}a^{15}+\frac{51\!\cdots\!00}{13\!\cdots\!59}a^{14}-\frac{13\!\cdots\!00}{13\!\cdots\!59}a^{13}+\frac{75\!\cdots\!16}{13\!\cdots\!59}a^{12}-\frac{284462633524552}{120799188155869}a^{11}+\frac{30\!\cdots\!24}{13\!\cdots\!59}a^{10}-\frac{20\!\cdots\!75}{13\!\cdots\!59}a^{9}-\frac{17\!\cdots\!29}{13\!\cdots\!59}a^{8}-\frac{35\!\cdots\!78}{13\!\cdots\!59}a^{7}+\frac{15\!\cdots\!44}{13\!\cdots\!59}a^{6}+\frac{30\!\cdots\!45}{13\!\cdots\!59}a^{5}+\frac{12\!\cdots\!27}{13\!\cdots\!59}a^{4}+\frac{29\!\cdots\!10}{13\!\cdots\!59}a^{3}+\frac{232296741154899}{13\!\cdots\!59}a^{2}-\frac{548532157004254}{13\!\cdots\!59}a-\frac{13\!\cdots\!15}{13\!\cdots\!59}$, $\frac{751910697883090}{13\!\cdots\!59}a^{20}+\frac{48237792128426}{13\!\cdots\!59}a^{19}+\frac{596363061463564}{13\!\cdots\!59}a^{18}-\frac{36\!\cdots\!30}{13\!\cdots\!59}a^{17}+\frac{55\!\cdots\!17}{13\!\cdots\!59}a^{16}-\frac{25\!\cdots\!63}{13\!\cdots\!59}a^{15}+\frac{65\!\cdots\!89}{13\!\cdots\!59}a^{14}-\frac{28\!\cdots\!84}{13\!\cdots\!59}a^{13}+\frac{18\!\cdots\!64}{120799188155869}a^{12}+\frac{96\!\cdots\!52}{13\!\cdots\!59}a^{11}+\frac{40\!\cdots\!19}{13\!\cdots\!59}a^{10}-\frac{60\!\cdots\!54}{13\!\cdots\!59}a^{9}-\frac{23\!\cdots\!75}{13\!\cdots\!59}a^{8}-\frac{18\!\cdots\!82}{13\!\cdots\!59}a^{7}+\frac{52\!\cdots\!42}{13\!\cdots\!59}a^{6}+\frac{30\!\cdots\!23}{13\!\cdots\!59}a^{5}-\frac{20\!\cdots\!62}{13\!\cdots\!59}a^{4}-\frac{61\!\cdots\!98}{13\!\cdots\!59}a^{3}-\frac{40\!\cdots\!38}{13\!\cdots\!59}a^{2}-\frac{12\!\cdots\!28}{13\!\cdots\!59}a+\frac{19\!\cdots\!04}{13\!\cdots\!59}$, $\frac{698923823931459}{13\!\cdots\!59}a^{20}+\frac{423494427091323}{13\!\cdots\!59}a^{19}+\frac{766440923049085}{13\!\cdots\!59}a^{18}-\frac{30\!\cdots\!33}{13\!\cdots\!59}a^{17}+\frac{35\!\cdots\!69}{13\!\cdots\!59}a^{16}-\frac{556920559768179}{13\!\cdots\!59}a^{15}+\frac{60\!\cdots\!19}{13\!\cdots\!59}a^{14}-\frac{23\!\cdots\!86}{13\!\cdots\!59}a^{13}+\frac{70\!\cdots\!70}{13\!\cdots\!59}a^{12}+\frac{11\!\cdots\!10}{120799188155869}a^{11}+\frac{46\!\cdots\!57}{13\!\cdots\!59}a^{10}-\frac{34\!\cdots\!26}{13\!\cdots\!59}a^{9}-\frac{39\!\cdots\!07}{13\!\cdots\!59}a^{8}-\frac{42\!\cdots\!24}{13\!\cdots\!59}a^{7}+\frac{32\!\cdots\!83}{13\!\cdots\!59}a^{6}+\frac{44\!\cdots\!15}{13\!\cdots\!59}a^{5}+\frac{81\!\cdots\!25}{13\!\cdots\!59}a^{4}-\frac{60\!\cdots\!93}{13\!\cdots\!59}a^{3}-\frac{78\!\cdots\!41}{13\!\cdots\!59}a^{2}-\frac{30\!\cdots\!61}{13\!\cdots\!59}a-\frac{17\!\cdots\!21}{13\!\cdots\!59}$
| 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: | \( 11721.4726991 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 11721.4726991 \cdot 1}{2\cdot\sqrt{23714597990105054274863104}}\cr\approx \mathstrut & 0.230819757541 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 + x^19 - 5*x^18 + 8*x^17 - 5*x^16 + 11*x^15 - 40*x^14 + 33*x^13 + 2*x^12 + 62*x^11 - 87*x^10 - 12*x^9 - 43*x^8 + 76*x^7 + 27*x^6 - 15*x^5 - 7*x^4 - 10*x^3 + 2*x^2 + 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 + x^19 - 5*x^18 + 8*x^17 - 5*x^16 + 11*x^15 - 40*x^14 + 33*x^13 + 2*x^12 + 62*x^11 - 87*x^10 - 12*x^9 - 43*x^8 + 76*x^7 + 27*x^6 - 15*x^5 - 7*x^4 - 10*x^3 + 2*x^2 + 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 + x^19 - 5*x^18 + 8*x^17 - 5*x^16 + 11*x^15 - 40*x^14 + 33*x^13 + 2*x^12 + 62*x^11 - 87*x^10 - 12*x^9 - 43*x^8 + 76*x^7 + 27*x^6 - 15*x^5 - 7*x^4 - 10*x^3 + 2*x^2 + 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 + x^19 - 5*x^18 + 8*x^17 - 5*x^16 + 11*x^15 - 40*x^14 + 33*x^13 + 2*x^12 + 62*x^11 - 87*x^10 - 12*x^9 - 43*x^8 + 76*x^7 + 27*x^6 - 15*x^5 - 7*x^4 - 10*x^3 + 2*x^2 + 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
$S_3\times S_7$ (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 $S_3\times S_7$ is not computed |
Intermediate fields
| 3.1.44.1, 7.1.207911.1 |
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 | R | $21$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.3.0.1}{3} }$ | R | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ | $21$ | ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.6.0.1}{6} }^{3}{,}\,{\href{/padicField/37.3.0.1}{3} }$ | R | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.21.14.1 | $x^{21} + 14 x^{18} + 87 x^{15} + 3 x^{14} - 14 x^{12} - 462 x^{11} + 1655 x^{9} + 4290 x^{8} + 3 x^{7} + 2982 x^{6} - 6090 x^{5} + 210 x^{4} - 1651 x^{3} + 1263 x^{2} + 87 x + 251$ | $3$ | $7$ | $14$ | 21T6 | $[\ ]_{3}^{14}$ |
|
\(11\)
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.5.0.1 | $x^{5} + 10 x^{2} + 9$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 11.10.5.2 | $x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(41\)
| $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.0.1 | $x^{4} + 23 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 41.4.0.1 | $x^{4} + 23 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 41.4.0.1 | $x^{4} + 23 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(461\)
| $\Q_{461}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{461}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |