Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 3*x^18 + 17*x^17 - 47*x^16 + 135*x^15 - 310*x^14 + 709*x^13 - 1319*x^12 + 2423*x^11 - 3757*x^10 + 5497*x^9 - 6561*x^8 + 7142*x^7 - 5699*x^6 + 4212*x^5 - 1490*x^4 + 482*x^3 + 408*x^2 - 210*x + 49)
gp: K = bnfinit(y^19 - 3*y^18 + 17*y^17 - 47*y^16 + 135*y^15 - 310*y^14 + 709*y^13 - 1319*y^12 + 2423*y^11 - 3757*y^10 + 5497*y^9 - 6561*y^8 + 7142*y^7 - 5699*y^6 + 4212*y^5 - 1490*y^4 + 482*y^3 + 408*y^2 - 210*y + 49, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 3*x^18 + 17*x^17 - 47*x^16 + 135*x^15 - 310*x^14 + 709*x^13 - 1319*x^12 + 2423*x^11 - 3757*x^10 + 5497*x^9 - 6561*x^8 + 7142*x^7 - 5699*x^6 + 4212*x^5 - 1490*x^4 + 482*x^3 + 408*x^2 - 210*x + 49);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 - 3*x^18 + 17*x^17 - 47*x^16 + 135*x^15 - 310*x^14 + 709*x^13 - 1319*x^12 + 2423*x^11 - 3757*x^10 + 5497*x^9 - 6561*x^8 + 7142*x^7 - 5699*x^6 + 4212*x^5 - 1490*x^4 + 482*x^3 + 408*x^2 - 210*x + 49)
\( x^{19} - 3 x^{18} + 17 x^{17} - 47 x^{16} + 135 x^{15} - 310 x^{14} + 709 x^{13} - 1319 x^{12} + \cdots + 49 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $19$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-5121210743359411191500170799\)
\(\medspace = -\,11^{9}\cdot 109^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(28.73\) | 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: | $11^{1/2}109^{1/2}\approx 34.62657938636157$ | ||
| Ramified primes: |
\(11\), \(109\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1199}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{11}a^{11}-\frac{3}{11}a^{10}+\frac{3}{11}a^{8}-\frac{4}{11}a^{7}-\frac{4}{11}a^{6}-\frac{4}{11}a^{5}-\frac{3}{11}a^{2}-\frac{2}{11}$, $\frac{1}{11}a^{12}+\frac{2}{11}a^{10}+\frac{3}{11}a^{9}+\frac{5}{11}a^{8}-\frac{5}{11}a^{7}-\frac{5}{11}a^{6}-\frac{1}{11}a^{5}-\frac{3}{11}a^{3}+\frac{2}{11}a^{2}-\frac{2}{11}a+\frac{5}{11}$, $\frac{1}{11}a^{13}-\frac{2}{11}a^{10}+\frac{5}{11}a^{9}+\frac{3}{11}a^{7}-\frac{4}{11}a^{6}-\frac{3}{11}a^{5}-\frac{3}{11}a^{4}+\frac{2}{11}a^{3}+\frac{4}{11}a^{2}+\frac{5}{11}a+\frac{4}{11}$, $\frac{1}{11}a^{14}-\frac{1}{11}a^{10}-\frac{2}{11}a^{8}-\frac{1}{11}a^{7}+\frac{2}{11}a^{4}+\frac{4}{11}a^{3}-\frac{1}{11}a^{2}+\frac{4}{11}a-\frac{4}{11}$, $\frac{1}{11}a^{15}-\frac{3}{11}a^{10}-\frac{2}{11}a^{9}+\frac{2}{11}a^{8}-\frac{4}{11}a^{7}-\frac{4}{11}a^{6}-\frac{2}{11}a^{5}+\frac{4}{11}a^{4}-\frac{1}{11}a^{3}+\frac{1}{11}a^{2}-\frac{4}{11}a-\frac{2}{11}$, $\frac{1}{77}a^{16}-\frac{2}{77}a^{15}-\frac{2}{77}a^{14}-\frac{1}{77}a^{13}+\frac{1}{77}a^{11}-\frac{4}{77}a^{10}-\frac{10}{77}a^{9}-\frac{25}{77}a^{8}-\frac{5}{11}a^{7}-\frac{4}{11}a^{6}-\frac{16}{77}a^{5}-\frac{3}{11}a^{4}+\frac{26}{77}a^{3}+\frac{24}{77}a^{2}-\frac{18}{77}a$, $\frac{1}{847}a^{17}+\frac{3}{847}a^{16}+\frac{37}{847}a^{15}+\frac{38}{847}a^{14}+\frac{30}{847}a^{13}+\frac{1}{847}a^{12}-\frac{6}{847}a^{11}+\frac{17}{77}a^{10}+\frac{387}{847}a^{9}-\frac{335}{847}a^{8}+\frac{43}{121}a^{7}+\frac{152}{847}a^{6}+\frac{417}{847}a^{5}+\frac{3}{77}a^{4}+\frac{20}{121}a^{3}+\frac{263}{847}a^{2}+\frac{393}{847}a+\frac{24}{121}$, $\frac{1}{51\!\cdots\!43}a^{18}-\frac{29701844258160}{51\!\cdots\!43}a^{17}-\frac{172604384349723}{51\!\cdots\!43}a^{16}+\frac{842653333060995}{51\!\cdots\!43}a^{15}-\frac{268650353179353}{73\!\cdots\!49}a^{14}-\frac{66104475505676}{73\!\cdots\!49}a^{13}+\frac{18\!\cdots\!36}{51\!\cdots\!43}a^{12}+\frac{94415154718519}{47\!\cdots\!13}a^{11}-\frac{52\!\cdots\!66}{51\!\cdots\!43}a^{10}+\frac{146979307371132}{12\!\cdots\!01}a^{9}-\frac{87\!\cdots\!59}{51\!\cdots\!43}a^{8}-\frac{11\!\cdots\!97}{51\!\cdots\!43}a^{7}+\frac{18\!\cdots\!88}{51\!\cdots\!43}a^{6}+\frac{20\!\cdots\!74}{47\!\cdots\!13}a^{5}+\frac{800344130348002}{73\!\cdots\!49}a^{4}-\frac{18\!\cdots\!83}{51\!\cdots\!43}a^{3}-\frac{39\!\cdots\!05}{51\!\cdots\!43}a^{2}+\frac{27\!\cdots\!21}{51\!\cdots\!43}a+\frac{209747834143810}{671786277085759}$
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 $1$
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: | $9$ | 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: |
$\frac{480957763708575}{51\!\cdots\!43}a^{18}-\frac{991638443767420}{51\!\cdots\!43}a^{17}+\frac{632566963033515}{47\!\cdots\!13}a^{16}-\frac{15\!\cdots\!77}{51\!\cdots\!43}a^{15}+\frac{46\!\cdots\!05}{51\!\cdots\!43}a^{14}-\frac{99\!\cdots\!48}{51\!\cdots\!43}a^{13}+\frac{21\!\cdots\!60}{47\!\cdots\!13}a^{12}-\frac{39\!\cdots\!22}{51\!\cdots\!43}a^{11}+\frac{74\!\cdots\!06}{51\!\cdots\!43}a^{10}-\frac{24\!\cdots\!02}{12\!\cdots\!01}a^{9}+\frac{15\!\cdots\!21}{51\!\cdots\!43}a^{8}-\frac{14\!\cdots\!60}{47\!\cdots\!13}a^{7}+\frac{18\!\cdots\!61}{51\!\cdots\!43}a^{6}-\frac{11\!\cdots\!81}{51\!\cdots\!43}a^{5}+\frac{11\!\cdots\!82}{51\!\cdots\!43}a^{4}-\frac{12\!\cdots\!31}{51\!\cdots\!43}a^{3}+\frac{19\!\cdots\!82}{51\!\cdots\!43}a^{2}+\frac{21\!\cdots\!70}{51\!\cdots\!43}a-\frac{70\!\cdots\!60}{73\!\cdots\!49}$, $\frac{1851042552248}{51\!\cdots\!43}a^{18}+\frac{125625103262404}{51\!\cdots\!43}a^{17}-\frac{54125383019280}{51\!\cdots\!43}a^{16}+\frac{13\!\cdots\!24}{51\!\cdots\!43}a^{15}-\frac{13\!\cdots\!50}{51\!\cdots\!43}a^{14}+\frac{736003497301876}{73\!\cdots\!49}a^{13}-\frac{71\!\cdots\!70}{51\!\cdots\!43}a^{12}+\frac{19\!\cdots\!25}{51\!\cdots\!43}a^{11}-\frac{97\!\cdots\!93}{51\!\cdots\!43}a^{10}+\frac{870158190067412}{12\!\cdots\!01}a^{9}-\frac{583372668838859}{51\!\cdots\!43}a^{8}+\frac{84\!\cdots\!68}{51\!\cdots\!43}a^{7}+\frac{10\!\cdots\!80}{51\!\cdots\!43}a^{6}-\frac{41\!\cdots\!98}{51\!\cdots\!43}a^{5}+\frac{18\!\cdots\!79}{51\!\cdots\!43}a^{4}+\frac{51\!\cdots\!75}{51\!\cdots\!43}a^{3}+\frac{16\!\cdots\!37}{73\!\cdots\!49}a^{2}+\frac{10\!\cdots\!98}{51\!\cdots\!43}a+\frac{57\!\cdots\!29}{73\!\cdots\!49}$, $\frac{674635759134301}{51\!\cdots\!43}a^{18}-\frac{19\!\cdots\!50}{51\!\cdots\!43}a^{17}+\frac{10\!\cdots\!12}{51\!\cdots\!43}a^{16}-\frac{40\!\cdots\!97}{73\!\cdots\!49}a^{15}+\frac{76\!\cdots\!94}{51\!\cdots\!43}a^{14}-\frac{17\!\cdots\!74}{51\!\cdots\!43}a^{13}+\frac{38\!\cdots\!31}{51\!\cdots\!43}a^{12}-\frac{67\!\cdots\!07}{51\!\cdots\!43}a^{11}+\frac{11\!\cdots\!55}{51\!\cdots\!43}a^{10}-\frac{40\!\cdots\!03}{12\!\cdots\!01}a^{9}+\frac{30\!\cdots\!79}{671786277085759}a^{8}-\frac{25\!\cdots\!93}{51\!\cdots\!43}a^{7}+\frac{23\!\cdots\!41}{51\!\cdots\!43}a^{6}-\frac{13\!\cdots\!73}{51\!\cdots\!43}a^{5}+\frac{76\!\cdots\!68}{51\!\cdots\!43}a^{4}-\frac{74\!\cdots\!87}{51\!\cdots\!43}a^{3}-\frac{52\!\cdots\!19}{51\!\cdots\!43}a^{2}+\frac{77\!\cdots\!00}{73\!\cdots\!49}a-\frac{15\!\cdots\!08}{73\!\cdots\!49}$, $\frac{362739174289}{88726489426421}a^{18}-\frac{779919834112}{975991383690631}a^{17}+\frac{35866304615171}{975991383690631}a^{16}-\frac{10941721988630}{975991383690631}a^{15}+\frac{64015318001696}{975991383690631}a^{14}+\frac{68984760927745}{975991383690631}a^{13}-\frac{83344706958745}{975991383690631}a^{12}+\frac{12\!\cdots\!86}{975991383690631}a^{11}-\frac{183129563556815}{88726489426421}a^{10}+\frac{18769913881570}{3242496291331}a^{9}-\frac{88\!\cdots\!76}{975991383690631}a^{8}+\frac{16\!\cdots\!68}{975991383690631}a^{7}-\frac{18\!\cdots\!88}{975991383690631}a^{6}+\frac{22\!\cdots\!59}{975991383690631}a^{5}-\frac{11\!\cdots\!28}{88726489426421}a^{4}+\frac{11\!\cdots\!43}{975991383690631}a^{3}-\frac{622709172678372}{975991383690631}a^{2}+\frac{398696558353305}{975991383690631}a+\frac{6754454721936}{139427340527233}$, $\frac{35706714982156}{73\!\cdots\!49}a^{18}-\frac{834323276199467}{51\!\cdots\!43}a^{17}+\frac{627051652913291}{73\!\cdots\!49}a^{16}-\frac{12\!\cdots\!58}{51\!\cdots\!43}a^{15}+\frac{35\!\cdots\!59}{51\!\cdots\!43}a^{14}-\frac{73\!\cdots\!12}{47\!\cdots\!13}a^{13}+\frac{18\!\cdots\!84}{51\!\cdots\!43}a^{12}-\frac{33\!\cdots\!32}{51\!\cdots\!43}a^{11}+\frac{61\!\cdots\!55}{51\!\cdots\!43}a^{10}-\frac{21\!\cdots\!47}{12\!\cdots\!01}a^{9}+\frac{13\!\cdots\!32}{51\!\cdots\!43}a^{8}-\frac{21\!\cdots\!97}{73\!\cdots\!49}a^{7}+\frac{15\!\cdots\!73}{51\!\cdots\!43}a^{6}-\frac{99\!\cdots\!31}{51\!\cdots\!43}a^{5}+\frac{62\!\cdots\!08}{51\!\cdots\!43}a^{4}+\frac{11\!\cdots\!07}{51\!\cdots\!43}a^{3}-\frac{73\!\cdots\!78}{51\!\cdots\!43}a^{2}+\frac{23\!\cdots\!33}{51\!\cdots\!43}a+\frac{43\!\cdots\!67}{73\!\cdots\!49}$, $\frac{157679833406801}{51\!\cdots\!43}a^{18}+\frac{477515698809011}{51\!\cdots\!43}a^{17}-\frac{158858621991614}{51\!\cdots\!43}a^{16}+\frac{76\!\cdots\!87}{51\!\cdots\!43}a^{15}-\frac{18\!\cdots\!65}{47\!\cdots\!13}a^{14}+\frac{90\!\cdots\!87}{73\!\cdots\!49}a^{13}-\frac{14\!\cdots\!39}{51\!\cdots\!43}a^{12}+\frac{35\!\cdots\!30}{51\!\cdots\!43}a^{11}-\frac{62\!\cdots\!62}{51\!\cdots\!43}a^{10}+\frac{26\!\cdots\!74}{12\!\cdots\!01}a^{9}-\frac{17\!\cdots\!63}{51\!\cdots\!43}a^{8}+\frac{24\!\cdots\!58}{51\!\cdots\!43}a^{7}-\frac{34\!\cdots\!66}{671786277085759}a^{6}+\frac{25\!\cdots\!60}{51\!\cdots\!43}a^{5}-\frac{13\!\cdots\!23}{51\!\cdots\!43}a^{4}+\frac{64\!\cdots\!35}{51\!\cdots\!43}a^{3}+\frac{17\!\cdots\!94}{51\!\cdots\!43}a^{2}-\frac{16\!\cdots\!38}{51\!\cdots\!43}a+\frac{34\!\cdots\!62}{73\!\cdots\!49}$, $\frac{48413760684207}{47\!\cdots\!13}a^{18}-\frac{12\!\cdots\!44}{51\!\cdots\!43}a^{17}+\frac{75\!\cdots\!34}{51\!\cdots\!43}a^{16}-\frac{18\!\cdots\!99}{51\!\cdots\!43}a^{15}+\frac{49\!\cdots\!67}{51\!\cdots\!43}a^{14}-\frac{10\!\cdots\!98}{51\!\cdots\!43}a^{13}+\frac{23\!\cdots\!15}{51\!\cdots\!43}a^{12}-\frac{37\!\cdots\!11}{51\!\cdots\!43}a^{11}+\frac{54\!\cdots\!09}{427500358145483}a^{10}-\frac{29\!\cdots\!13}{171852303440543}a^{9}+\frac{11\!\cdots\!73}{51\!\cdots\!43}a^{8}-\frac{95\!\cdots\!21}{51\!\cdots\!43}a^{7}+\frac{75\!\cdots\!78}{51\!\cdots\!43}a^{6}+\frac{29\!\cdots\!51}{51\!\cdots\!43}a^{5}-\frac{76\!\cdots\!10}{47\!\cdots\!13}a^{4}+\frac{45\!\cdots\!94}{51\!\cdots\!43}a^{3}-\frac{47\!\cdots\!29}{51\!\cdots\!43}a^{2}+\frac{87\!\cdots\!61}{51\!\cdots\!43}a+\frac{45\!\cdots\!14}{73\!\cdots\!49}$, $\frac{783454543370942}{51\!\cdots\!43}a^{18}-\frac{16\!\cdots\!85}{51\!\cdots\!43}a^{17}+\frac{10\!\cdots\!95}{51\!\cdots\!43}a^{16}-\frac{21\!\cdots\!23}{47\!\cdots\!13}a^{15}+\frac{63\!\cdots\!50}{51\!\cdots\!43}a^{14}-\frac{13\!\cdots\!06}{51\!\cdots\!43}a^{13}+\frac{29\!\cdots\!51}{51\!\cdots\!43}a^{12}-\frac{46\!\cdots\!71}{51\!\cdots\!43}a^{11}+\frac{82\!\cdots\!96}{51\!\cdots\!43}a^{10}-\frac{25\!\cdots\!87}{12\!\cdots\!01}a^{9}+\frac{14\!\cdots\!79}{51\!\cdots\!43}a^{8}-\frac{11\!\cdots\!22}{51\!\cdots\!43}a^{7}+\frac{10\!\cdots\!50}{51\!\cdots\!43}a^{6}-\frac{16\!\cdots\!85}{51\!\cdots\!43}a^{5}+\frac{89\!\cdots\!93}{51\!\cdots\!43}a^{4}+\frac{20\!\cdots\!92}{47\!\cdots\!13}a^{3}-\frac{52\!\cdots\!68}{51\!\cdots\!43}a^{2}-\frac{76\!\cdots\!84}{51\!\cdots\!43}a-\frac{11\!\cdots\!02}{73\!\cdots\!49}$, $\frac{235602446643171}{51\!\cdots\!43}a^{18}-\frac{19\!\cdots\!01}{51\!\cdots\!43}a^{17}+\frac{690768906766464}{47\!\cdots\!13}a^{16}-\frac{30\!\cdots\!12}{51\!\cdots\!43}a^{15}+\frac{12\!\cdots\!19}{73\!\cdots\!49}a^{14}-\frac{30\!\cdots\!61}{73\!\cdots\!49}a^{13}+\frac{43\!\cdots\!44}{47\!\cdots\!13}a^{12}-\frac{10\!\cdots\!27}{51\!\cdots\!43}a^{11}+\frac{18\!\cdots\!39}{51\!\cdots\!43}a^{10}-\frac{70\!\cdots\!84}{12\!\cdots\!01}a^{9}+\frac{44\!\cdots\!14}{51\!\cdots\!43}a^{8}-\frac{53\!\cdots\!30}{47\!\cdots\!13}a^{7}+\frac{61\!\cdots\!42}{51\!\cdots\!43}a^{6}-\frac{54\!\cdots\!65}{51\!\cdots\!43}a^{5}+\frac{46\!\cdots\!64}{73\!\cdots\!49}a^{4}-\frac{14\!\cdots\!97}{51\!\cdots\!43}a^{3}-\frac{17\!\cdots\!42}{51\!\cdots\!43}a^{2}+\frac{35\!\cdots\!56}{51\!\cdots\!43}a-\frac{24\!\cdots\!77}{73\!\cdots\!49}$
| 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: | \( 2481320.5796 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 2481320.5796 \cdot 1}{2\cdot\sqrt{5121210743359411191500170799}}\cr\approx \mathstrut & 0.52919455720 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^19 - 3*x^18 + 17*x^17 - 47*x^16 + 135*x^15 - 310*x^14 + 709*x^13 - 1319*x^12 + 2423*x^11 - 3757*x^10 + 5497*x^9 - 6561*x^8 + 7142*x^7 - 5699*x^6 + 4212*x^5 - 1490*x^4 + 482*x^3 + 408*x^2 - 210*x + 49)
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^19 - 3*x^18 + 17*x^17 - 47*x^16 + 135*x^15 - 310*x^14 + 709*x^13 - 1319*x^12 + 2423*x^11 - 3757*x^10 + 5497*x^9 - 6561*x^8 + 7142*x^7 - 5699*x^6 + 4212*x^5 - 1490*x^4 + 482*x^3 + 408*x^2 - 210*x + 49, 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^19 - 3*x^18 + 17*x^17 - 47*x^16 + 135*x^15 - 310*x^14 + 709*x^13 - 1319*x^12 + 2423*x^11 - 3757*x^10 + 5497*x^9 - 6561*x^8 + 7142*x^7 - 5699*x^6 + 4212*x^5 - 1490*x^4 + 482*x^3 + 408*x^2 - 210*x + 49);
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^19 - 3*x^18 + 17*x^17 - 47*x^16 + 135*x^15 - 310*x^14 + 709*x^13 - 1319*x^12 + 2423*x^11 - 3757*x^10 + 5497*x^9 - 6561*x^8 + 7142*x^7 - 5699*x^6 + 4212*x^5 - 1490*x^4 + 482*x^3 + 408*x^2 - 210*x + 49);
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 solvable group of order 38 |
| The 11 conjugacy class representatives for $D_{19}$ |
| Character table for $D_{19}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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
| Galois closure: | 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 | $19$ | $19$ | $19$ | ${\href{/padicField/7.2.0.1}{2} }^{9}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | $19$ | $19$ | $19$ | ${\href{/padicField/23.2.0.1}{2} }^{9}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{9}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $19$ | ${\href{/padicField/37.2.0.1}{2} }^{9}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $19$ | ${\href{/padicField/43.2.0.1}{2} }^{9}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{9}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{9}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{9}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(109\)
| $\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |