Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 200*x^13 - 235*x^12 + 14183*x^11 + 39397*x^10 - 404770*x^9 - 1625940*x^8 + 4071591*x^7 + 23923079*x^6 - 3550930*x^5 - 131623425*x^4 - 118897750*x^3 + 169055000*x^2 + 288950000*x + 110453125)
gp: K = bnfinit(y^15 - y^14 - 200*y^13 - 235*y^12 + 14183*y^11 + 39397*y^10 - 404770*y^9 - 1625940*y^8 + 4071591*y^7 + 23923079*y^6 - 3550930*y^5 - 131623425*y^4 - 118897750*y^3 + 169055000*y^2 + 288950000*y + 110453125, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 - 200*x^13 - 235*x^12 + 14183*x^11 + 39397*x^10 - 404770*x^9 - 1625940*x^8 + 4071591*x^7 + 23923079*x^6 - 3550930*x^5 - 131623425*x^4 - 118897750*x^3 + 169055000*x^2 + 288950000*x + 110453125);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - x^14 - 200*x^13 - 235*x^12 + 14183*x^11 + 39397*x^10 - 404770*x^9 - 1625940*x^8 + 4071591*x^7 + 23923079*x^6 - 3550930*x^5 - 131623425*x^4 - 118897750*x^3 + 169055000*x^2 + 288950000*x + 110453125)
\( x^{15} - x^{14} - 200 x^{13} - 235 x^{12} + 14183 x^{11} + 39397 x^{10} - 404770 x^{9} + \cdots + 110453125 \)
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: |
\(4640873420279330256301704361074121\)
\(\medspace = 19^{10}\cdot 31^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(175.57\) | 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: | $19^{2/3}31^{14/15}\approx 175.5658249685855$ | ||
| Ramified primes: |
\(19\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $15$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(589=19\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{589}(1,·)$, $\chi_{589}(258,·)$, $\chi_{589}(419,·)$, $\chi_{589}(438,·)$, $\chi_{589}(7,·)$, $\chi_{589}(39,·)$, $\chi_{589}(273,·)$, $\chi_{589}(45,·)$, $\chi_{589}(505,·)$, $\chi_{589}(144,·)$, $\chi_{589}(49,·)$, $\chi_{589}(577,·)$, $\chi_{589}(343,·)$, $\chi_{589}(121,·)$, $\chi_{589}(315,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}-\frac{1}{5}a$, $\frac{1}{5}a^{6}-\frac{1}{5}a^{2}$, $\frac{1}{5}a^{7}-\frac{1}{5}a^{3}$, $\frac{1}{5}a^{8}-\frac{1}{5}a^{4}$, $\frac{1}{5}a^{9}-\frac{1}{5}a$, $\frac{1}{25}a^{10}-\frac{2}{25}a^{6}+\frac{1}{25}a^{2}$, $\frac{1}{125}a^{11}-\frac{2}{25}a^{8}-\frac{2}{125}a^{7}-\frac{1}{25}a^{6}+\frac{2}{25}a^{4}+\frac{26}{125}a^{3}+\frac{1}{25}a^{2}-\frac{1}{5}a$, $\frac{1}{625}a^{12}-\frac{1}{625}a^{11}+\frac{2}{125}a^{10}-\frac{2}{125}a^{9}+\frac{58}{625}a^{8}+\frac{47}{625}a^{7}+\frac{12}{125}a^{6}+\frac{12}{125}a^{5}+\frac{91}{625}a^{4}-\frac{71}{625}a^{3}-\frac{19}{125}a^{2}-\frac{11}{25}a+\frac{1}{5}$, $\frac{1}{15625}a^{13}+\frac{9}{15625}a^{12}-\frac{2}{3125}a^{11}+\frac{13}{3125}a^{10}-\frac{542}{15625}a^{9}-\frac{148}{15625}a^{8}+\frac{32}{625}a^{7}-\frac{98}{3125}a^{6}-\frac{184}{15625}a^{5}-\frac{2761}{15625}a^{4}+\frac{862}{3125}a^{3}-\frac{27}{625}a^{2}+\frac{2}{125}a-\frac{6}{25}$, $\frac{1}{23\!\cdots\!75}a^{14}-\frac{42\!\cdots\!96}{23\!\cdots\!75}a^{13}-\frac{57\!\cdots\!36}{47\!\cdots\!75}a^{12}+\frac{68\!\cdots\!98}{47\!\cdots\!75}a^{11}+\frac{20\!\cdots\!83}{23\!\cdots\!75}a^{10}+\frac{15\!\cdots\!62}{23\!\cdots\!75}a^{9}-\frac{29\!\cdots\!92}{47\!\cdots\!75}a^{8}+\frac{85\!\cdots\!27}{47\!\cdots\!75}a^{7}-\frac{19\!\cdots\!09}{23\!\cdots\!75}a^{6}+\frac{48\!\cdots\!34}{23\!\cdots\!75}a^{5}+\frac{23\!\cdots\!73}{47\!\cdots\!75}a^{4}-\frac{44\!\cdots\!89}{95\!\cdots\!75}a^{3}-\frac{15\!\cdots\!96}{19\!\cdots\!75}a^{2}+\frac{66\!\cdots\!67}{38\!\cdots\!75}a+\frac{34\!\cdots\!81}{76\!\cdots\!75}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
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: |
$\frac{20\!\cdots\!78}{23\!\cdots\!75}a^{14}-\frac{70\!\cdots\!63}{23\!\cdots\!75}a^{13}-\frac{75\!\cdots\!58}{47\!\cdots\!75}a^{12}+\frac{92\!\cdots\!44}{47\!\cdots\!75}a^{11}+\frac{26\!\cdots\!24}{23\!\cdots\!75}a^{10}+\frac{14\!\cdots\!86}{23\!\cdots\!75}a^{9}-\frac{16\!\cdots\!26}{47\!\cdots\!75}a^{8}-\frac{26\!\cdots\!19}{47\!\cdots\!75}a^{7}+\frac{10\!\cdots\!73}{23\!\cdots\!75}a^{6}+\frac{23\!\cdots\!77}{23\!\cdots\!75}a^{5}-\frac{97\!\cdots\!31}{47\!\cdots\!75}a^{4}-\frac{56\!\cdots\!17}{95\!\cdots\!75}a^{3}+\frac{23\!\cdots\!37}{19\!\cdots\!75}a^{2}+\frac{37\!\cdots\!51}{38\!\cdots\!75}a+\frac{39\!\cdots\!93}{76\!\cdots\!75}$, $\frac{14\!\cdots\!32}{23\!\cdots\!75}a^{14}+\frac{43\!\cdots\!78}{23\!\cdots\!75}a^{13}-\frac{11\!\cdots\!57}{47\!\cdots\!75}a^{12}-\frac{16\!\cdots\!39}{47\!\cdots\!75}a^{11}+\frac{53\!\cdots\!81}{23\!\cdots\!75}a^{10}+\frac{57\!\cdots\!09}{23\!\cdots\!75}a^{9}-\frac{34\!\cdots\!84}{47\!\cdots\!75}a^{8}-\frac{35\!\cdots\!61}{47\!\cdots\!75}a^{7}+\frac{19\!\cdots\!12}{23\!\cdots\!75}a^{6}+\frac{23\!\cdots\!63}{23\!\cdots\!75}a^{5}-\frac{86\!\cdots\!19}{47\!\cdots\!75}a^{4}-\frac{51\!\cdots\!38}{95\!\cdots\!75}a^{3}-\frac{32\!\cdots\!57}{19\!\cdots\!75}a^{2}+\frac{31\!\cdots\!29}{38\!\cdots\!75}a+\frac{44\!\cdots\!37}{76\!\cdots\!75}$, $\frac{35\!\cdots\!41}{23\!\cdots\!75}a^{14}-\frac{47\!\cdots\!36}{23\!\cdots\!75}a^{13}-\frac{14\!\cdots\!66}{47\!\cdots\!75}a^{12}-\frac{10\!\cdots\!07}{47\!\cdots\!75}a^{11}+\frac{52\!\cdots\!03}{23\!\cdots\!75}a^{10}+\frac{11\!\cdots\!67}{23\!\cdots\!75}a^{9}-\frac{31\!\cdots\!17}{47\!\cdots\!75}a^{8}-\frac{10\!\cdots\!43}{47\!\cdots\!75}a^{7}+\frac{19\!\cdots\!06}{23\!\cdots\!75}a^{6}+\frac{76\!\cdots\!69}{23\!\cdots\!75}a^{5}-\frac{15\!\cdots\!22}{47\!\cdots\!75}a^{4}-\frac{17\!\cdots\!44}{95\!\cdots\!75}a^{3}-\frac{47\!\cdots\!16}{19\!\cdots\!75}a^{2}+\frac{11\!\cdots\!52}{38\!\cdots\!75}a+\frac{14\!\cdots\!31}{76\!\cdots\!75}$, $\frac{89\!\cdots\!39}{47\!\cdots\!75}a^{14}-\frac{20\!\cdots\!84}{47\!\cdots\!75}a^{13}-\frac{34\!\cdots\!56}{95\!\cdots\!75}a^{12}+\frac{82\!\cdots\!77}{95\!\cdots\!75}a^{11}+\frac{12\!\cdots\!62}{47\!\cdots\!75}a^{10}+\frac{20\!\cdots\!98}{47\!\cdots\!75}a^{9}-\frac{75\!\cdots\!69}{95\!\cdots\!75}a^{8}-\frac{20\!\cdots\!47}{95\!\cdots\!75}a^{7}+\frac{45\!\cdots\!99}{47\!\cdots\!75}a^{6}+\frac{15\!\cdots\!11}{47\!\cdots\!75}a^{5}-\frac{74\!\cdots\!44}{19\!\cdots\!75}a^{4}-\frac{37\!\cdots\!92}{19\!\cdots\!75}a^{3}-\frac{99\!\cdots\!53}{38\!\cdots\!75}a^{2}+\frac{24\!\cdots\!72}{76\!\cdots\!75}a+\frac{32\!\cdots\!62}{15\!\cdots\!75}$, $\frac{26\!\cdots\!07}{23\!\cdots\!75}a^{14}-\frac{47\!\cdots\!22}{23\!\cdots\!75}a^{13}-\frac{10\!\cdots\!67}{47\!\cdots\!75}a^{12}-\frac{36\!\cdots\!39}{47\!\cdots\!75}a^{11}+\frac{37\!\cdots\!31}{23\!\cdots\!75}a^{10}+\frac{71\!\cdots\!34}{23\!\cdots\!75}a^{9}-\frac{22\!\cdots\!64}{47\!\cdots\!75}a^{8}-\frac{65\!\cdots\!11}{47\!\cdots\!75}a^{7}+\frac{13\!\cdots\!37}{23\!\cdots\!75}a^{6}+\frac{50\!\cdots\!38}{23\!\cdots\!75}a^{5}-\frac{10\!\cdots\!54}{47\!\cdots\!75}a^{4}-\frac{11\!\cdots\!93}{95\!\cdots\!75}a^{3}-\frac{38\!\cdots\!02}{19\!\cdots\!75}a^{2}+\frac{77\!\cdots\!74}{38\!\cdots\!75}a+\frac{10\!\cdots\!77}{76\!\cdots\!75}$, $\frac{32\!\cdots\!82}{23\!\cdots\!75}a^{14}-\frac{15\!\cdots\!97}{23\!\cdots\!75}a^{13}-\frac{11\!\cdots\!27}{47\!\cdots\!75}a^{12}+\frac{29\!\cdots\!86}{47\!\cdots\!75}a^{11}+\frac{40\!\cdots\!06}{23\!\cdots\!75}a^{10}-\frac{24\!\cdots\!91}{23\!\cdots\!75}a^{9}-\frac{24\!\cdots\!69}{47\!\cdots\!75}a^{8}-\frac{13\!\cdots\!86}{47\!\cdots\!75}a^{7}+\frac{15\!\cdots\!37}{23\!\cdots\!75}a^{6}+\frac{18\!\cdots\!38}{23\!\cdots\!75}a^{5}-\frac{15\!\cdots\!64}{47\!\cdots\!75}a^{4}-\frac{51\!\cdots\!48}{95\!\cdots\!75}a^{3}+\frac{64\!\cdots\!53}{19\!\cdots\!75}a^{2}+\frac{36\!\cdots\!69}{38\!\cdots\!75}a+\frac{32\!\cdots\!42}{76\!\cdots\!75}$, $\frac{85\!\cdots\!66}{23\!\cdots\!75}a^{14}-\frac{30\!\cdots\!11}{23\!\cdots\!75}a^{13}-\frac{32\!\cdots\!31}{47\!\cdots\!75}a^{12}-\frac{64\!\cdots\!82}{47\!\cdots\!75}a^{11}+\frac{99\!\cdots\!03}{23\!\cdots\!75}a^{10}+\frac{38\!\cdots\!67}{23\!\cdots\!75}a^{9}-\frac{39\!\cdots\!37}{47\!\cdots\!75}a^{8}-\frac{24\!\cdots\!93}{47\!\cdots\!75}a^{7}-\frac{84\!\cdots\!69}{23\!\cdots\!75}a^{6}+\frac{97\!\cdots\!19}{23\!\cdots\!75}a^{5}+\frac{33\!\cdots\!38}{47\!\cdots\!75}a^{4}-\frac{23\!\cdots\!64}{95\!\cdots\!75}a^{3}-\frac{30\!\cdots\!71}{19\!\cdots\!75}a^{2}-\frac{52\!\cdots\!93}{38\!\cdots\!75}a-\frac{29\!\cdots\!34}{76\!\cdots\!75}$, $\frac{16\!\cdots\!02}{23\!\cdots\!75}a^{14}+\frac{42\!\cdots\!08}{23\!\cdots\!75}a^{13}-\frac{63\!\cdots\!07}{47\!\cdots\!75}a^{12}-\frac{30\!\cdots\!29}{47\!\cdots\!75}a^{11}+\frac{18\!\cdots\!41}{23\!\cdots\!75}a^{10}+\frac{13\!\cdots\!74}{23\!\cdots\!75}a^{9}-\frac{43\!\cdots\!64}{47\!\cdots\!75}a^{8}-\frac{72\!\cdots\!46}{47\!\cdots\!75}a^{7}-\frac{58\!\cdots\!68}{23\!\cdots\!75}a^{6}+\frac{23\!\cdots\!18}{23\!\cdots\!75}a^{5}+\frac{16\!\cdots\!61}{47\!\cdots\!75}a^{4}+\frac{14\!\cdots\!67}{95\!\cdots\!75}a^{3}-\frac{11\!\cdots\!37}{19\!\cdots\!75}a^{2}-\frac{32\!\cdots\!96}{38\!\cdots\!75}a-\frac{24\!\cdots\!98}{76\!\cdots\!75}$, $\frac{10\!\cdots\!36}{47\!\cdots\!75}a^{14}-\frac{20\!\cdots\!46}{47\!\cdots\!75}a^{13}-\frac{43\!\cdots\!93}{95\!\cdots\!75}a^{12}-\frac{12\!\cdots\!17}{95\!\cdots\!75}a^{11}+\frac{15\!\cdots\!13}{47\!\cdots\!75}a^{10}+\frac{29\!\cdots\!87}{47\!\cdots\!75}a^{9}-\frac{93\!\cdots\!28}{95\!\cdots\!75}a^{8}-\frac{27\!\cdots\!28}{95\!\cdots\!75}a^{7}+\frac{56\!\cdots\!01}{47\!\cdots\!75}a^{6}+\frac{21\!\cdots\!09}{47\!\cdots\!75}a^{5}-\frac{45\!\cdots\!34}{95\!\cdots\!75}a^{4}-\frac{98\!\cdots\!01}{38\!\cdots\!75}a^{3}-\frac{31\!\cdots\!52}{76\!\cdots\!75}a^{2}+\frac{32\!\cdots\!41}{76\!\cdots\!75}a+\frac{43\!\cdots\!69}{15\!\cdots\!75}$, $\frac{67\!\cdots\!98}{23\!\cdots\!75}a^{14}-\frac{50\!\cdots\!58}{23\!\cdots\!75}a^{13}-\frac{23\!\cdots\!03}{47\!\cdots\!75}a^{12}+\frac{12\!\cdots\!29}{47\!\cdots\!75}a^{11}+\frac{84\!\cdots\!59}{23\!\cdots\!75}a^{10}-\frac{26\!\cdots\!49}{23\!\cdots\!75}a^{9}-\frac{58\!\cdots\!91}{47\!\cdots\!75}a^{8}+\frac{71\!\cdots\!71}{47\!\cdots\!75}a^{7}+\frac{48\!\cdots\!68}{23\!\cdots\!75}a^{6}+\frac{10\!\cdots\!07}{23\!\cdots\!75}a^{5}-\frac{63\!\cdots\!96}{47\!\cdots\!75}a^{4}-\frac{13\!\cdots\!72}{95\!\cdots\!75}a^{3}+\frac{34\!\cdots\!92}{19\!\cdots\!75}a^{2}+\frac{13\!\cdots\!91}{38\!\cdots\!75}a+\frac{10\!\cdots\!63}{76\!\cdots\!75}$, $\frac{48\!\cdots\!14}{47\!\cdots\!75}a^{14}+\frac{50\!\cdots\!66}{47\!\cdots\!75}a^{13}-\frac{18\!\cdots\!96}{95\!\cdots\!75}a^{12}-\frac{61\!\cdots\!48}{95\!\cdots\!75}a^{11}+\frac{57\!\cdots\!12}{47\!\cdots\!75}a^{10}+\frac{29\!\cdots\!23}{47\!\cdots\!75}a^{9}-\frac{20\!\cdots\!39}{95\!\cdots\!75}a^{8}-\frac{17\!\cdots\!97}{95\!\cdots\!75}a^{7}-\frac{53\!\cdots\!01}{47\!\cdots\!75}a^{6}+\frac{69\!\cdots\!11}{47\!\cdots\!75}a^{5}+\frac{57\!\cdots\!23}{19\!\cdots\!75}a^{4}-\frac{15\!\cdots\!12}{19\!\cdots\!75}a^{3}-\frac{24\!\cdots\!08}{38\!\cdots\!75}a^{2}-\frac{41\!\cdots\!48}{76\!\cdots\!75}a-\frac{20\!\cdots\!03}{15\!\cdots\!75}$, $\frac{20\!\cdots\!87}{23\!\cdots\!75}a^{14}-\frac{56\!\cdots\!02}{23\!\cdots\!75}a^{13}-\frac{80\!\cdots\!32}{47\!\cdots\!75}a^{12}+\frac{37\!\cdots\!76}{47\!\cdots\!75}a^{11}+\frac{28\!\cdots\!46}{23\!\cdots\!75}a^{10}+\frac{34\!\cdots\!69}{23\!\cdots\!75}a^{9}-\frac{17\!\cdots\!04}{47\!\cdots\!75}a^{8}-\frac{38\!\cdots\!26}{47\!\cdots\!75}a^{7}+\frac{10\!\cdots\!17}{23\!\cdots\!75}a^{6}+\frac{31\!\cdots\!83}{23\!\cdots\!75}a^{5}-\frac{97\!\cdots\!99}{47\!\cdots\!75}a^{4}-\frac{75\!\cdots\!68}{95\!\cdots\!75}a^{3}+\frac{55\!\cdots\!23}{19\!\cdots\!75}a^{2}+\frac{49\!\cdots\!04}{38\!\cdots\!75}a+\frac{57\!\cdots\!22}{76\!\cdots\!75}$, $\frac{14\!\cdots\!07}{47\!\cdots\!75}a^{14}-\frac{71\!\cdots\!17}{47\!\cdots\!75}a^{13}-\frac{57\!\cdots\!53}{95\!\cdots\!75}a^{12}+\frac{15\!\cdots\!76}{95\!\cdots\!75}a^{11}+\frac{23\!\cdots\!06}{47\!\cdots\!75}a^{10}-\frac{15\!\cdots\!51}{47\!\cdots\!75}a^{9}-\frac{18\!\cdots\!22}{95\!\cdots\!75}a^{8}-\frac{16\!\cdots\!11}{95\!\cdots\!75}a^{7}+\frac{16\!\cdots\!62}{47\!\cdots\!75}a^{6}+\frac{32\!\cdots\!18}{47\!\cdots\!75}a^{5}-\frac{40\!\cdots\!37}{19\!\cdots\!75}a^{4}-\frac{11\!\cdots\!11}{19\!\cdots\!75}a^{3}+\frac{39\!\cdots\!76}{38\!\cdots\!75}a^{2}+\frac{92\!\cdots\!21}{76\!\cdots\!75}a+\frac{10\!\cdots\!21}{15\!\cdots\!75}$, $\frac{64\!\cdots\!81}{23\!\cdots\!75}a^{14}+\frac{61\!\cdots\!74}{23\!\cdots\!75}a^{13}-\frac{24\!\cdots\!06}{47\!\cdots\!75}a^{12}-\frac{79\!\cdots\!87}{47\!\cdots\!75}a^{11}+\frac{77\!\cdots\!48}{23\!\cdots\!75}a^{10}+\frac{39\!\cdots\!97}{23\!\cdots\!75}a^{9}-\frac{28\!\cdots\!97}{47\!\cdots\!75}a^{8}-\frac{23\!\cdots\!63}{47\!\cdots\!75}a^{7}-\frac{64\!\cdots\!54}{23\!\cdots\!75}a^{6}+\frac{93\!\cdots\!79}{23\!\cdots\!75}a^{5}+\frac{36\!\cdots\!73}{47\!\cdots\!75}a^{4}-\frac{27\!\cdots\!29}{95\!\cdots\!75}a^{3}-\frac{32\!\cdots\!06}{19\!\cdots\!75}a^{2}-\frac{44\!\cdots\!43}{38\!\cdots\!75}a-\frac{11\!\cdots\!04}{76\!\cdots\!75}$
| 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: | \( 4307906685437.854 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 4307906685437.854 \cdot 3}{2\cdot\sqrt{4640873420279330256301704361074121}}\cr\approx \mathstrut & 3.10819043610185 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 200*x^13 - 235*x^12 + 14183*x^11 + 39397*x^10 - 404770*x^9 - 1625940*x^8 + 4071591*x^7 + 23923079*x^6 - 3550930*x^5 - 131623425*x^4 - 118897750*x^3 + 169055000*x^2 + 288950000*x + 110453125)
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 - 200*x^13 - 235*x^12 + 14183*x^11 + 39397*x^10 - 404770*x^9 - 1625940*x^8 + 4071591*x^7 + 23923079*x^6 - 3550930*x^5 - 131623425*x^4 - 118897750*x^3 + 169055000*x^2 + 288950000*x + 110453125, 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 - 200*x^13 - 235*x^12 + 14183*x^11 + 39397*x^10 - 404770*x^9 - 1625940*x^8 + 4071591*x^7 + 23923079*x^6 - 3550930*x^5 - 131623425*x^4 - 118897750*x^3 + 169055000*x^2 + 288950000*x + 110453125);
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 - 200*x^13 - 235*x^12 + 14183*x^11 + 39397*x^10 - 404770*x^9 - 1625940*x^8 + 4071591*x^7 + 23923079*x^6 - 3550930*x^5 - 131623425*x^4 - 118897750*x^3 + 169055000*x^2 + 288950000*x + 110453125);
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 cyclic group of order 15 |
| The 15 conjugacy class representatives for $C_{15}$ |
| Character table for $C_{15}$ |
Intermediate fields
| 3.3.346921.2, 5.5.923521.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $15$ | $15$ | ${\href{/padicField/5.1.0.1}{1} }^{15}$ | $15$ | $15$ | $15$ | $15$ | R | $15$ | $15$ | R | ${\href{/padicField/37.3.0.1}{3} }^{5}$ | ${\href{/padicField/41.5.0.1}{5} }^{3}$ | $15$ | $15$ | $15$ | $15$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| 19.15.10.3 | $x^{15} + 3610 x^{9} + 3258025 x^{3} + 715592611$ | $3$ | $5$ | $10$ | $C_{15}$ | $[\ ]_{3}^{5}$ |
|
\(31\)
| 31.15.14.1 | $x^{15} + 31$ | $15$ | $1$ | $14$ | $C_{15}$ | $[\ ]_{15}$ |