Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 - 30*x^28 + 30*x^27 + 404*x^26 - 404*x^25 - 3223*x^24 + 3223*x^23 + 16927*x^22 - 16927*x^21 - 61503*x^20 + 61503*x^19 + 158101*x^18 - 158101*x^17 - 288950*x^16 + 288950*x^15 + 371908*x^14 - 371908*x^13 - 329002*x^12 + 329002*x^11 + 191674*x^10 - 191674*x^9 - 68664*x^8 + 68664*x^7 + 13548*x^6 - 13548*x^5 - 1208*x^4 + 1208*x^3 + 32*x^2 - 32*x + 1)
gp: K = bnfinit(y^30 - y^29 - 30*y^28 + 30*y^27 + 404*y^26 - 404*y^25 - 3223*y^24 + 3223*y^23 + 16927*y^22 - 16927*y^21 - 61503*y^20 + 61503*y^19 + 158101*y^18 - 158101*y^17 - 288950*y^16 + 288950*y^15 + 371908*y^14 - 371908*y^13 - 329002*y^12 + 329002*y^11 + 191674*y^10 - 191674*y^9 - 68664*y^8 + 68664*y^7 + 13548*y^6 - 13548*y^5 - 1208*y^4 + 1208*y^3 + 32*y^2 - 32*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - x^29 - 30*x^28 + 30*x^27 + 404*x^26 - 404*x^25 - 3223*x^24 + 3223*x^23 + 16927*x^22 - 16927*x^21 - 61503*x^20 + 61503*x^19 + 158101*x^18 - 158101*x^17 - 288950*x^16 + 288950*x^15 + 371908*x^14 - 371908*x^13 - 329002*x^12 + 329002*x^11 + 191674*x^10 - 191674*x^9 - 68664*x^8 + 68664*x^7 + 13548*x^6 - 13548*x^5 - 1208*x^4 + 1208*x^3 + 32*x^2 - 32*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - x^29 - 30*x^28 + 30*x^27 + 404*x^26 - 404*x^25 - 3223*x^24 + 3223*x^23 + 16927*x^22 - 16927*x^21 - 61503*x^20 + 61503*x^19 + 158101*x^18 - 158101*x^17 - 288950*x^16 + 288950*x^15 + 371908*x^14 - 371908*x^13 - 329002*x^12 + 329002*x^11 + 191674*x^10 - 191674*x^9 - 68664*x^8 + 68664*x^7 + 13548*x^6 - 13548*x^5 - 1208*x^4 + 1208*x^3 + 32*x^2 - 32*x + 1)
\( x^{30} - x^{29} - 30 x^{28} + 30 x^{27} + 404 x^{26} - 404 x^{25} - 3223 x^{24} + 3223 x^{23} + 16927 x^{22} + \cdots + 1 \)
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: | $[30, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(254863675513583304379979153001217903140700917991797\)
\(\medspace = 3^{15}\cdot 31^{29}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(47.89\) | 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: | $3^{1/2}31^{29/30}\approx 47.88618783549088$ | ||
| Ramified primes: |
\(3\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{93}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $30$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(93=3\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{93}(64,·)$, $\chi_{93}(1,·)$, $\chi_{93}(67,·)$, $\chi_{93}(4,·)$, $\chi_{93}(68,·)$, $\chi_{93}(7,·)$, $\chi_{93}(10,·)$, $\chi_{93}(11,·)$, $\chi_{93}(76,·)$, $\chi_{93}(77,·)$, $\chi_{93}(16,·)$, $\chi_{93}(17,·)$, $\chi_{93}(82,·)$, $\chi_{93}(19,·)$, $\chi_{93}(86,·)$, $\chi_{93}(23,·)$, $\chi_{93}(25,·)$, $\chi_{93}(26,·)$, $\chi_{93}(28,·)$, $\chi_{93}(29,·)$, $\chi_{93}(70,·)$, $\chi_{93}(65,·)$, $\chi_{93}(40,·)$, $\chi_{93}(92,·)$, $\chi_{93}(44,·)$, $\chi_{93}(49,·)$, $\chi_{93}(83,·)$, $\chi_{93}(53,·)$, $\chi_{93}(89,·)$, $\chi_{93}(74,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
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: | $29$ | 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: |
$a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}-1015a^{3}+a^{2}+29a-2$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+a^{8}+9867a^{7}-8a^{6}-3289a^{5}+20a^{4}+506a^{3}-16a^{2}-23a+2$, $a^{27}-27a^{25}+a^{24}+323a^{23}-24a^{22}-2254a^{21}+251a^{20}+10165a^{19}-1500a^{18}-31008a^{17}+5644a^{16}+65077a^{15}-13888a^{14}-93719a^{13}+22478a^{12}+90751a^{11}-23464a^{10}-56749a^{9}+15068a^{8}+21427a^{7}-5472a^{6}-4383a^{5}+976a^{4}+399a^{3}-64a^{2}-11a+1$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+2$, $a^{27}-27a^{25}+323a^{23}-2254a^{21}+10165a^{19}-31008a^{17}+65076a^{15}-93704a^{13}+a^{12}+90662a^{11}-12a^{10}-56485a^{9}+53a^{8}+21021a^{7}-104a^{6}-4082a^{5}+85a^{4}+313a^{3}-20a^{2}-4a+1$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{29}-30a^{27}+a^{26}+404a^{25}-26a^{24}-3224a^{23}+299a^{22}+16950a^{21}-2001a^{20}-61733a^{19}+8624a^{18}+159412a^{17}-25006a^{16}-293641a^{15}+49454a^{14}+382841a^{13}-66146a^{12}-345656a^{11}+58004a^{10}+207844a^{9}-31372a^{8}-78081a^{7}+9392a^{6}+16459a^{5}-1296a^{4}-1574a^{3}+65a^{2}+40a-2$, $a^{27}-27a^{25}+323a^{23}-2254a^{21}+10165a^{19}-31008a^{17}+65076a^{15}-93704a^{13}+90662a^{11}-56485a^{9}-a^{8}+21021a^{7}+8a^{6}-4082a^{5}-20a^{4}+313a^{3}+16a^{2}-4a-1$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a+1$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-1$, $a^{17}-17a^{15}+a^{14}+119a^{13}-14a^{12}-442a^{11}+77a^{10}+935a^{9}-210a^{8}-1122a^{7}+294a^{6}+714a^{5}-196a^{4}-204a^{3}+49a^{2}+17a-2$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a+1$, $a^{29}-29a^{27}+377a^{25}-a^{24}-2899a^{23}+24a^{22}+14651a^{21}-252a^{20}-51129a^{19}+1519a^{18}+125971a^{17}-5796a^{16}-220133a^{15}+14554a^{14}+270181a^{13}-24221a^{12}-227240a^{11}+26258a^{10}+125488a^{9}-17786a^{8}-42527a^{7}+7020a^{6}+7981a^{5}-1457a^{4}-714a^{3}+133a^{2}+26a-4$, $a^{2}-2$, $a^{22}-22a^{20}+209a^{18}-1122a^{16}+3740a^{14}-8008a^{12}+11011a^{10}-9438a^{8}+4719a^{6}-1210a^{4}+121a^{2}-2$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-2$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}+a^{7}-8008a^{6}-7a^{5}+1716a^{4}+14a^{3}-144a^{2}-7a+2$, $a^{19}-19a^{17}+152a^{15}-665a^{13}+a^{12}+1729a^{11}-12a^{10}-2717a^{9}+54a^{8}+2508a^{7}-112a^{6}-1254a^{5}+105a^{4}+285a^{3}-36a^{2}-19a+2$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+a^{4}+819a^{3}-4a^{2}-27a+2$, $a^{25}-25a^{23}+275a^{21}+a^{20}-1750a^{19}-20a^{18}+7126a^{17}+170a^{16}-19397a^{15}-800a^{14}+35819a^{13}+2275a^{12}-44641a^{11}-4004a^{10}+36674a^{9}+4290a^{8}-18953a^{7}-2640a^{6}+5642a^{5}+825a^{4}-799a^{3}-100a^{2}+31a+1$, $a^{28}-a^{27}-28a^{26}+27a^{25}+350a^{24}-324a^{23}-2576a^{22}+2277a^{21}+12397a^{20}-10395a^{19}-40964a^{18}+32319a^{17}+94962a^{16}-69768a^{15}-155040a^{14}+104652a^{13}+176358a^{12}-107406a^{11}-136136a^{10}+72930a^{9}+68068a^{8}-30888a^{7}-20384a^{6}+7371a^{5}+3185a^{4}-819a^{3}-196a^{2}+27a+2$, $a^{12}-12a^{10}+55a^{8}-120a^{6}+126a^{4}-56a^{2}+7$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12397a^{20}-40964a^{18}+94962a^{16}-155040a^{14}+176358a^{12}-136136a^{10}+68068a^{8}-20384a^{6}+3185a^{4}+a^{3}-196a^{2}-3a+2$, $a^{29}-30a^{27}+a^{26}+404a^{25}-27a^{24}-3223a^{23}+323a^{22}+16927a^{21}-2253a^{20}-61503a^{19}+10144a^{18}+158101a^{17}-30820a^{16}-288950a^{15}+64142a^{14}+371908a^{13}-90898a^{12}-329002a^{11}+85460a^{10}+191674a^{9}-50676a^{8}-68664a^{7}+17392a^{6}+13548a^{5}-2992a^{4}-1208a^{3}+193a^{2}+33a-3$, $a^{29}-29a^{27}+a^{26}+377a^{25}-27a^{24}-2899a^{23}+323a^{22}+14650a^{21}-2253a^{20}-51108a^{19}+10144a^{18}+125782a^{17}-30820a^{16}-219182a^{15}+64142a^{14}+267256a^{13}-90898a^{12}-221596a^{11}+85460a^{10}+118744a^{9}-50676a^{8}-37776a^{7}+17392a^{6}+6177a^{5}-2992a^{4}-389a^{3}+193a^{2}+6a-3$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-651a^{3}+28a$, $a^{22}-22a^{20}+209a^{18}-1122a^{16}+3740a^{14}-8007a^{12}+10999a^{10}+a^{9}-9384a^{8}-9a^{7}+4607a^{6}+27a^{5}-1105a^{4}-30a^{3}+85a^{2}+9a$, $a^{9}-9a^{7}+27a^{5}-30a^{3}-a^{2}+9a+2$
| 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: | \( 3165460564789336.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{30}\cdot(2\pi)^{0}\cdot 3165460564789336.0 \cdot 1}{2\cdot\sqrt{254863675513583304379979153001217903140700917991797}}\cr\approx \mathstrut & 0.106451751314542 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 - 30*x^28 + 30*x^27 + 404*x^26 - 404*x^25 - 3223*x^24 + 3223*x^23 + 16927*x^22 - 16927*x^21 - 61503*x^20 + 61503*x^19 + 158101*x^18 - 158101*x^17 - 288950*x^16 + 288950*x^15 + 371908*x^14 - 371908*x^13 - 329002*x^12 + 329002*x^11 + 191674*x^10 - 191674*x^9 - 68664*x^8 + 68664*x^7 + 13548*x^6 - 13548*x^5 - 1208*x^4 + 1208*x^3 + 32*x^2 - 32*x + 1)
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 - x^29 - 30*x^28 + 30*x^27 + 404*x^26 - 404*x^25 - 3223*x^24 + 3223*x^23 + 16927*x^22 - 16927*x^21 - 61503*x^20 + 61503*x^19 + 158101*x^18 - 158101*x^17 - 288950*x^16 + 288950*x^15 + 371908*x^14 - 371908*x^13 - 329002*x^12 + 329002*x^11 + 191674*x^10 - 191674*x^9 - 68664*x^8 + 68664*x^7 + 13548*x^6 - 13548*x^5 - 1208*x^4 + 1208*x^3 + 32*x^2 - 32*x + 1, 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 - x^29 - 30*x^28 + 30*x^27 + 404*x^26 - 404*x^25 - 3223*x^24 + 3223*x^23 + 16927*x^22 - 16927*x^21 - 61503*x^20 + 61503*x^19 + 158101*x^18 - 158101*x^17 - 288950*x^16 + 288950*x^15 + 371908*x^14 - 371908*x^13 - 329002*x^12 + 329002*x^11 + 191674*x^10 - 191674*x^9 - 68664*x^8 + 68664*x^7 + 13548*x^6 - 13548*x^5 - 1208*x^4 + 1208*x^3 + 32*x^2 - 32*x + 1);
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 - x^29 - 30*x^28 + 30*x^27 + 404*x^26 - 404*x^25 - 3223*x^24 + 3223*x^23 + 16927*x^22 - 16927*x^21 - 61503*x^20 + 61503*x^19 + 158101*x^18 - 158101*x^17 - 288950*x^16 + 288950*x^15 + 371908*x^14 - 371908*x^13 - 329002*x^12 + 329002*x^11 + 191674*x^10 - 191674*x^9 - 68664*x^8 + 68664*x^7 + 13548*x^6 - 13548*x^5 - 1208*x^4 + 1208*x^3 + 32*x^2 - 32*x + 1);
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
| \(\Q(\sqrt{93}) \), 3.3.961.1, 5.5.923521.1, 6.6.772987077.1, 10.10.6424828185043053.1, \(\Q(\zeta_{31})^+\) |
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 | ${\href{/padicField/2.10.0.1}{10} }^{3}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{5}$ | $15^{2}$ | $15^{2}$ | $30$ | $15^{2}$ | $15^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{6}$ | ${\href{/padicField/29.5.0.1}{5} }^{6}$ | R | ${\href{/padicField/37.6.0.1}{6} }^{5}$ | $30$ | $30$ | ${\href{/padicField/47.10.0.1}{10} }^{3}$ | $15^{2}$ | $30$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $30$ | $2$ | $15$ | $15$ | |||
|
\(31\)
| Deg $30$ | $30$ | $1$ | $29$ |