Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 2*x^29 - 13*x^28 + 22*x^27 + 174*x^26 - 256*x^25 - 157*x^24 + 458*x^23 + 5658*x^22 - 6464*x^21 + 45434*x^20 - 23206*x^19 + 322087*x^18 - 245272*x^17 + 2223136*x^16 - 1463962*x^15 + 13159287*x^14 - 5855334*x^13 + 62967354*x^12 - 19483620*x^11 + 244339329*x^10 - 69901890*x^9 + 742997779*x^8 - 222989416*x^7 + 1686428990*x^6 - 498857406*x^5 + 2661692327*x^4 - 653724964*x^3 + 2600745170*x^2 - 385672188*x + 1205881601)
gp: K = bnfinit(y^30 - 2*y^29 - 13*y^28 + 22*y^27 + 174*y^26 - 256*y^25 - 157*y^24 + 458*y^23 + 5658*y^22 - 6464*y^21 + 45434*y^20 - 23206*y^19 + 322087*y^18 - 245272*y^17 + 2223136*y^16 - 1463962*y^15 + 13159287*y^14 - 5855334*y^13 + 62967354*y^12 - 19483620*y^11 + 244339329*y^10 - 69901890*y^9 + 742997779*y^8 - 222989416*y^7 + 1686428990*y^6 - 498857406*y^5 + 2661692327*y^4 - 653724964*y^3 + 2600745170*y^2 - 385672188*y + 1205881601, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 2*x^29 - 13*x^28 + 22*x^27 + 174*x^26 - 256*x^25 - 157*x^24 + 458*x^23 + 5658*x^22 - 6464*x^21 + 45434*x^20 - 23206*x^19 + 322087*x^18 - 245272*x^17 + 2223136*x^16 - 1463962*x^15 + 13159287*x^14 - 5855334*x^13 + 62967354*x^12 - 19483620*x^11 + 244339329*x^10 - 69901890*x^9 + 742997779*x^8 - 222989416*x^7 + 1686428990*x^6 - 498857406*x^5 + 2661692327*x^4 - 653724964*x^3 + 2600745170*x^2 - 385672188*x + 1205881601);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 2*x^29 - 13*x^28 + 22*x^27 + 174*x^26 - 256*x^25 - 157*x^24 + 458*x^23 + 5658*x^22 - 6464*x^21 + 45434*x^20 - 23206*x^19 + 322087*x^18 - 245272*x^17 + 2223136*x^16 - 1463962*x^15 + 13159287*x^14 - 5855334*x^13 + 62967354*x^12 - 19483620*x^11 + 244339329*x^10 - 69901890*x^9 + 742997779*x^8 - 222989416*x^7 + 1686428990*x^6 - 498857406*x^5 + 2661692327*x^4 - 653724964*x^3 + 2600745170*x^2 - 385672188*x + 1205881601)
\( x^{30} - 2 x^{29} - 13 x^{28} + 22 x^{27} + 174 x^{26} - 256 x^{25} - 157 x^{24} + 458 x^{23} + \cdots + 1205881601 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-27652541257338422096297668839356545284021085927702003712\)
\(\medspace = -\,2^{45}\cdot 7^{20}\cdot 11^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(70.48\) | 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: | $2^{3/2}7^{2/3}11^{4/5}\approx 70.47869191420928$ | ||
| Ramified primes: |
\(2\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $30$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(616=2^{3}\cdot 7\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{616}(1,·)$, $\chi_{616}(603,·)$, $\chi_{616}(515,·)$, $\chi_{616}(449,·)$, $\chi_{616}(9,·)$, $\chi_{616}(267,·)$, $\chi_{616}(499,·)$, $\chi_{616}(401,·)$, $\chi_{616}(323,·)$, $\chi_{616}(529,·)$, $\chi_{616}(345,·)$, $\chi_{616}(225,·)$, $\chi_{616}(25,·)$, $\chi_{616}(331,·)$, $\chi_{616}(155,·)$, $\chi_{616}(361,·)$, $\chi_{616}(289,·)$, $\chi_{616}(67,·)$, $\chi_{616}(291,·)$, $\chi_{616}(113,·)$, $\chi_{616}(163,·)$, $\chi_{616}(81,·)$, $\chi_{616}(379,·)$, $\chi_{616}(169,·)$, $\chi_{616}(555,·)$, $\chi_{616}(177,·)$, $\chi_{616}(179,·)$, $\chi_{616}(235,·)$, $\chi_{616}(137,·)$, $\chi_{616}(443,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{16384}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{43}a^{25}-\frac{3}{43}a^{24}-\frac{14}{43}a^{23}-\frac{11}{43}a^{22}+\frac{3}{43}a^{21}-\frac{8}{43}a^{20}-\frac{21}{43}a^{19}-\frac{8}{43}a^{18}-\frac{18}{43}a^{17}-\frac{8}{43}a^{16}-\frac{4}{43}a^{15}+\frac{17}{43}a^{14}-\frac{6}{43}a^{13}-\frac{20}{43}a^{12}-\frac{3}{43}a^{11}-\frac{6}{43}a^{10}-\frac{10}{43}a^{9}+\frac{20}{43}a^{8}-\frac{13}{43}a^{7}-\frac{14}{43}a^{6}-\frac{14}{43}a^{5}-\frac{18}{43}a^{4}-\frac{20}{43}a^{3}-\frac{16}{43}a^{2}-\frac{10}{43}a-\frac{11}{43}$, $\frac{1}{43}a^{26}+\frac{20}{43}a^{24}-\frac{10}{43}a^{23}+\frac{13}{43}a^{22}+\frac{1}{43}a^{21}-\frac{2}{43}a^{20}+\frac{15}{43}a^{19}+\frac{1}{43}a^{18}-\frac{19}{43}a^{17}+\frac{15}{43}a^{16}+\frac{5}{43}a^{15}+\frac{2}{43}a^{14}+\frac{5}{43}a^{13}-\frac{20}{43}a^{12}-\frac{15}{43}a^{11}+\frac{15}{43}a^{10}-\frac{10}{43}a^{9}+\frac{4}{43}a^{8}-\frac{10}{43}a^{7}-\frac{13}{43}a^{6}-\frac{17}{43}a^{5}+\frac{12}{43}a^{4}+\frac{10}{43}a^{3}-\frac{15}{43}a^{2}+\frac{2}{43}a+\frac{10}{43}$, $\frac{1}{43}a^{27}+\frac{7}{43}a^{24}-\frac{8}{43}a^{23}+\frac{6}{43}a^{22}-\frac{19}{43}a^{21}+\frac{3}{43}a^{20}-\frac{9}{43}a^{19}+\frac{12}{43}a^{18}-\frac{12}{43}a^{17}-\frac{7}{43}a^{16}-\frac{4}{43}a^{15}+\frac{9}{43}a^{14}+\frac{14}{43}a^{13}-\frac{2}{43}a^{12}-\frac{11}{43}a^{11}-\frac{19}{43}a^{10}-\frac{11}{43}a^{9}+\frac{20}{43}a^{8}-\frac{11}{43}a^{7}+\frac{5}{43}a^{6}-\frac{9}{43}a^{5}-\frac{17}{43}a^{4}-\frac{2}{43}a^{3}+\frac{21}{43}a^{2}-\frac{5}{43}a+\frac{5}{43}$, $\frac{1}{180175181801}a^{28}-\frac{157019561}{180175181801}a^{27}+\frac{1985005653}{180175181801}a^{26}-\frac{305886139}{180175181801}a^{25}+\frac{62287195924}{180175181801}a^{24}+\frac{3072955712}{180175181801}a^{23}+\frac{74487560585}{180175181801}a^{22}-\frac{36133755933}{180175181801}a^{21}-\frac{77269191945}{180175181801}a^{20}-\frac{32517971040}{180175181801}a^{19}-\frac{15319936459}{180175181801}a^{18}+\frac{28736711641}{180175181801}a^{17}+\frac{33810517844}{180175181801}a^{16}+\frac{43431815719}{180175181801}a^{15}-\frac{15766427892}{180175181801}a^{14}+\frac{66619162112}{180175181801}a^{13}+\frac{63466494868}{180175181801}a^{12}-\frac{25807278881}{180175181801}a^{11}-\frac{88274015730}{180175181801}a^{10}+\frac{83936878599}{180175181801}a^{9}-\frac{6209520978}{180175181801}a^{8}+\frac{33736557218}{180175181801}a^{7}+\frac{74998442536}{180175181801}a^{6}-\frac{28520140898}{180175181801}a^{5}+\frac{31005092690}{180175181801}a^{4}+\frac{53048364337}{180175181801}a^{3}-\frac{30404652651}{180175181801}a^{2}-\frac{71579929061}{180175181801}a+\frac{46276336215}{180175181801}$, $\frac{1}{12\!\cdots\!99}a^{29}+\frac{19\!\cdots\!63}{12\!\cdots\!99}a^{28}+\frac{68\!\cdots\!59}{12\!\cdots\!99}a^{27}-\frac{10\!\cdots\!77}{12\!\cdots\!99}a^{26}+\frac{65\!\cdots\!86}{12\!\cdots\!99}a^{25}-\frac{12\!\cdots\!58}{12\!\cdots\!99}a^{24}+\frac{58\!\cdots\!95}{12\!\cdots\!99}a^{23}+\frac{48\!\cdots\!03}{12\!\cdots\!99}a^{22}+\frac{40\!\cdots\!37}{12\!\cdots\!99}a^{21}-\frac{44\!\cdots\!57}{12\!\cdots\!99}a^{20}+\frac{56\!\cdots\!41}{12\!\cdots\!99}a^{19}-\frac{44\!\cdots\!58}{12\!\cdots\!99}a^{18}-\frac{10\!\cdots\!46}{28\!\cdots\!93}a^{17}+\frac{21\!\cdots\!77}{12\!\cdots\!99}a^{16}+\frac{54\!\cdots\!37}{12\!\cdots\!99}a^{15}+\frac{28\!\cdots\!51}{12\!\cdots\!99}a^{14}-\frac{33\!\cdots\!27}{12\!\cdots\!99}a^{13}+\frac{53\!\cdots\!34}{12\!\cdots\!99}a^{12}-\frac{33\!\cdots\!65}{12\!\cdots\!99}a^{11}-\frac{34\!\cdots\!38}{12\!\cdots\!99}a^{10}+\frac{25\!\cdots\!65}{12\!\cdots\!99}a^{9}-\frac{33\!\cdots\!75}{12\!\cdots\!99}a^{8}-\frac{14\!\cdots\!98}{12\!\cdots\!99}a^{7}+\frac{28\!\cdots\!04}{12\!\cdots\!99}a^{6}+\frac{32\!\cdots\!03}{12\!\cdots\!99}a^{5}-\frac{43\!\cdots\!54}{12\!\cdots\!99}a^{4}+\frac{53\!\cdots\!58}{12\!\cdots\!99}a^{3}+\frac{25\!\cdots\!84}{12\!\cdots\!99}a^{2}-\frac{38\!\cdots\!42}{12\!\cdots\!99}a-\frac{45\!\cdots\!13}{12\!\cdots\!99}$
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
$C_{2}\times C_{2}\times C_{2}\times C_{63010}$, which has order $504080$ (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: |
$\frac{25\!\cdots\!44}{22\!\cdots\!97}a^{29}-\frac{54\!\cdots\!18}{22\!\cdots\!97}a^{28}-\frac{38\!\cdots\!72}{22\!\cdots\!97}a^{27}+\frac{73\!\cdots\!04}{22\!\cdots\!97}a^{26}+\frac{51\!\cdots\!70}{22\!\cdots\!97}a^{25}-\frac{87\!\cdots\!16}{22\!\cdots\!97}a^{24}-\frac{13\!\cdots\!04}{22\!\cdots\!97}a^{23}+\frac{32\!\cdots\!83}{22\!\cdots\!97}a^{22}+\frac{16\!\cdots\!22}{22\!\cdots\!97}a^{21}-\frac{24\!\cdots\!42}{22\!\cdots\!97}a^{20}+\frac{89\!\cdots\!14}{22\!\cdots\!97}a^{19}-\frac{18\!\cdots\!12}{22\!\cdots\!97}a^{18}+\frac{61\!\cdots\!20}{22\!\cdots\!97}a^{17}-\frac{60\!\cdots\!01}{22\!\cdots\!97}a^{16}+\frac{44\!\cdots\!84}{22\!\cdots\!97}a^{15}-\frac{28\!\cdots\!06}{22\!\cdots\!97}a^{14}+\frac{24\!\cdots\!00}{22\!\cdots\!97}a^{13}-\frac{96\!\cdots\!10}{22\!\cdots\!97}a^{12}+\frac{10\!\cdots\!86}{22\!\cdots\!97}a^{11}-\frac{31\!\cdots\!96}{22\!\cdots\!97}a^{10}+\frac{38\!\cdots\!76}{22\!\cdots\!97}a^{9}-\frac{11\!\cdots\!98}{22\!\cdots\!97}a^{8}+\frac{10\!\cdots\!42}{22\!\cdots\!97}a^{7}-\frac{29\!\cdots\!41}{22\!\cdots\!97}a^{6}+\frac{18\!\cdots\!38}{22\!\cdots\!97}a^{5}-\frac{39\!\cdots\!48}{22\!\cdots\!97}a^{4}+\frac{20\!\cdots\!42}{22\!\cdots\!97}a^{3}-\frac{68\!\cdots\!61}{22\!\cdots\!97}a^{2}+\frac{98\!\cdots\!96}{22\!\cdots\!97}a+\frac{67\!\cdots\!94}{53\!\cdots\!79}$, $\frac{11\!\cdots\!68}{22\!\cdots\!97}a^{29}-\frac{53\!\cdots\!28}{22\!\cdots\!97}a^{28}-\frac{97\!\cdots\!58}{22\!\cdots\!97}a^{27}+\frac{12\!\cdots\!77}{22\!\cdots\!97}a^{26}+\frac{33\!\cdots\!50}{22\!\cdots\!97}a^{25}-\frac{18\!\cdots\!57}{22\!\cdots\!97}a^{24}+\frac{20\!\cdots\!54}{22\!\cdots\!97}a^{23}+\frac{15\!\cdots\!80}{22\!\cdots\!97}a^{22}-\frac{12\!\cdots\!38}{22\!\cdots\!97}a^{21}-\frac{81\!\cdots\!47}{22\!\cdots\!97}a^{20}+\frac{14\!\cdots\!90}{22\!\cdots\!97}a^{19}+\frac{33\!\cdots\!70}{22\!\cdots\!97}a^{18}-\frac{50\!\cdots\!48}{22\!\cdots\!97}a^{17}+\frac{52\!\cdots\!60}{22\!\cdots\!97}a^{16}+\frac{36\!\cdots\!90}{22\!\cdots\!97}a^{15}+\frac{52\!\cdots\!13}{22\!\cdots\!97}a^{14}+\frac{82\!\cdots\!40}{22\!\cdots\!97}a^{13}+\frac{39\!\cdots\!76}{22\!\cdots\!97}a^{12}+\frac{56\!\cdots\!44}{22\!\cdots\!97}a^{11}+\frac{25\!\cdots\!46}{22\!\cdots\!97}a^{10}+\frac{24\!\cdots\!62}{22\!\cdots\!97}a^{9}+\frac{11\!\cdots\!96}{22\!\cdots\!97}a^{8}+\frac{76\!\cdots\!78}{22\!\cdots\!97}a^{7}+\frac{41\!\cdots\!55}{22\!\cdots\!97}a^{6}+\frac{11\!\cdots\!16}{22\!\cdots\!97}a^{5}+\frac{10\!\cdots\!25}{22\!\cdots\!97}a^{4}+\frac{31\!\cdots\!62}{22\!\cdots\!97}a^{3}+\frac{16\!\cdots\!20}{22\!\cdots\!97}a^{2}-\frac{11\!\cdots\!26}{22\!\cdots\!97}a+\frac{28\!\cdots\!26}{53\!\cdots\!79}$, $\frac{58\!\cdots\!28}{53\!\cdots\!31}a^{29}-\frac{11\!\cdots\!19}{53\!\cdots\!31}a^{28}-\frac{87\!\cdots\!32}{53\!\cdots\!31}a^{27}+\frac{16\!\cdots\!84}{53\!\cdots\!31}a^{26}+\frac{11\!\cdots\!60}{53\!\cdots\!31}a^{25}-\frac{46\!\cdots\!91}{12\!\cdots\!17}a^{24}-\frac{26\!\cdots\!22}{53\!\cdots\!31}a^{23}+\frac{23\!\cdots\!43}{12\!\cdots\!17}a^{22}+\frac{32\!\cdots\!20}{53\!\cdots\!31}a^{21}-\frac{62\!\cdots\!06}{53\!\cdots\!31}a^{20}+\frac{22\!\cdots\!64}{53\!\cdots\!31}a^{19}+\frac{10\!\cdots\!27}{53\!\cdots\!31}a^{18}+\frac{13\!\cdots\!22}{53\!\cdots\!31}a^{17}-\frac{20\!\cdots\!44}{12\!\cdots\!17}a^{16}+\frac{10\!\cdots\!74}{53\!\cdots\!31}a^{15}-\frac{19\!\cdots\!60}{53\!\cdots\!31}a^{14}+\frac{53\!\cdots\!56}{53\!\cdots\!31}a^{13}+\frac{51\!\cdots\!01}{53\!\cdots\!31}a^{12}+\frac{24\!\cdots\!32}{53\!\cdots\!31}a^{11}+\frac{68\!\cdots\!38}{53\!\cdots\!31}a^{10}+\frac{88\!\cdots\!30}{53\!\cdots\!31}a^{9}+\frac{31\!\cdots\!24}{53\!\cdots\!31}a^{8}+\frac{23\!\cdots\!24}{53\!\cdots\!31}a^{7}+\frac{10\!\cdots\!36}{53\!\cdots\!31}a^{6}+\frac{42\!\cdots\!22}{53\!\cdots\!31}a^{5}+\frac{30\!\cdots\!17}{53\!\cdots\!31}a^{4}+\frac{43\!\cdots\!72}{53\!\cdots\!31}a^{3}+\frac{53\!\cdots\!61}{53\!\cdots\!31}a^{2}+\frac{16\!\cdots\!16}{53\!\cdots\!31}a+\frac{45\!\cdots\!30}{53\!\cdots\!31}$, $\frac{39\!\cdots\!10}{53\!\cdots\!31}a^{29}-\frac{85\!\cdots\!30}{53\!\cdots\!31}a^{28}-\frac{58\!\cdots\!40}{53\!\cdots\!31}a^{27}+\frac{12\!\cdots\!48}{53\!\cdots\!31}a^{26}+\frac{77\!\cdots\!88}{53\!\cdots\!31}a^{25}-\frac{14\!\cdots\!29}{53\!\cdots\!31}a^{24}-\frac{18\!\cdots\!58}{53\!\cdots\!31}a^{23}+\frac{63\!\cdots\!15}{53\!\cdots\!31}a^{22}+\frac{22\!\cdots\!20}{53\!\cdots\!31}a^{21}-\frac{43\!\cdots\!06}{53\!\cdots\!31}a^{20}+\frac{14\!\cdots\!50}{53\!\cdots\!31}a^{19}+\frac{92\!\cdots\!59}{53\!\cdots\!31}a^{18}+\frac{89\!\cdots\!66}{53\!\cdots\!31}a^{17}-\frac{85\!\cdots\!28}{53\!\cdots\!31}a^{16}+\frac{69\!\cdots\!78}{53\!\cdots\!31}a^{15}-\frac{34\!\cdots\!53}{53\!\cdots\!31}a^{14}+\frac{36\!\cdots\!08}{53\!\cdots\!31}a^{13}-\frac{21\!\cdots\!69}{12\!\cdots\!17}a^{12}+\frac{16\!\cdots\!72}{53\!\cdots\!31}a^{11}-\frac{15\!\cdots\!90}{53\!\cdots\!31}a^{10}+\frac{58\!\cdots\!42}{53\!\cdots\!31}a^{9}-\frac{26\!\cdots\!78}{53\!\cdots\!31}a^{8}+\frac{15\!\cdots\!94}{53\!\cdots\!31}a^{7}+\frac{50\!\cdots\!12}{53\!\cdots\!31}a^{6}+\frac{65\!\cdots\!40}{12\!\cdots\!17}a^{5}+\frac{63\!\cdots\!69}{53\!\cdots\!31}a^{4}+\frac{29\!\cdots\!00}{53\!\cdots\!31}a^{3}+\frac{18\!\cdots\!85}{53\!\cdots\!31}a^{2}+\frac{12\!\cdots\!30}{53\!\cdots\!31}a+\frac{19\!\cdots\!82}{53\!\cdots\!31}$, $\frac{79\!\cdots\!64}{53\!\cdots\!31}a^{29}-\frac{17\!\cdots\!89}{53\!\cdots\!31}a^{28}-\frac{11\!\cdots\!80}{53\!\cdots\!31}a^{27}+\frac{23\!\cdots\!28}{53\!\cdots\!31}a^{26}+\frac{15\!\cdots\!40}{53\!\cdots\!31}a^{25}-\frac{28\!\cdots\!84}{53\!\cdots\!31}a^{24}-\frac{41\!\cdots\!56}{53\!\cdots\!31}a^{23}+\frac{11\!\cdots\!64}{53\!\cdots\!31}a^{22}+\frac{48\!\cdots\!08}{53\!\cdots\!31}a^{21}-\frac{81\!\cdots\!80}{53\!\cdots\!31}a^{20}+\frac{27\!\cdots\!04}{53\!\cdots\!31}a^{19}-\frac{30\!\cdots\!44}{53\!\cdots\!31}a^{18}+\frac{18\!\cdots\!20}{53\!\cdots\!31}a^{17}-\frac{18\!\cdots\!44}{53\!\cdots\!31}a^{16}+\frac{13\!\cdots\!72}{53\!\cdots\!31}a^{15}-\frac{82\!\cdots\!38}{53\!\cdots\!31}a^{14}+\frac{72\!\cdots\!32}{53\!\cdots\!31}a^{13}-\frac{27\!\cdots\!16}{53\!\cdots\!31}a^{12}+\frac{32\!\cdots\!40}{53\!\cdots\!31}a^{11}-\frac{81\!\cdots\!64}{53\!\cdots\!31}a^{10}+\frac{26\!\cdots\!36}{12\!\cdots\!17}a^{9}-\frac{27\!\cdots\!24}{53\!\cdots\!31}a^{8}+\frac{30\!\cdots\!28}{53\!\cdots\!31}a^{7}-\frac{61\!\cdots\!52}{53\!\cdots\!31}a^{6}+\frac{54\!\cdots\!56}{53\!\cdots\!31}a^{5}-\frac{45\!\cdots\!12}{53\!\cdots\!31}a^{4}+\frac{57\!\cdots\!52}{53\!\cdots\!31}a^{3}+\frac{10\!\cdots\!20}{53\!\cdots\!31}a^{2}+\frac{24\!\cdots\!52}{53\!\cdots\!31}a+\frac{19\!\cdots\!52}{53\!\cdots\!31}$, $\frac{39\!\cdots\!88}{53\!\cdots\!31}a^{29}-\frac{80\!\cdots\!50}{53\!\cdots\!31}a^{28}-\frac{58\!\cdots\!84}{53\!\cdots\!31}a^{27}+\frac{10\!\cdots\!90}{53\!\cdots\!31}a^{26}+\frac{78\!\cdots\!18}{53\!\cdots\!31}a^{25}-\frac{12\!\cdots\!74}{53\!\cdots\!31}a^{24}-\frac{20\!\cdots\!66}{53\!\cdots\!31}a^{23}+\frac{47\!\cdots\!29}{53\!\cdots\!31}a^{22}+\frac{24\!\cdots\!56}{53\!\cdots\!31}a^{21}-\frac{35\!\cdots\!18}{53\!\cdots\!31}a^{20}+\frac{13\!\cdots\!52}{53\!\cdots\!31}a^{19}-\frac{20\!\cdots\!74}{53\!\cdots\!31}a^{18}+\frac{92\!\cdots\!44}{53\!\cdots\!31}a^{17}-\frac{86\!\cdots\!58}{53\!\cdots\!31}a^{16}+\frac{67\!\cdots\!70}{53\!\cdots\!31}a^{15}-\frac{39\!\cdots\!05}{53\!\cdots\!31}a^{14}+\frac{36\!\cdots\!70}{53\!\cdots\!31}a^{13}-\frac{12\!\cdots\!87}{53\!\cdots\!31}a^{12}+\frac{16\!\cdots\!96}{53\!\cdots\!31}a^{11}-\frac{39\!\cdots\!82}{53\!\cdots\!31}a^{10}+\frac{57\!\cdots\!36}{53\!\cdots\!31}a^{9}-\frac{14\!\cdots\!01}{53\!\cdots\!31}a^{8}+\frac{15\!\cdots\!82}{53\!\cdots\!31}a^{7}-\frac{38\!\cdots\!00}{53\!\cdots\!31}a^{6}+\frac{27\!\cdots\!24}{53\!\cdots\!31}a^{5}-\frac{52\!\cdots\!56}{53\!\cdots\!31}a^{4}+\frac{30\!\cdots\!10}{53\!\cdots\!31}a^{3}-\frac{39\!\cdots\!02}{53\!\cdots\!31}a^{2}+\frac{13\!\cdots\!50}{53\!\cdots\!31}a+\frac{47\!\cdots\!15}{53\!\cdots\!31}$, $\frac{95\!\cdots\!02}{12\!\cdots\!99}a^{29}-\frac{22\!\cdots\!66}{12\!\cdots\!99}a^{28}-\frac{14\!\cdots\!56}{12\!\cdots\!99}a^{27}+\frac{28\!\cdots\!52}{12\!\cdots\!99}a^{26}+\frac{20\!\cdots\!58}{12\!\cdots\!99}a^{25}-\frac{32\!\cdots\!90}{12\!\cdots\!99}a^{24}-\frac{63\!\cdots\!72}{12\!\cdots\!99}a^{23}+\frac{59\!\cdots\!61}{12\!\cdots\!99}a^{22}+\frac{71\!\cdots\!46}{12\!\cdots\!99}a^{21}-\frac{74\!\cdots\!86}{12\!\cdots\!99}a^{20}+\frac{27\!\cdots\!48}{12\!\cdots\!99}a^{19}-\frac{42\!\cdots\!74}{12\!\cdots\!99}a^{18}+\frac{56\!\cdots\!76}{28\!\cdots\!93}a^{17}-\frac{81\!\cdots\!41}{28\!\cdots\!93}a^{16}+\frac{15\!\cdots\!08}{12\!\cdots\!99}a^{15}-\frac{21\!\cdots\!27}{12\!\cdots\!99}a^{14}+\frac{89\!\cdots\!72}{12\!\cdots\!99}a^{13}-\frac{10\!\cdots\!20}{12\!\cdots\!99}a^{12}+\frac{37\!\cdots\!06}{12\!\cdots\!99}a^{11}-\frac{47\!\cdots\!80}{12\!\cdots\!99}a^{10}+\frac{12\!\cdots\!96}{12\!\cdots\!99}a^{9}-\frac{18\!\cdots\!96}{12\!\cdots\!99}a^{8}+\frac{77\!\cdots\!28}{28\!\cdots\!93}a^{7}-\frac{55\!\cdots\!47}{12\!\cdots\!99}a^{6}+\frac{63\!\cdots\!52}{12\!\cdots\!99}a^{5}-\frac{11\!\cdots\!58}{12\!\cdots\!99}a^{4}+\frac{76\!\cdots\!34}{12\!\cdots\!99}a^{3}-\frac{14\!\cdots\!41}{12\!\cdots\!99}a^{2}+\frac{43\!\cdots\!06}{12\!\cdots\!99}a-\frac{90\!\cdots\!25}{12\!\cdots\!99}$, $\frac{18\!\cdots\!50}{12\!\cdots\!99}a^{29}-\frac{38\!\cdots\!22}{12\!\cdots\!99}a^{28}-\frac{27\!\cdots\!78}{12\!\cdots\!99}a^{27}+\frac{57\!\cdots\!43}{12\!\cdots\!99}a^{26}+\frac{35\!\cdots\!72}{12\!\cdots\!99}a^{25}-\frac{71\!\cdots\!73}{12\!\cdots\!99}a^{24}-\frac{67\!\cdots\!46}{12\!\cdots\!99}a^{23}+\frac{36\!\cdots\!21}{12\!\cdots\!99}a^{22}+\frac{93\!\cdots\!56}{12\!\cdots\!99}a^{21}-\frac{23\!\cdots\!91}{12\!\cdots\!99}a^{20}+\frac{77\!\cdots\!58}{12\!\cdots\!99}a^{19}+\frac{40\!\cdots\!56}{12\!\cdots\!99}a^{18}+\frac{41\!\cdots\!96}{12\!\cdots\!99}a^{17}-\frac{28\!\cdots\!51}{12\!\cdots\!99}a^{16}+\frac{33\!\cdots\!78}{12\!\cdots\!99}a^{15}-\frac{58\!\cdots\!06}{12\!\cdots\!99}a^{14}+\frac{17\!\cdots\!08}{12\!\cdots\!99}a^{13}+\frac{21\!\cdots\!72}{12\!\cdots\!99}a^{12}+\frac{80\!\cdots\!66}{12\!\cdots\!99}a^{11}+\frac{27\!\cdots\!98}{12\!\cdots\!99}a^{10}+\frac{29\!\cdots\!42}{12\!\cdots\!99}a^{9}+\frac{13\!\cdots\!96}{12\!\cdots\!99}a^{8}+\frac{79\!\cdots\!50}{12\!\cdots\!99}a^{7}+\frac{46\!\cdots\!38}{12\!\cdots\!99}a^{6}+\frac{14\!\cdots\!12}{12\!\cdots\!99}a^{5}+\frac{12\!\cdots\!01}{12\!\cdots\!99}a^{4}+\frac{14\!\cdots\!48}{12\!\cdots\!99}a^{3}+\frac{22\!\cdots\!07}{12\!\cdots\!99}a^{2}+\frac{55\!\cdots\!16}{12\!\cdots\!99}a+\frac{18\!\cdots\!58}{12\!\cdots\!99}$, $\frac{17\!\cdots\!00}{12\!\cdots\!99}a^{29}-\frac{36\!\cdots\!05}{12\!\cdots\!99}a^{28}-\frac{26\!\cdots\!34}{12\!\cdots\!99}a^{27}+\frac{47\!\cdots\!53}{12\!\cdots\!99}a^{26}+\frac{36\!\cdots\!10}{12\!\cdots\!99}a^{25}-\frac{55\!\cdots\!17}{12\!\cdots\!99}a^{24}-\frac{10\!\cdots\!42}{12\!\cdots\!99}a^{23}+\frac{17\!\cdots\!96}{12\!\cdots\!99}a^{22}+\frac{11\!\cdots\!78}{12\!\cdots\!99}a^{21}-\frac{14\!\cdots\!71}{12\!\cdots\!99}a^{20}+\frac{55\!\cdots\!86}{12\!\cdots\!99}a^{19}-\frac{24\!\cdots\!66}{12\!\cdots\!99}a^{18}+\frac{41\!\cdots\!96}{12\!\cdots\!99}a^{17}-\frac{42\!\cdots\!96}{12\!\cdots\!99}a^{16}+\frac{29\!\cdots\!58}{12\!\cdots\!99}a^{15}-\frac{21\!\cdots\!73}{12\!\cdots\!99}a^{14}+\frac{16\!\cdots\!48}{12\!\cdots\!99}a^{13}-\frac{82\!\cdots\!56}{12\!\cdots\!99}a^{12}+\frac{70\!\cdots\!12}{12\!\cdots\!99}a^{11}-\frac{32\!\cdots\!38}{12\!\cdots\!99}a^{10}+\frac{24\!\cdots\!38}{12\!\cdots\!99}a^{9}-\frac{12\!\cdots\!28}{12\!\cdots\!99}a^{8}+\frac{64\!\cdots\!86}{12\!\cdots\!99}a^{7}-\frac{36\!\cdots\!93}{12\!\cdots\!99}a^{6}+\frac{11\!\cdots\!52}{12\!\cdots\!99}a^{5}-\frac{65\!\cdots\!23}{12\!\cdots\!99}a^{4}+\frac{13\!\cdots\!54}{12\!\cdots\!99}a^{3}-\frac{63\!\cdots\!60}{12\!\cdots\!99}a^{2}+\frac{62\!\cdots\!50}{12\!\cdots\!99}a-\frac{21\!\cdots\!97}{12\!\cdots\!99}$, $\frac{13\!\cdots\!86}{12\!\cdots\!99}a^{29}-\frac{30\!\cdots\!26}{12\!\cdots\!99}a^{28}-\frac{20\!\cdots\!48}{12\!\cdots\!99}a^{27}+\frac{42\!\cdots\!50}{12\!\cdots\!99}a^{26}+\frac{27\!\cdots\!20}{12\!\cdots\!99}a^{25}-\frac{51\!\cdots\!67}{12\!\cdots\!99}a^{24}-\frac{15\!\cdots\!52}{28\!\cdots\!93}a^{23}+\frac{20\!\cdots\!55}{12\!\cdots\!99}a^{22}+\frac{82\!\cdots\!30}{12\!\cdots\!99}a^{21}-\frac{14\!\cdots\!66}{12\!\cdots\!99}a^{20}+\frac{50\!\cdots\!80}{12\!\cdots\!99}a^{19}-\frac{30\!\cdots\!47}{12\!\cdots\!99}a^{18}+\frac{31\!\cdots\!78}{12\!\cdots\!99}a^{17}-\frac{32\!\cdots\!97}{12\!\cdots\!99}a^{16}+\frac{24\!\cdots\!60}{12\!\cdots\!99}a^{15}-\frac{13\!\cdots\!94}{12\!\cdots\!99}a^{14}+\frac{12\!\cdots\!02}{12\!\cdots\!99}a^{13}-\frac{43\!\cdots\!90}{12\!\cdots\!99}a^{12}+\frac{13\!\cdots\!62}{28\!\cdots\!93}a^{11}-\frac{11\!\cdots\!64}{12\!\cdots\!99}a^{10}+\frac{20\!\cdots\!66}{12\!\cdots\!99}a^{9}-\frac{32\!\cdots\!99}{12\!\cdots\!99}a^{8}+\frac{55\!\cdots\!62}{12\!\cdots\!99}a^{7}-\frac{56\!\cdots\!99}{12\!\cdots\!99}a^{6}+\frac{10\!\cdots\!30}{12\!\cdots\!99}a^{5}+\frac{52\!\cdots\!09}{12\!\cdots\!99}a^{4}+\frac{11\!\cdots\!24}{12\!\cdots\!99}a^{3}+\frac{39\!\cdots\!36}{12\!\cdots\!99}a^{2}+\frac{49\!\cdots\!52}{12\!\cdots\!99}a+\frac{51\!\cdots\!56}{12\!\cdots\!99}$, $\frac{69\!\cdots\!94}{12\!\cdots\!99}a^{29}-\frac{39\!\cdots\!96}{12\!\cdots\!99}a^{28}-\frac{44\!\cdots\!10}{12\!\cdots\!99}a^{27}+\frac{97\!\cdots\!41}{12\!\cdots\!99}a^{26}-\frac{11\!\cdots\!20}{12\!\cdots\!99}a^{25}-\frac{14\!\cdots\!14}{12\!\cdots\!99}a^{24}+\frac{16\!\cdots\!50}{12\!\cdots\!99}a^{23}+\frac{27\!\cdots\!78}{28\!\cdots\!93}a^{22}-\frac{10\!\cdots\!90}{12\!\cdots\!99}a^{21}-\frac{62\!\cdots\!01}{12\!\cdots\!99}a^{20}+\frac{10\!\cdots\!72}{12\!\cdots\!99}a^{19}+\frac{24\!\cdots\!47}{12\!\cdots\!99}a^{18}-\frac{10\!\cdots\!78}{12\!\cdots\!99}a^{17}+\frac{82\!\cdots\!30}{28\!\cdots\!93}a^{16}+\frac{23\!\cdots\!62}{12\!\cdots\!99}a^{15}+\frac{39\!\cdots\!79}{12\!\cdots\!99}a^{14}+\frac{37\!\cdots\!82}{12\!\cdots\!99}a^{13}+\frac{29\!\cdots\!72}{12\!\cdots\!99}a^{12}+\frac{29\!\cdots\!52}{12\!\cdots\!99}a^{11}+\frac{19\!\cdots\!50}{12\!\cdots\!99}a^{10}+\frac{14\!\cdots\!28}{12\!\cdots\!99}a^{9}+\frac{20\!\cdots\!93}{28\!\cdots\!93}a^{8}+\frac{46\!\cdots\!74}{12\!\cdots\!99}a^{7}+\frac{32\!\cdots\!33}{12\!\cdots\!99}a^{6}+\frac{69\!\cdots\!56}{12\!\cdots\!99}a^{5}+\frac{81\!\cdots\!00}{12\!\cdots\!99}a^{4}+\frac{59\!\cdots\!64}{12\!\cdots\!99}a^{3}+\frac{13\!\cdots\!63}{12\!\cdots\!99}a^{2}-\frac{95\!\cdots\!22}{12\!\cdots\!99}a+\frac{10\!\cdots\!48}{12\!\cdots\!99}$, $\frac{51\!\cdots\!30}{12\!\cdots\!99}a^{29}-\frac{11\!\cdots\!04}{12\!\cdots\!99}a^{28}-\frac{77\!\cdots\!56}{12\!\cdots\!99}a^{27}+\frac{14\!\cdots\!05}{12\!\cdots\!99}a^{26}+\frac{10\!\cdots\!56}{12\!\cdots\!99}a^{25}-\frac{17\!\cdots\!27}{12\!\cdots\!99}a^{24}-\frac{28\!\cdots\!24}{12\!\cdots\!99}a^{23}+\frac{63\!\cdots\!73}{12\!\cdots\!99}a^{22}+\frac{32\!\cdots\!54}{12\!\cdots\!99}a^{21}-\frac{49\!\cdots\!05}{12\!\cdots\!99}a^{20}+\frac{17\!\cdots\!98}{12\!\cdots\!99}a^{19}-\frac{53\!\cdots\!56}{12\!\cdots\!99}a^{18}+\frac{12\!\cdots\!64}{12\!\cdots\!99}a^{17}-\frac{12\!\cdots\!06}{12\!\cdots\!99}a^{16}+\frac{87\!\cdots\!64}{12\!\cdots\!99}a^{15}-\frac{62\!\cdots\!56}{12\!\cdots\!99}a^{14}+\frac{10\!\cdots\!88}{28\!\cdots\!93}a^{13}-\frac{23\!\cdots\!46}{12\!\cdots\!99}a^{12}+\frac{20\!\cdots\!24}{12\!\cdots\!99}a^{11}-\frac{83\!\cdots\!22}{12\!\cdots\!99}a^{10}+\frac{73\!\cdots\!20}{12\!\cdots\!99}a^{9}-\frac{30\!\cdots\!42}{12\!\cdots\!99}a^{8}+\frac{19\!\cdots\!00}{12\!\cdots\!99}a^{7}-\frac{80\!\cdots\!42}{12\!\cdots\!99}a^{6}+\frac{35\!\cdots\!20}{12\!\cdots\!99}a^{5}-\frac{12\!\cdots\!60}{12\!\cdots\!99}a^{4}+\frac{92\!\cdots\!28}{28\!\cdots\!93}a^{3}-\frac{74\!\cdots\!90}{12\!\cdots\!99}a^{2}+\frac{18\!\cdots\!38}{12\!\cdots\!99}a+\frac{18\!\cdots\!99}{12\!\cdots\!99}$, $\frac{14\!\cdots\!68}{12\!\cdots\!99}a^{29}-\frac{28\!\cdots\!27}{12\!\cdots\!99}a^{28}-\frac{20\!\cdots\!14}{12\!\cdots\!99}a^{27}+\frac{44\!\cdots\!95}{12\!\cdots\!99}a^{26}+\frac{26\!\cdots\!90}{12\!\cdots\!99}a^{25}-\frac{56\!\cdots\!96}{12\!\cdots\!99}a^{24}-\frac{49\!\cdots\!20}{12\!\cdots\!99}a^{23}+\frac{30\!\cdots\!81}{12\!\cdots\!99}a^{22}+\frac{67\!\cdots\!34}{12\!\cdots\!99}a^{21}-\frac{18\!\cdots\!23}{12\!\cdots\!99}a^{20}+\frac{58\!\cdots\!86}{12\!\cdots\!99}a^{19}+\frac{41\!\cdots\!73}{12\!\cdots\!99}a^{18}+\frac{29\!\cdots\!22}{12\!\cdots\!99}a^{17}-\frac{19\!\cdots\!48}{12\!\cdots\!99}a^{16}+\frac{25\!\cdots\!76}{12\!\cdots\!99}a^{15}-\frac{17\!\cdots\!69}{12\!\cdots\!99}a^{14}+\frac{12\!\cdots\!04}{12\!\cdots\!99}a^{13}+\frac{33\!\cdots\!21}{12\!\cdots\!99}a^{12}+\frac{58\!\cdots\!76}{12\!\cdots\!99}a^{11}+\frac{29\!\cdots\!84}{12\!\cdots\!99}a^{10}+\frac{21\!\cdots\!24}{12\!\cdots\!99}a^{9}+\frac{13\!\cdots\!28}{12\!\cdots\!99}a^{8}+\frac{58\!\cdots\!22}{12\!\cdots\!99}a^{7}+\frac{47\!\cdots\!29}{12\!\cdots\!99}a^{6}+\frac{10\!\cdots\!10}{12\!\cdots\!99}a^{5}+\frac{12\!\cdots\!68}{12\!\cdots\!99}a^{4}+\frac{10\!\cdots\!42}{12\!\cdots\!99}a^{3}+\frac{21\!\cdots\!09}{12\!\cdots\!99}a^{2}+\frac{31\!\cdots\!22}{12\!\cdots\!99}a+\frac{17\!\cdots\!76}{12\!\cdots\!99}$, $\frac{43\!\cdots\!08}{12\!\cdots\!99}a^{29}-\frac{98\!\cdots\!75}{12\!\cdots\!99}a^{28}-\frac{65\!\cdots\!96}{12\!\cdots\!99}a^{27}+\frac{14\!\cdots\!44}{12\!\cdots\!99}a^{26}+\frac{87\!\cdots\!70}{12\!\cdots\!99}a^{25}-\frac{18\!\cdots\!64}{12\!\cdots\!99}a^{24}-\frac{23\!\cdots\!56}{12\!\cdots\!99}a^{23}+\frac{84\!\cdots\!95}{12\!\cdots\!99}a^{22}+\frac{25\!\cdots\!58}{12\!\cdots\!99}a^{21}-\frac{56\!\cdots\!06}{12\!\cdots\!99}a^{20}+\frac{15\!\cdots\!78}{12\!\cdots\!99}a^{19}+\frac{29\!\cdots\!28}{12\!\cdots\!99}a^{18}+\frac{88\!\cdots\!40}{12\!\cdots\!99}a^{17}-\frac{10\!\cdots\!09}{12\!\cdots\!99}a^{16}+\frac{71\!\cdots\!60}{12\!\cdots\!99}a^{15}-\frac{37\!\cdots\!00}{12\!\cdots\!99}a^{14}+\frac{35\!\cdots\!28}{12\!\cdots\!99}a^{13}-\frac{98\!\cdots\!94}{12\!\cdots\!99}a^{12}+\frac{15\!\cdots\!98}{12\!\cdots\!99}a^{11}-\frac{17\!\cdots\!24}{12\!\cdots\!99}a^{10}+\frac{54\!\cdots\!00}{12\!\cdots\!99}a^{9}-\frac{22\!\cdots\!30}{12\!\cdots\!99}a^{8}+\frac{14\!\cdots\!98}{12\!\cdots\!99}a^{7}+\frac{14\!\cdots\!39}{12\!\cdots\!99}a^{6}+\frac{23\!\cdots\!78}{12\!\cdots\!99}a^{5}+\frac{10\!\cdots\!68}{12\!\cdots\!99}a^{4}+\frac{20\!\cdots\!94}{12\!\cdots\!99}a^{3}+\frac{28\!\cdots\!67}{12\!\cdots\!99}a^{2}+\frac{37\!\cdots\!76}{12\!\cdots\!99}a+\frac{28\!\cdots\!95}{12\!\cdots\!99}$
| 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: | \( 4697581952.048968 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 4697581952.048968 \cdot 504080}{2\cdot\sqrt{27652541257338422096297668839356545284021085927702003712}}\cr\approx \mathstrut & 0.211433762006357 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 - 2*x^29 - 13*x^28 + 22*x^27 + 174*x^26 - 256*x^25 - 157*x^24 + 458*x^23 + 5658*x^22 - 6464*x^21 + 45434*x^20 - 23206*x^19 + 322087*x^18 - 245272*x^17 + 2223136*x^16 - 1463962*x^15 + 13159287*x^14 - 5855334*x^13 + 62967354*x^12 - 19483620*x^11 + 244339329*x^10 - 69901890*x^9 + 742997779*x^8 - 222989416*x^7 + 1686428990*x^6 - 498857406*x^5 + 2661692327*x^4 - 653724964*x^3 + 2600745170*x^2 - 385672188*x + 1205881601)
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^30 - 2*x^29 - 13*x^28 + 22*x^27 + 174*x^26 - 256*x^25 - 157*x^24 + 458*x^23 + 5658*x^22 - 6464*x^21 + 45434*x^20 - 23206*x^19 + 322087*x^18 - 245272*x^17 + 2223136*x^16 - 1463962*x^15 + 13159287*x^14 - 5855334*x^13 + 62967354*x^12 - 19483620*x^11 + 244339329*x^10 - 69901890*x^9 + 742997779*x^8 - 222989416*x^7 + 1686428990*x^6 - 498857406*x^5 + 2661692327*x^4 - 653724964*x^3 + 2600745170*x^2 - 385672188*x + 1205881601, 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^30 - 2*x^29 - 13*x^28 + 22*x^27 + 174*x^26 - 256*x^25 - 157*x^24 + 458*x^23 + 5658*x^22 - 6464*x^21 + 45434*x^20 - 23206*x^19 + 322087*x^18 - 245272*x^17 + 2223136*x^16 - 1463962*x^15 + 13159287*x^14 - 5855334*x^13 + 62967354*x^12 - 19483620*x^11 + 244339329*x^10 - 69901890*x^9 + 742997779*x^8 - 222989416*x^7 + 1686428990*x^6 - 498857406*x^5 + 2661692327*x^4 - 653724964*x^3 + 2600745170*x^2 - 385672188*x + 1205881601);
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^30 - 2*x^29 - 13*x^28 + 22*x^27 + 174*x^26 - 256*x^25 - 157*x^24 + 458*x^23 + 5658*x^22 - 6464*x^21 + 45434*x^20 - 23206*x^19 + 322087*x^18 - 245272*x^17 + 2223136*x^16 - 1463962*x^15 + 13159287*x^14 - 5855334*x^13 + 62967354*x^12 - 19483620*x^11 + 244339329*x^10 - 69901890*x^9 + 742997779*x^8 - 222989416*x^7 + 1686428990*x^6 - 498857406*x^5 + 2661692327*x^4 - 653724964*x^3 + 2600745170*x^2 - 385672188*x + 1205881601);
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 30 |
| The 30 conjugacy class representatives for $C_{30}$ |
| Character table for $C_{30}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.0.1229312.1, 10.0.7024111812608.1, 15.15.886528337182930278529.1 |
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 | R | $15^{2}$ | $30$ | R | R | ${\href{/padicField/13.10.0.1}{10} }^{3}$ | $15^{2}$ | $15^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }^{3}$ | $30$ | $30$ | ${\href{/padicField/41.5.0.1}{5} }^{6}$ | ${\href{/padicField/43.1.0.1}{1} }^{30}$ | $30$ | $30$ | $15^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $30$ | $2$ | $15$ | $45$ | |||
|
\(7\)
| Deg $30$ | $3$ | $10$ | $20$ | |||
|
\(11\)
| 11.15.12.1 | $x^{15} + 10 x^{13} + 45 x^{12} + 40 x^{11} + 393 x^{10} + 890 x^{9} + 750 x^{8} - 3970 x^{7} + 9610 x^{6} + 13085 x^{5} + 151045 x^{4} + 26525 x^{3} + 116170 x^{2} - 53795 x + 67662$ | $5$ | $3$ | $12$ | $C_{15}$ | $[\ ]_{5}^{3}$ |
| 11.15.12.1 | $x^{15} + 10 x^{13} + 45 x^{12} + 40 x^{11} + 393 x^{10} + 890 x^{9} + 750 x^{8} - 3970 x^{7} + 9610 x^{6} + 13085 x^{5} + 151045 x^{4} + 26525 x^{3} + 116170 x^{2} - 53795 x + 67662$ | $5$ | $3$ | $12$ | $C_{15}$ | $[\ ]_{5}^{3}$ |