Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 4452*x^13 - 50296*x^12 + 5784843*x^11 + 91964349*x^10 - 3060924423*x^9 - 54750997444*x^8 + 781942057449*x^7 + 14664282257229*x^6 - 102907697118067*x^5 - 1901998615798914*x^4 + 6977459265924253*x^3 + 113854569421380357*x^2 - 202156325407652112*x - 2402311520555338669)
gp: K = bnfinit(y^15 - y^14 - 4452*y^13 - 50296*y^12 + 5784843*y^11 + 91964349*y^10 - 3060924423*y^9 - 54750997444*y^8 + 781942057449*y^7 + 14664282257229*y^6 - 102907697118067*y^5 - 1901998615798914*y^4 + 6977459265924253*y^3 + 113854569421380357*y^2 - 202156325407652112*y - 2402311520555338669, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 - 4452*x^13 - 50296*x^12 + 5784843*x^11 + 91964349*x^10 - 3060924423*x^9 - 54750997444*x^8 + 781942057449*x^7 + 14664282257229*x^6 - 102907697118067*x^5 - 1901998615798914*x^4 + 6977459265924253*x^3 + 113854569421380357*x^2 - 202156325407652112*x - 2402311520555338669);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - x^14 - 4452*x^13 - 50296*x^12 + 5784843*x^11 + 91964349*x^10 - 3060924423*x^9 - 54750997444*x^8 + 781942057449*x^7 + 14664282257229*x^6 - 102907697118067*x^5 - 1901998615798914*x^4 + 6977459265924253*x^3 + 113854569421380357*x^2 - 202156325407652112*x - 2402311520555338669)
\( x^{15} - x^{14} - 4452 x^{13} - 50296 x^{12} + 5784843 x^{11} + 91964349 x^{10} - 3060924423 x^{9} + \cdots - 24\!\cdots\!69 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[15, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(691985919132121715398802656928813567369454726910003125\)
\(\medspace = 3^{12}\cdot 5^{5}\cdot 79^{4}\cdot 401^{7}\cdot 50329^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(3884.54\) | 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: | $3^{16/9}5^{1/2}79^{2/3}401^{1/2}50329^{2/3}\approx 7923243.703822074$ | ||
| Ramified primes: |
\(3\), \(5\), \(79\), \(401\), \(50329\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2005}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13}a^{13}+\frac{3}{13}a^{12}+\frac{4}{13}a^{11}-\frac{2}{13}a^{10}-\frac{4}{13}a^{9}-\frac{3}{13}a^{8}-\frac{2}{13}a^{7}-\frac{2}{13}a^{6}+\frac{3}{13}a^{5}-\frac{3}{13}a^{4}-\frac{2}{13}a^{3}+\frac{2}{13}a^{2}+\frac{1}{13}a-\frac{2}{13}$, $\frac{1}{24\!\cdots\!81}a^{14}+\frac{13\!\cdots\!05}{24\!\cdots\!81}a^{13}-\frac{97\!\cdots\!19}{24\!\cdots\!81}a^{12}-\frac{46\!\cdots\!00}{24\!\cdots\!81}a^{11}+\frac{13\!\cdots\!93}{24\!\cdots\!81}a^{10}+\frac{11\!\cdots\!12}{24\!\cdots\!81}a^{9}-\frac{73\!\cdots\!72}{24\!\cdots\!81}a^{8}-\frac{10\!\cdots\!49}{24\!\cdots\!81}a^{7}-\frac{88\!\cdots\!02}{24\!\cdots\!81}a^{6}-\frac{22\!\cdots\!82}{66\!\cdots\!13}a^{5}-\frac{58\!\cdots\!53}{24\!\cdots\!81}a^{4}-\frac{81\!\cdots\!74}{24\!\cdots\!81}a^{3}+\frac{42\!\cdots\!08}{24\!\cdots\!81}a^{2}+\frac{74\!\cdots\!36}{24\!\cdots\!81}a+\frac{13\!\cdots\!74}{18\!\cdots\!37}$
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
$C_{3}$, which has order $3$ (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: | $14$ | 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: |
$\frac{10\!\cdots\!83}{24\!\cdots\!81}a^{14}+\frac{47\!\cdots\!26}{24\!\cdots\!81}a^{13}-\frac{44\!\cdots\!67}{24\!\cdots\!81}a^{12}-\frac{77\!\cdots\!36}{24\!\cdots\!81}a^{11}+\frac{52\!\cdots\!16}{24\!\cdots\!81}a^{10}+\frac{12\!\cdots\!23}{24\!\cdots\!81}a^{9}-\frac{21\!\cdots\!30}{24\!\cdots\!81}a^{8}-\frac{63\!\cdots\!54}{24\!\cdots\!81}a^{7}+\frac{27\!\cdots\!24}{24\!\cdots\!81}a^{6}+\frac{29\!\cdots\!78}{50\!\cdots\!01}a^{5}+\frac{14\!\cdots\!68}{24\!\cdots\!81}a^{4}-\frac{10\!\cdots\!23}{18\!\cdots\!37}a^{3}-\frac{46\!\cdots\!05}{24\!\cdots\!81}a^{2}+\frac{45\!\cdots\!94}{24\!\cdots\!81}a+\frac{21\!\cdots\!52}{24\!\cdots\!81}$, $\frac{10\!\cdots\!83}{24\!\cdots\!81}a^{14}+\frac{47\!\cdots\!26}{24\!\cdots\!81}a^{13}-\frac{44\!\cdots\!67}{24\!\cdots\!81}a^{12}-\frac{77\!\cdots\!36}{24\!\cdots\!81}a^{11}+\frac{52\!\cdots\!16}{24\!\cdots\!81}a^{10}+\frac{12\!\cdots\!23}{24\!\cdots\!81}a^{9}-\frac{21\!\cdots\!30}{24\!\cdots\!81}a^{8}-\frac{63\!\cdots\!54}{24\!\cdots\!81}a^{7}+\frac{27\!\cdots\!24}{24\!\cdots\!81}a^{6}+\frac{29\!\cdots\!78}{50\!\cdots\!01}a^{5}+\frac{14\!\cdots\!68}{24\!\cdots\!81}a^{4}-\frac{10\!\cdots\!23}{18\!\cdots\!37}a^{3}-\frac{46\!\cdots\!05}{24\!\cdots\!81}a^{2}+\frac{45\!\cdots\!94}{24\!\cdots\!81}a+\frac{20\!\cdots\!71}{24\!\cdots\!81}$, $\frac{37\!\cdots\!19}{24\!\cdots\!81}a^{14}-\frac{21\!\cdots\!62}{24\!\cdots\!81}a^{13}-\frac{15\!\cdots\!85}{24\!\cdots\!81}a^{12}+\frac{72\!\cdots\!23}{24\!\cdots\!81}a^{11}+\frac{26\!\cdots\!17}{24\!\cdots\!81}a^{10}-\frac{82\!\cdots\!79}{24\!\cdots\!81}a^{9}-\frac{21\!\cdots\!54}{24\!\cdots\!81}a^{8}+\frac{40\!\cdots\!16}{24\!\cdots\!81}a^{7}+\frac{83\!\cdots\!11}{24\!\cdots\!81}a^{6}-\frac{19\!\cdots\!39}{50\!\cdots\!01}a^{5}-\frac{15\!\cdots\!87}{24\!\cdots\!81}a^{4}+\frac{73\!\cdots\!01}{18\!\cdots\!37}a^{3}+\frac{12\!\cdots\!72}{24\!\cdots\!81}a^{2}-\frac{33\!\cdots\!07}{24\!\cdots\!81}a-\frac{34\!\cdots\!57}{24\!\cdots\!81}$, $\frac{10\!\cdots\!02}{24\!\cdots\!81}a^{14}+\frac{25\!\cdots\!64}{24\!\cdots\!81}a^{13}-\frac{46\!\cdots\!52}{24\!\cdots\!81}a^{12}-\frac{70\!\cdots\!13}{24\!\cdots\!81}a^{11}+\frac{55\!\cdots\!33}{24\!\cdots\!81}a^{10}+\frac{11\!\cdots\!44}{24\!\cdots\!81}a^{9}-\frac{23\!\cdots\!84}{24\!\cdots\!81}a^{8}-\frac{59\!\cdots\!38}{24\!\cdots\!81}a^{7}+\frac{35\!\cdots\!35}{24\!\cdots\!81}a^{6}+\frac{27\!\cdots\!39}{50\!\cdots\!01}a^{5}-\frac{10\!\cdots\!19}{24\!\cdots\!81}a^{4}-\frac{97\!\cdots\!22}{18\!\cdots\!37}a^{3}-\frac{34\!\cdots\!33}{24\!\cdots\!81}a^{2}+\frac{42\!\cdots\!87}{24\!\cdots\!81}a+\frac{17\!\cdots\!14}{24\!\cdots\!81}$, $\frac{99\!\cdots\!19}{24\!\cdots\!81}a^{14}+\frac{62\!\cdots\!37}{24\!\cdots\!81}a^{13}-\frac{69\!\cdots\!46}{24\!\cdots\!81}a^{12}-\frac{10\!\cdots\!66}{24\!\cdots\!81}a^{11}-\frac{65\!\cdots\!43}{18\!\cdots\!37}a^{10}+\frac{60\!\cdots\!67}{24\!\cdots\!81}a^{9}+\frac{82\!\cdots\!77}{24\!\cdots\!81}a^{8}-\frac{15\!\cdots\!22}{24\!\cdots\!81}a^{7}-\frac{25\!\cdots\!16}{24\!\cdots\!81}a^{6}+\frac{48\!\cdots\!12}{66\!\cdots\!13}a^{5}+\frac{33\!\cdots\!96}{24\!\cdots\!81}a^{4}-\frac{10\!\cdots\!68}{24\!\cdots\!81}a^{3}-\frac{19\!\cdots\!90}{24\!\cdots\!81}a^{2}+\frac{27\!\cdots\!53}{24\!\cdots\!81}a+\frac{39\!\cdots\!04}{24\!\cdots\!81}$, $\frac{36\!\cdots\!34}{18\!\cdots\!37}a^{14}+\frac{52\!\cdots\!48}{24\!\cdots\!81}a^{13}-\frac{19\!\cdots\!27}{24\!\cdots\!81}a^{12}-\frac{48\!\cdots\!72}{24\!\cdots\!81}a^{11}+\frac{17\!\cdots\!04}{24\!\cdots\!81}a^{10}+\frac{62\!\cdots\!66}{24\!\cdots\!81}a^{9}-\frac{24\!\cdots\!24}{24\!\cdots\!81}a^{8}-\frac{24\!\cdots\!13}{24\!\cdots\!81}a^{7}-\frac{89\!\cdots\!91}{24\!\cdots\!81}a^{6}+\frac{10\!\cdots\!45}{66\!\cdots\!13}a^{5}+\frac{24\!\cdots\!08}{24\!\cdots\!81}a^{4}-\frac{28\!\cdots\!12}{24\!\cdots\!81}a^{3}-\frac{20\!\cdots\!52}{24\!\cdots\!81}a^{2}+\frac{77\!\cdots\!82}{24\!\cdots\!81}a+\frac{52\!\cdots\!22}{24\!\cdots\!81}$, $\frac{14\!\cdots\!99}{24\!\cdots\!81}a^{14}-\frac{23\!\cdots\!90}{24\!\cdots\!81}a^{13}-\frac{59\!\cdots\!94}{24\!\cdots\!81}a^{12}+\frac{20\!\cdots\!70}{24\!\cdots\!81}a^{11}+\frac{61\!\cdots\!37}{18\!\cdots\!37}a^{10}+\frac{84\!\cdots\!70}{24\!\cdots\!81}a^{9}-\frac{45\!\cdots\!23}{24\!\cdots\!81}a^{8}-\frac{85\!\cdots\!77}{24\!\cdots\!81}a^{7}+\frac{12\!\cdots\!04}{24\!\cdots\!81}a^{6}+\frac{43\!\cdots\!79}{66\!\cdots\!13}a^{5}-\frac{17\!\cdots\!30}{24\!\cdots\!81}a^{4}-\frac{53\!\cdots\!79}{24\!\cdots\!81}a^{3}+\frac{10\!\cdots\!47}{24\!\cdots\!81}a^{2}-\frac{43\!\cdots\!51}{24\!\cdots\!81}a-\frac{22\!\cdots\!05}{24\!\cdots\!81}$, $\frac{14\!\cdots\!68}{24\!\cdots\!81}a^{14}+\frac{11\!\cdots\!17}{24\!\cdots\!81}a^{13}-\frac{65\!\cdots\!28}{24\!\cdots\!81}a^{12}-\frac{13\!\cdots\!82}{24\!\cdots\!81}a^{11}+\frac{74\!\cdots\!38}{24\!\cdots\!81}a^{10}+\frac{15\!\cdots\!43}{18\!\cdots\!37}a^{9}-\frac{27\!\cdots\!13}{24\!\cdots\!81}a^{8}-\frac{10\!\cdots\!54}{24\!\cdots\!81}a^{7}+\frac{22\!\cdots\!30}{24\!\cdots\!81}a^{6}+\frac{64\!\cdots\!21}{66\!\cdots\!13}a^{5}+\frac{56\!\cdots\!56}{24\!\cdots\!81}a^{4}-\frac{23\!\cdots\!92}{24\!\cdots\!81}a^{3}-\frac{10\!\cdots\!44}{24\!\cdots\!81}a^{2}+\frac{80\!\cdots\!96}{24\!\cdots\!81}a+\frac{40\!\cdots\!15}{24\!\cdots\!81}$, $\frac{34\!\cdots\!27}{24\!\cdots\!81}a^{14}+\frac{26\!\cdots\!33}{24\!\cdots\!81}a^{13}-\frac{15\!\cdots\!03}{24\!\cdots\!81}a^{12}-\frac{30\!\cdots\!87}{24\!\cdots\!81}a^{11}+\frac{17\!\cdots\!50}{24\!\cdots\!81}a^{10}+\frac{46\!\cdots\!77}{24\!\cdots\!81}a^{9}-\frac{64\!\cdots\!44}{24\!\cdots\!81}a^{8}-\frac{24\!\cdots\!63}{24\!\cdots\!81}a^{7}+\frac{54\!\cdots\!76}{24\!\cdots\!81}a^{6}+\frac{11\!\cdots\!64}{50\!\cdots\!01}a^{5}+\frac{12\!\cdots\!95}{24\!\cdots\!81}a^{4}-\frac{40\!\cdots\!79}{18\!\cdots\!37}a^{3}-\frac{23\!\cdots\!42}{24\!\cdots\!81}a^{2}+\frac{18\!\cdots\!97}{24\!\cdots\!81}a+\frac{92\!\cdots\!02}{24\!\cdots\!81}$, $\frac{39\!\cdots\!22}{24\!\cdots\!81}a^{14}-\frac{65\!\cdots\!13}{24\!\cdots\!81}a^{13}-\frac{16\!\cdots\!29}{24\!\cdots\!81}a^{12}+\frac{58\!\cdots\!97}{24\!\cdots\!81}a^{11}+\frac{22\!\cdots\!61}{24\!\cdots\!81}a^{10}+\frac{23\!\cdots\!64}{24\!\cdots\!81}a^{9}-\frac{12\!\cdots\!54}{24\!\cdots\!81}a^{8}-\frac{23\!\cdots\!44}{24\!\cdots\!81}a^{7}+\frac{34\!\cdots\!23}{24\!\cdots\!81}a^{6}+\frac{12\!\cdots\!74}{66\!\cdots\!13}a^{5}-\frac{48\!\cdots\!85}{24\!\cdots\!81}a^{4}-\frac{14\!\cdots\!79}{24\!\cdots\!81}a^{3}+\frac{23\!\cdots\!86}{18\!\cdots\!37}a^{2}-\frac{12\!\cdots\!72}{24\!\cdots\!81}a-\frac{61\!\cdots\!77}{24\!\cdots\!81}$, $\frac{20\!\cdots\!02}{78\!\cdots\!51}a^{14}-\frac{33\!\cdots\!41}{78\!\cdots\!51}a^{13}-\frac{86\!\cdots\!14}{78\!\cdots\!51}a^{12}+\frac{30\!\cdots\!81}{78\!\cdots\!51}a^{11}+\frac{11\!\cdots\!63}{78\!\cdots\!51}a^{10}+\frac{12\!\cdots\!14}{78\!\cdots\!51}a^{9}-\frac{65\!\cdots\!44}{78\!\cdots\!51}a^{8}-\frac{12\!\cdots\!94}{78\!\cdots\!51}a^{7}+\frac{18\!\cdots\!92}{78\!\cdots\!51}a^{6}+\frac{63\!\cdots\!96}{21\!\cdots\!23}a^{5}-\frac{24\!\cdots\!38}{78\!\cdots\!51}a^{4}-\frac{77\!\cdots\!30}{78\!\cdots\!51}a^{3}+\frac{15\!\cdots\!39}{78\!\cdots\!51}a^{2}-\frac{62\!\cdots\!29}{78\!\cdots\!51}a-\frac{32\!\cdots\!47}{78\!\cdots\!51}$, $\frac{68\!\cdots\!28}{24\!\cdots\!81}a^{14}-\frac{10\!\cdots\!85}{24\!\cdots\!81}a^{13}-\frac{29\!\cdots\!65}{24\!\cdots\!81}a^{12}+\frac{68\!\cdots\!35}{24\!\cdots\!81}a^{11}+\frac{38\!\cdots\!68}{24\!\cdots\!81}a^{10}+\frac{80\!\cdots\!60}{24\!\cdots\!81}a^{9}-\frac{22\!\cdots\!44}{24\!\cdots\!81}a^{8}-\frac{60\!\cdots\!21}{24\!\cdots\!81}a^{7}+\frac{62\!\cdots\!96}{24\!\cdots\!81}a^{6}+\frac{32\!\cdots\!67}{66\!\cdots\!13}a^{5}-\frac{87\!\cdots\!51}{24\!\cdots\!81}a^{4}-\frac{57\!\cdots\!95}{24\!\cdots\!81}a^{3}+\frac{56\!\cdots\!89}{24\!\cdots\!81}a^{2}-\frac{16\!\cdots\!19}{24\!\cdots\!81}a-\frac{11\!\cdots\!32}{24\!\cdots\!81}$, $\frac{52\!\cdots\!36}{66\!\cdots\!13}a^{14}-\frac{14\!\cdots\!40}{66\!\cdots\!13}a^{13}-\frac{19\!\cdots\!96}{66\!\cdots\!13}a^{12}+\frac{26\!\cdots\!60}{66\!\cdots\!13}a^{11}+\frac{23\!\cdots\!44}{66\!\cdots\!13}a^{10}-\frac{14\!\cdots\!68}{66\!\cdots\!13}a^{9}-\frac{92\!\cdots\!88}{50\!\cdots\!01}a^{8}+\frac{39\!\cdots\!27}{66\!\cdots\!13}a^{7}+\frac{23\!\cdots\!53}{50\!\cdots\!01}a^{6}-\frac{56\!\cdots\!64}{66\!\cdots\!13}a^{5}-\frac{29\!\cdots\!94}{50\!\cdots\!01}a^{4}+\frac{51\!\cdots\!80}{66\!\cdots\!13}a^{3}+\frac{22\!\cdots\!42}{66\!\cdots\!13}a^{2}-\frac{22\!\cdots\!61}{66\!\cdots\!13}a-\frac{46\!\cdots\!34}{66\!\cdots\!13}$, $\frac{15\!\cdots\!66}{18\!\cdots\!37}a^{14}+\frac{21\!\cdots\!15}{18\!\cdots\!37}a^{13}-\frac{63\!\cdots\!58}{18\!\cdots\!37}a^{12}-\frac{17\!\cdots\!41}{18\!\cdots\!37}a^{11}+\frac{59\!\cdots\!25}{18\!\cdots\!37}a^{10}+\frac{23\!\cdots\!62}{18\!\cdots\!37}a^{9}-\frac{10\!\cdots\!36}{18\!\cdots\!37}a^{8}-\frac{97\!\cdots\!35}{18\!\cdots\!37}a^{7}-\frac{34\!\cdots\!64}{18\!\cdots\!37}a^{6}+\frac{45\!\cdots\!97}{50\!\cdots\!01}a^{5}+\frac{10\!\cdots\!37}{18\!\cdots\!37}a^{4}-\frac{12\!\cdots\!40}{18\!\cdots\!37}a^{3}-\frac{87\!\cdots\!54}{18\!\cdots\!37}a^{2}+\frac{34\!\cdots\!54}{18\!\cdots\!37}a+\frac{23\!\cdots\!86}{18\!\cdots\!37}$
| 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: | \( 5820768023810000000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 5820768023810000000000 \cdot 3}{2\cdot\sqrt{691985919132121715398802656928813567369454726910003125}}\cr\approx \mathstrut & 0.343932215241233 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 4452*x^13 - 50296*x^12 + 5784843*x^11 + 91964349*x^10 - 3060924423*x^9 - 54750997444*x^8 + 781942057449*x^7 + 14664282257229*x^6 - 102907697118067*x^5 - 1901998615798914*x^4 + 6977459265924253*x^3 + 113854569421380357*x^2 - 202156325407652112*x - 2402311520555338669)
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^15 - x^14 - 4452*x^13 - 50296*x^12 + 5784843*x^11 + 91964349*x^10 - 3060924423*x^9 - 54750997444*x^8 + 781942057449*x^7 + 14664282257229*x^6 - 102907697118067*x^5 - 1901998615798914*x^4 + 6977459265924253*x^3 + 113854569421380357*x^2 - 202156325407652112*x - 2402311520555338669, 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^15 - x^14 - 4452*x^13 - 50296*x^12 + 5784843*x^11 + 91964349*x^10 - 3060924423*x^9 - 54750997444*x^8 + 781942057449*x^7 + 14664282257229*x^6 - 102907697118067*x^5 - 1901998615798914*x^4 + 6977459265924253*x^3 + 113854569421380357*x^2 - 202156325407652112*x - 2402311520555338669);
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^15 - x^14 - 4452*x^13 - 50296*x^12 + 5784843*x^11 + 91964349*x^10 - 3060924423*x^9 - 54750997444*x^8 + 781942057449*x^7 + 14664282257229*x^6 - 102907697118067*x^5 - 1901998615798914*x^4 + 6977459265924253*x^3 + 113854569421380357*x^2 - 202156325407652112*x - 2402311520555338669);
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
$C_3^4:D_{10}$ (as 15T43):
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 1620 |
| The 24 conjugacy class representatives for $C_3^4:D_{10}$ |
| Character table for $C_3^4:D_{10}$ is not computed |
Intermediate fields
| 5.5.160801.1 |
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 15 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 45 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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.5.0.1}{5} }$ | R | R | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.5.0.1}{5} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.5.0.1}{5} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.5.0.1}{5} }^{3}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
| 3.6.8.10 | $x^{6} + 9 x^{4} + 42 x^{3} + 441$ | $3$ | $2$ | $8$ | $S_3\times C_3$ | $[2, 2]^{2}$ | |
| 3.6.0.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(5\)
| 5.5.0.1 | $x^{5} + 4 x + 3$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(79\)
| $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.6.4.2 | $x^{6} - 6162 x^{3} + 18723$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(401\)
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $2$ | $3$ | $3$ | ||||
| Deg $6$ | $2$ | $3$ | $3$ | ||||
|
\(50329\)
| $\Q_{50329}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{50329}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{50329}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $3$ | $1$ | $2$ | ||||
| Deg $3$ | $3$ | $1$ | $2$ |