Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 138*x^13 + 459*x^12 + 6294*x^11 - 29158*x^10 - 90469*x^9 + 640304*x^8 - 282922*x^7 - 3320427*x^6 + 5075634*x^5 + 1483772*x^4 - 5986612*x^3 + 1193096*x^2 + 1830056*x - 504860)
gp: K = bnfinit(y^15 - 2*y^14 - 138*y^13 + 459*y^12 + 6294*y^11 - 29158*y^10 - 90469*y^9 + 640304*y^8 - 282922*y^7 - 3320427*y^6 + 5075634*y^5 + 1483772*y^4 - 5986612*y^3 + 1193096*y^2 + 1830056*y - 504860, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 2*x^14 - 138*x^13 + 459*x^12 + 6294*x^11 - 29158*x^10 - 90469*x^9 + 640304*x^8 - 282922*x^7 - 3320427*x^6 + 5075634*x^5 + 1483772*x^4 - 5986612*x^3 + 1193096*x^2 + 1830056*x - 504860);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 2*x^14 - 138*x^13 + 459*x^12 + 6294*x^11 - 29158*x^10 - 90469*x^9 + 640304*x^8 - 282922*x^7 - 3320427*x^6 + 5075634*x^5 + 1483772*x^4 - 5986612*x^3 + 1193096*x^2 + 1830056*x - 504860)
\( x^{15} - 2 x^{14} - 138 x^{13} + 459 x^{12} + 6294 x^{11} - 29158 x^{10} - 90469 x^{9} + 640304 x^{8} + \cdots - 504860 \)
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: |
\(7360671657636832731250000000000\)
\(\medspace = 2^{10}\cdot 5^{14}\cdot 19^{8}\cdot 37^{5}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(114.23\) | 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: | $2^{2/3}5^{13/10}19^{4/5}37^{1/2}\approx 824.9951818970933$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{13}+\frac{1}{4}a^{10}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{73\!\cdots\!32}a^{14}-\frac{94\!\cdots\!49}{85\!\cdots\!62}a^{13}+\frac{73\!\cdots\!61}{36\!\cdots\!66}a^{12}-\frac{81\!\cdots\!51}{73\!\cdots\!32}a^{11}+\frac{14\!\cdots\!27}{36\!\cdots\!66}a^{10}+\frac{69\!\cdots\!89}{36\!\cdots\!66}a^{9}+\frac{32\!\cdots\!41}{73\!\cdots\!32}a^{8}+\frac{73\!\cdots\!79}{18\!\cdots\!33}a^{7}+\frac{34\!\cdots\!73}{36\!\cdots\!66}a^{6}-\frac{15\!\cdots\!61}{73\!\cdots\!32}a^{5}-\frac{17\!\cdots\!87}{36\!\cdots\!66}a^{4}-\frac{14\!\cdots\!60}{18\!\cdots\!33}a^{3}+\frac{14\!\cdots\!89}{36\!\cdots\!66}a^{2}-\frac{33\!\cdots\!36}{18\!\cdots\!33}a-\frac{59\!\cdots\!11}{18\!\cdots\!33}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $14$ | 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: |
$\frac{97\!\cdots\!61}{33\!\cdots\!14}a^{14}-\frac{34\!\cdots\!61}{16\!\cdots\!07}a^{13}-\frac{13\!\cdots\!53}{33\!\cdots\!14}a^{12}+\frac{13\!\cdots\!12}{16\!\cdots\!07}a^{11}+\frac{64\!\cdots\!97}{33\!\cdots\!14}a^{10}-\frac{99\!\cdots\!19}{16\!\cdots\!07}a^{9}-\frac{11\!\cdots\!59}{33\!\cdots\!14}a^{8}+\frac{23\!\cdots\!52}{16\!\cdots\!07}a^{7}+\frac{33\!\cdots\!43}{33\!\cdots\!14}a^{6}-\frac{13\!\cdots\!25}{16\!\cdots\!07}a^{5}+\frac{13\!\cdots\!29}{33\!\cdots\!14}a^{4}+\frac{15\!\cdots\!06}{16\!\cdots\!07}a^{3}-\frac{86\!\cdots\!06}{16\!\cdots\!07}a^{2}-\frac{54\!\cdots\!72}{16\!\cdots\!07}a+\frac{18\!\cdots\!91}{16\!\cdots\!07}$, $\frac{10\!\cdots\!20}{16\!\cdots\!07}a^{14}-\frac{76\!\cdots\!28}{16\!\cdots\!07}a^{13}-\frac{14\!\cdots\!98}{16\!\cdots\!07}a^{12}+\frac{30\!\cdots\!23}{16\!\cdots\!07}a^{11}+\frac{71\!\cdots\!83}{16\!\cdots\!07}a^{10}-\frac{22\!\cdots\!45}{16\!\cdots\!07}a^{9}-\frac{12\!\cdots\!16}{16\!\cdots\!07}a^{8}+\frac{52\!\cdots\!59}{16\!\cdots\!07}a^{7}+\frac{37\!\cdots\!01}{16\!\cdots\!07}a^{6}-\frac{30\!\cdots\!76}{16\!\cdots\!07}a^{5}+\frac{14\!\cdots\!00}{16\!\cdots\!07}a^{4}+\frac{34\!\cdots\!88}{16\!\cdots\!07}a^{3}-\frac{19\!\cdots\!47}{16\!\cdots\!07}a^{2}-\frac{11\!\cdots\!38}{16\!\cdots\!07}a+\frac{42\!\cdots\!71}{16\!\cdots\!07}$, $\frac{31\!\cdots\!71}{73\!\cdots\!32}a^{14}-\frac{25\!\cdots\!55}{85\!\cdots\!62}a^{13}-\frac{21\!\cdots\!33}{36\!\cdots\!66}a^{12}+\frac{87\!\cdots\!03}{73\!\cdots\!32}a^{11}+\frac{10\!\cdots\!45}{36\!\cdots\!66}a^{10}-\frac{32\!\cdots\!35}{36\!\cdots\!66}a^{9}-\frac{36\!\cdots\!33}{73\!\cdots\!32}a^{8}+\frac{38\!\cdots\!07}{18\!\cdots\!33}a^{7}+\frac{54\!\cdots\!07}{36\!\cdots\!66}a^{6}-\frac{90\!\cdots\!19}{73\!\cdots\!32}a^{5}+\frac{21\!\cdots\!01}{36\!\cdots\!66}a^{4}+\frac{25\!\cdots\!78}{18\!\cdots\!33}a^{3}-\frac{28\!\cdots\!23}{36\!\cdots\!66}a^{2}-\frac{87\!\cdots\!30}{18\!\cdots\!33}a+\frac{30\!\cdots\!46}{18\!\cdots\!33}$, $\frac{19\!\cdots\!05}{36\!\cdots\!66}a^{14}-\frac{32\!\cdots\!13}{85\!\cdots\!62}a^{13}-\frac{13\!\cdots\!03}{18\!\cdots\!33}a^{12}+\frac{27\!\cdots\!72}{18\!\cdots\!33}a^{11}+\frac{65\!\cdots\!10}{18\!\cdots\!33}a^{10}-\frac{40\!\cdots\!81}{36\!\cdots\!66}a^{9}-\frac{23\!\cdots\!61}{36\!\cdots\!66}a^{8}+\frac{96\!\cdots\!51}{36\!\cdots\!66}a^{7}+\frac{34\!\cdots\!29}{18\!\cdots\!33}a^{6}-\frac{28\!\cdots\!08}{18\!\cdots\!33}a^{5}+\frac{13\!\cdots\!24}{18\!\cdots\!33}a^{4}+\frac{64\!\cdots\!09}{36\!\cdots\!66}a^{3}-\frac{17\!\cdots\!28}{18\!\cdots\!33}a^{2}-\frac{11\!\cdots\!89}{18\!\cdots\!33}a+\frac{38\!\cdots\!52}{18\!\cdots\!33}$, $\frac{22\!\cdots\!75}{36\!\cdots\!66}a^{14}-\frac{73\!\cdots\!77}{17\!\cdots\!24}a^{13}-\frac{31\!\cdots\!59}{36\!\cdots\!66}a^{12}+\frac{62\!\cdots\!59}{36\!\cdots\!66}a^{11}+\frac{29\!\cdots\!33}{73\!\cdots\!32}a^{10}-\frac{46\!\cdots\!55}{36\!\cdots\!66}a^{9}-\frac{26\!\cdots\!31}{36\!\cdots\!66}a^{8}+\frac{22\!\cdots\!41}{73\!\cdots\!32}a^{7}+\frac{39\!\cdots\!87}{18\!\cdots\!33}a^{6}-\frac{64\!\cdots\!29}{36\!\cdots\!66}a^{5}+\frac{61\!\cdots\!71}{73\!\cdots\!32}a^{4}+\frac{73\!\cdots\!15}{36\!\cdots\!66}a^{3}-\frac{40\!\cdots\!03}{36\!\cdots\!66}a^{2}-\frac{25\!\cdots\!07}{36\!\cdots\!66}a+\frac{43\!\cdots\!63}{18\!\cdots\!33}$, $\frac{12\!\cdots\!71}{73\!\cdots\!32}a^{14}-\frac{11\!\cdots\!51}{85\!\cdots\!62}a^{13}-\frac{44\!\cdots\!97}{18\!\cdots\!33}a^{12}+\frac{36\!\cdots\!51}{73\!\cdots\!32}a^{11}+\frac{43\!\cdots\!41}{36\!\cdots\!66}a^{10}-\frac{67\!\cdots\!90}{18\!\cdots\!33}a^{9}-\frac{15\!\cdots\!69}{73\!\cdots\!32}a^{8}+\frac{15\!\cdots\!47}{18\!\cdots\!33}a^{7}+\frac{10\!\cdots\!62}{18\!\cdots\!33}a^{6}-\frac{37\!\cdots\!79}{73\!\cdots\!32}a^{5}+\frac{92\!\cdots\!91}{36\!\cdots\!66}a^{4}+\frac{21\!\cdots\!09}{36\!\cdots\!66}a^{3}-\frac{12\!\cdots\!37}{36\!\cdots\!66}a^{2}-\frac{36\!\cdots\!63}{18\!\cdots\!33}a+\frac{13\!\cdots\!01}{18\!\cdots\!33}$, $\frac{38\!\cdots\!27}{73\!\cdots\!32}a^{14}-\frac{28\!\cdots\!11}{85\!\cdots\!62}a^{13}-\frac{26\!\cdots\!05}{36\!\cdots\!66}a^{12}+\frac{10\!\cdots\!81}{73\!\cdots\!32}a^{11}+\frac{64\!\cdots\!94}{18\!\cdots\!33}a^{10}-\frac{19\!\cdots\!96}{18\!\cdots\!33}a^{9}-\frac{45\!\cdots\!65}{73\!\cdots\!32}a^{8}+\frac{46\!\cdots\!79}{18\!\cdots\!33}a^{7}+\frac{72\!\cdots\!87}{36\!\cdots\!66}a^{6}-\frac{10\!\cdots\!09}{73\!\cdots\!32}a^{5}+\frac{11\!\cdots\!28}{18\!\cdots\!33}a^{4}+\frac{61\!\cdots\!71}{36\!\cdots\!66}a^{3}-\frac{30\!\cdots\!23}{36\!\cdots\!66}a^{2}-\frac{10\!\cdots\!51}{18\!\cdots\!33}a+\frac{35\!\cdots\!73}{18\!\cdots\!33}$, $\frac{35\!\cdots\!27}{73\!\cdots\!32}a^{14}-\frac{54\!\cdots\!61}{17\!\cdots\!24}a^{13}-\frac{12\!\cdots\!09}{18\!\cdots\!33}a^{12}+\frac{97\!\cdots\!07}{73\!\cdots\!32}a^{11}+\frac{23\!\cdots\!93}{73\!\cdots\!32}a^{10}-\frac{18\!\cdots\!52}{18\!\cdots\!33}a^{9}-\frac{42\!\cdots\!05}{73\!\cdots\!32}a^{8}+\frac{17\!\cdots\!03}{73\!\cdots\!32}a^{7}+\frac{65\!\cdots\!99}{36\!\cdots\!66}a^{6}-\frac{10\!\cdots\!47}{73\!\cdots\!32}a^{5}+\frac{45\!\cdots\!63}{73\!\cdots\!32}a^{4}+\frac{57\!\cdots\!41}{36\!\cdots\!66}a^{3}-\frac{15\!\cdots\!61}{18\!\cdots\!33}a^{2}-\frac{19\!\cdots\!53}{36\!\cdots\!66}a+\frac{33\!\cdots\!14}{18\!\cdots\!33}$, $\frac{31\!\cdots\!51}{36\!\cdots\!66}a^{14}-\frac{51\!\cdots\!75}{85\!\cdots\!62}a^{13}-\frac{43\!\cdots\!73}{36\!\cdots\!66}a^{12}+\frac{43\!\cdots\!83}{18\!\cdots\!33}a^{11}+\frac{10\!\cdots\!19}{18\!\cdots\!33}a^{10}-\frac{31\!\cdots\!62}{18\!\cdots\!33}a^{9}-\frac{36\!\cdots\!67}{36\!\cdots\!66}a^{8}+\frac{15\!\cdots\!47}{36\!\cdots\!66}a^{7}+\frac{10\!\cdots\!35}{36\!\cdots\!66}a^{6}-\frac{44\!\cdots\!09}{18\!\cdots\!33}a^{5}+\frac{21\!\cdots\!60}{18\!\cdots\!33}a^{4}+\frac{50\!\cdots\!83}{18\!\cdots\!33}a^{3}-\frac{27\!\cdots\!94}{18\!\cdots\!33}a^{2}-\frac{17\!\cdots\!94}{18\!\cdots\!33}a+\frac{60\!\cdots\!43}{18\!\cdots\!33}$, $\frac{71\!\cdots\!84}{18\!\cdots\!33}a^{14}-\frac{11\!\cdots\!44}{42\!\cdots\!31}a^{13}-\frac{99\!\cdots\!34}{18\!\cdots\!33}a^{12}+\frac{20\!\cdots\!28}{18\!\cdots\!33}a^{11}+\frac{47\!\cdots\!86}{18\!\cdots\!33}a^{10}-\frac{14\!\cdots\!96}{18\!\cdots\!33}a^{9}-\frac{83\!\cdots\!23}{18\!\cdots\!33}a^{8}+\frac{35\!\cdots\!51}{18\!\cdots\!33}a^{7}+\frac{25\!\cdots\!82}{18\!\cdots\!33}a^{6}-\frac{20\!\cdots\!81}{18\!\cdots\!33}a^{5}+\frac{97\!\cdots\!73}{18\!\cdots\!33}a^{4}+\frac{23\!\cdots\!19}{18\!\cdots\!33}a^{3}-\frac{12\!\cdots\!97}{18\!\cdots\!33}a^{2}-\frac{79\!\cdots\!14}{18\!\cdots\!33}a+\frac{27\!\cdots\!93}{18\!\cdots\!33}$, $\frac{29\!\cdots\!35}{36\!\cdots\!66}a^{14}+\frac{72\!\cdots\!53}{17\!\cdots\!24}a^{13}-\frac{51\!\cdots\!75}{36\!\cdots\!66}a^{12}-\frac{10\!\cdots\!85}{18\!\cdots\!33}a^{11}+\frac{12\!\cdots\!33}{73\!\cdots\!32}a^{10}+\frac{47\!\cdots\!03}{18\!\cdots\!33}a^{9}-\frac{36\!\cdots\!63}{36\!\cdots\!66}a^{8}-\frac{30\!\cdots\!25}{73\!\cdots\!32}a^{7}+\frac{38\!\cdots\!96}{18\!\cdots\!33}a^{6}+\frac{76\!\cdots\!23}{18\!\cdots\!33}a^{5}-\frac{76\!\cdots\!57}{73\!\cdots\!32}a^{4}+\frac{17\!\cdots\!66}{18\!\cdots\!33}a^{3}+\frac{11\!\cdots\!11}{36\!\cdots\!66}a^{2}-\frac{15\!\cdots\!23}{36\!\cdots\!66}a+\frac{15\!\cdots\!43}{18\!\cdots\!33}$, $\frac{35\!\cdots\!65}{73\!\cdots\!32}a^{14}-\frac{59\!\cdots\!35}{17\!\cdots\!24}a^{13}-\frac{12\!\cdots\!80}{18\!\cdots\!33}a^{12}+\frac{99\!\cdots\!81}{73\!\cdots\!32}a^{11}+\frac{23\!\cdots\!51}{73\!\cdots\!32}a^{10}-\frac{18\!\cdots\!06}{18\!\cdots\!33}a^{9}-\frac{41\!\cdots\!27}{73\!\cdots\!32}a^{8}+\frac{17\!\cdots\!01}{73\!\cdots\!32}a^{7}+\frac{61\!\cdots\!13}{36\!\cdots\!66}a^{6}-\frac{10\!\cdots\!81}{73\!\cdots\!32}a^{5}+\frac{49\!\cdots\!89}{73\!\cdots\!32}a^{4}+\frac{57\!\cdots\!57}{36\!\cdots\!66}a^{3}-\frac{16\!\cdots\!55}{18\!\cdots\!33}a^{2}-\frac{19\!\cdots\!17}{36\!\cdots\!66}a+\frac{36\!\cdots\!56}{18\!\cdots\!33}$, $\frac{48\!\cdots\!95}{73\!\cdots\!32}a^{14}-\frac{55\!\cdots\!35}{85\!\cdots\!62}a^{13}-\frac{33\!\cdots\!53}{36\!\cdots\!66}a^{12}+\frac{75\!\cdots\!81}{73\!\cdots\!32}a^{11}+\frac{54\!\cdots\!87}{18\!\cdots\!33}a^{10}-\frac{10\!\cdots\!15}{18\!\cdots\!33}a^{9}+\frac{25\!\cdots\!99}{73\!\cdots\!32}a^{8}+\frac{19\!\cdots\!57}{18\!\cdots\!33}a^{7}-\frac{92\!\cdots\!39}{36\!\cdots\!66}a^{6}-\frac{35\!\cdots\!65}{73\!\cdots\!32}a^{5}+\frac{32\!\cdots\!67}{18\!\cdots\!33}a^{4}-\frac{42\!\cdots\!37}{36\!\cdots\!66}a^{3}-\frac{80\!\cdots\!85}{36\!\cdots\!66}a^{2}+\frac{72\!\cdots\!13}{18\!\cdots\!33}a+\frac{15\!\cdots\!21}{18\!\cdots\!33}$, $\frac{16\!\cdots\!20}{18\!\cdots\!33}a^{14}-\frac{12\!\cdots\!81}{17\!\cdots\!24}a^{13}-\frac{23\!\cdots\!72}{18\!\cdots\!33}a^{12}+\frac{97\!\cdots\!53}{36\!\cdots\!66}a^{11}+\frac{44\!\cdots\!27}{73\!\cdots\!32}a^{10}-\frac{70\!\cdots\!43}{36\!\cdots\!66}a^{9}-\frac{19\!\cdots\!80}{18\!\cdots\!33}a^{8}+\frac{33\!\cdots\!21}{73\!\cdots\!32}a^{7}+\frac{99\!\cdots\!63}{36\!\cdots\!66}a^{6}-\frac{96\!\cdots\!85}{36\!\cdots\!66}a^{5}+\frac{10\!\cdots\!69}{73\!\cdots\!32}a^{4}+\frac{98\!\cdots\!45}{36\!\cdots\!66}a^{3}-\frac{62\!\cdots\!25}{36\!\cdots\!66}a^{2}-\frac{29\!\cdots\!55}{36\!\cdots\!66}a+\frac{55\!\cdots\!19}{18\!\cdots\!33}$
| 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: | \( 276528967045 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 276528967045 \cdot 1}{2\cdot\sqrt{7360671657636832731250000000000}}\cr\approx \mathstrut & 1.66994377036061 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 138*x^13 + 459*x^12 + 6294*x^11 - 29158*x^10 - 90469*x^9 + 640304*x^8 - 282922*x^7 - 3320427*x^6 + 5075634*x^5 + 1483772*x^4 - 5986612*x^3 + 1193096*x^2 + 1830056*x - 504860)
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 - 2*x^14 - 138*x^13 + 459*x^12 + 6294*x^11 - 29158*x^10 - 90469*x^9 + 640304*x^8 - 282922*x^7 - 3320427*x^6 + 5075634*x^5 + 1483772*x^4 - 5986612*x^3 + 1193096*x^2 + 1830056*x - 504860, 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 - 2*x^14 - 138*x^13 + 459*x^12 + 6294*x^11 - 29158*x^10 - 90469*x^9 + 640304*x^8 - 282922*x^7 - 3320427*x^6 + 5075634*x^5 + 1483772*x^4 - 5986612*x^3 + 1193096*x^2 + 1830056*x - 504860);
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 - 2*x^14 - 138*x^13 + 459*x^12 + 6294*x^11 - 29158*x^10 - 90469*x^9 + 640304*x^8 - 282922*x^7 - 3320427*x^6 + 5075634*x^5 + 1483772*x^4 - 5986612*x^3 + 1193096*x^2 + 1830056*x - 504860);
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
$C_5^2:D_6$ (as 15T18):
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 300 |
| The 14 conjugacy class representatives for $((C_5^2 : C_3):C_2):C_2$ |
| Character table for $((C_5^2 : C_3):C_2):C_2$ |
Intermediate fields
| 3.3.148.1 |
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 sibling: | data not computed |
| Degree 25 sibling: | data not computed |
| Degree 30 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 | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.3.0.1}{3} }^{5}$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | R | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{5}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{5}$ | R | ${\href{/padicField/41.3.0.1}{3} }^{5}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.10.12.10 | $x^{10} + 20 x^{7} - 200 x^{6} + 10 x^{5} + 100 x^{4} + 100 x^{2} + 25$ | $5$ | $2$ | $12$ | $D_{10}$ | $[3/2]_{2}^{2}$ | |
|
\(19\)
| 19.5.0.1 | $x^{5} + 5 x + 17$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 19.10.8.3 | $x^{10} - 5510 x^{5} - 1650131$ | $5$ | $2$ | $8$ | $D_5\times C_5$ | $[\ ]_{5}^{10}$ | |
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |