Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 30*x^28 - x^27 + 405*x^26 + 27*x^25 - 3250*x^24 - 324*x^23 + 17250*x^22 + 2278*x^21 - 63756*x^20 - 10416*x^19 + 168244*x^18 + 32508*x^17 - 319752*x^16 - 70720*x^15 + 435915*x^14 + 107592*x^13 - 419353*x^12 - 113139*x^11 + 275835*x^10 + 79936*x^9 - 117504*x^8 - 36027*x^7 + 29442*x^6 + 9423*x^5 - 3555*x^4 - 1173*x^3 + 108*x^2 + 36*x + 1)
gp: K = bnfinit(y^30 - 30*y^28 - y^27 + 405*y^26 + 27*y^25 - 3250*y^24 - 324*y^23 + 17250*y^22 + 2278*y^21 - 63756*y^20 - 10416*y^19 + 168244*y^18 + 32508*y^17 - 319752*y^16 - 70720*y^15 + 435915*y^14 + 107592*y^13 - 419353*y^12 - 113139*y^11 + 275835*y^10 + 79936*y^9 - 117504*y^8 - 36027*y^7 + 29442*y^6 + 9423*y^5 - 3555*y^4 - 1173*y^3 + 108*y^2 + 36*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 30*x^28 - x^27 + 405*x^26 + 27*x^25 - 3250*x^24 - 324*x^23 + 17250*x^22 + 2278*x^21 - 63756*x^20 - 10416*x^19 + 168244*x^18 + 32508*x^17 - 319752*x^16 - 70720*x^15 + 435915*x^14 + 107592*x^13 - 419353*x^12 - 113139*x^11 + 275835*x^10 + 79936*x^9 - 117504*x^8 - 36027*x^7 + 29442*x^6 + 9423*x^5 - 3555*x^4 - 1173*x^3 + 108*x^2 + 36*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 30*x^28 - x^27 + 405*x^26 + 27*x^25 - 3250*x^24 - 324*x^23 + 17250*x^22 + 2278*x^21 - 63756*x^20 - 10416*x^19 + 168244*x^18 + 32508*x^17 - 319752*x^16 - 70720*x^15 + 435915*x^14 + 107592*x^13 - 419353*x^12 - 113139*x^11 + 275835*x^10 + 79936*x^9 - 117504*x^8 - 36027*x^7 + 29442*x^6 + 9423*x^5 - 3555*x^4 - 1173*x^3 + 108*x^2 + 36*x + 1)
\( x^{30} - 30 x^{28} - x^{27} + 405 x^{26} + 27 x^{25} - 3250 x^{24} - 324 x^{23} + 17250 x^{22} + \cdots + 1 \)
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: |
\(38731022422755868071217069332926190761458900976553\)
\(\medspace = 3^{45}\cdot 11^{27}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(44.97\) | 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: | $3^{3/2}11^{9/10}\approx 44.971285159244935$ | ||
| Ramified primes: |
\(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{33}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $30$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(99=3^{2}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{99}(64,·)$, $\chi_{99}(1,·)$, $\chi_{99}(2,·)$, $\chi_{99}(67,·)$, $\chi_{99}(4,·)$, $\chi_{99}(68,·)$, $\chi_{99}(65,·)$, $\chi_{99}(8,·)$, $\chi_{99}(74,·)$, $\chi_{99}(98,·)$, $\chi_{99}(16,·)$, $\chi_{99}(17,·)$, $\chi_{99}(82,·)$, $\chi_{99}(83,·)$, $\chi_{99}(25,·)$, $\chi_{99}(91,·)$, $\chi_{99}(29,·)$, $\chi_{99}(31,·)$, $\chi_{99}(32,·)$, $\chi_{99}(97,·)$, $\chi_{99}(34,·)$, $\chi_{99}(35,·)$, $\chi_{99}(37,·)$, $\chi_{99}(41,·)$, $\chi_{99}(49,·)$, $\chi_{99}(50,·)$, $\chi_{99}(58,·)$, $\chi_{99}(95,·)$, $\chi_{99}(70,·)$, $\chi_{99}(62,·)$$\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: |
$a^{22}-22a^{20}+209a^{18}-1122a^{16}+3740a^{14}-8008a^{12}+11011a^{10}-9438a^{8}+4719a^{6}-1210a^{4}+121a^{2}-2$, $a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a-1$, $a^{27}-27a^{25}+324a^{23}-2278a^{21}+10416a^{19}+a^{18}-32508a^{17}-18a^{16}+70720a^{15}+135a^{14}-107592a^{13}-547a^{12}+113139a^{11}+1299a^{10}-79936a^{9}-1836a^{8}+36027a^{7}+1498a^{6}-9423a^{5}-645a^{4}+1174a^{3}+117a^{2}-39a-3$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}+a^{9}-1782a^{8}-9a^{7}+1386a^{6}+27a^{5}-540a^{4}-30a^{3}+81a^{2}+9a-1$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-2$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a+1$, $a^{27}-27a^{25}+324a^{23}-2278a^{21}+10416a^{19}+a^{18}-32508a^{17}-18a^{16}+70720a^{15}+135a^{14}-107592a^{13}-547a^{12}+113139a^{11}+1299a^{10}-79936a^{9}-1836a^{8}+36027a^{7}+1498a^{6}-9423a^{5}-645a^{4}+1173a^{3}+117a^{2}-36a-4$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a-1$, $a^{6}-6a^{4}+9a^{2}-3$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+1$, $a^{24}-24a^{22}-a^{21}+252a^{20}+21a^{19}-1520a^{18}-189a^{17}+5814a^{16}+952a^{15}-14688a^{14}-2940a^{13}+24751a^{12}+5733a^{11}-27444a^{10}-7006a^{9}+19251a^{8}+5139a^{7}-7896a^{6}-2052a^{5}+1611a^{4}+355a^{3}-108a^{2}-12a$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}+8645a^{18}-25194a^{16}+50388a^{14}-68952a^{12}+63206a^{10}-37180a^{8}+a^{7}+13013a^{6}-7a^{5}-2366a^{4}+14a^{3}+169a^{2}-7a-3$, $a^{7}-7a^{5}+14a^{3}-7a+1$, $a^{27}-28a^{25}+349a^{23}-2552a^{21}+12145a^{19}-39445a^{17}+89165a^{15}-140470a^{13}+152034a^{11}-109539a^{9}-a^{8}+49685a^{7}+8a^{6}-12831a^{5}-20a^{4}+1526a^{3}+16a^{2}-42a-2$, $a^{2}-1$, $a^{4}-4a^{2}+3$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}-1015a^{3}+29a+1$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+3$, $a^{22}-22a^{20}+209a^{18}-1122a^{16}+3741a^{14}-8022a^{12}+a^{11}+11088a^{10}-11a^{9}-9648a^{8}+44a^{7}+5013a^{6}-77a^{5}-1406a^{4}+55a^{3}+170a^{2}-11a-4$, $a^{5}-5a^{3}+5a$, $a^{5}-5a^{3}+5a+1$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}+a^{10}-16445a^{9}-10a^{8}+9867a^{7}+35a^{6}-3289a^{5}-50a^{4}+506a^{3}+25a^{2}-23a-3$, $a^{7}-7a^{5}+14a^{3}-7a$, $a^{28}-28a^{26}+350a^{24}-2577a^{22}+12419a^{20}-41173a^{18}+a^{17}+96084a^{16}-17a^{15}-158780a^{14}+119a^{13}+184366a^{12}-443a^{11}-147146a^{10}+946a^{9}+77496a^{8}-1167a^{7}-25068a^{6}+798a^{5}+4345a^{4}-273a^{3}-292a^{2}+36a+2$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+a^{13}+2275a^{12}-13a^{11}-4004a^{10}+65a^{9}+4290a^{8}-156a^{7}-2640a^{6}+182a^{5}+825a^{4}-91a^{3}-100a^{2}+13a+1$, $a^{26}-26a^{24}+299a^{22}-2003a^{20}+8666a^{18}-25382a^{16}+51323a^{14}-71774a^{12}+68510a^{10}-43315a^{8}+17178a^{6}-3867a^{4}+399a^{2}-9$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}+a^{4}-1015a^{3}-4a^{2}+29a+2$, $a^{28}-29a^{26}-a^{25}+376a^{24}+26a^{23}-2874a^{22}-298a^{21}+14376a^{20}+1980a^{19}-49380a^{18}-8436a^{17}+118864a^{16}+24072a^{15}-200888a^{14}-46648a^{13}+235027a^{12}+60944a^{11}-184327a^{10}-52196a^{9}+91519a^{8}+27750a^{7}-26030a^{6}-8313a^{5}+3496a^{4}+1166a^{3}-129a^{2}-42a$, $a^{28}-27a^{26}+a^{25}+324a^{24}-25a^{23}-2277a^{22}+275a^{21}+10395a^{20}-1749a^{19}-32319a^{18}+7107a^{17}+69769a^{16}-19245a^{15}-104668a^{14}+35154a^{13}+107510a^{12}-42913a^{11}-73281a^{10}+33968a^{9}+31539a^{8}-16488a^{7}-8016a^{6}+4458a^{5}+1125a^{4}-555a^{3}-82a^{2}+17a+2$
| 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: | \( 1458908175124897.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{30}\cdot(2\pi)^{0}\cdot 1458908175124897.5 \cdot 1}{2\cdot\sqrt{38731022422755868071217069332926190761458900976553}}\cr\approx \mathstrut & 0.125854384871860 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 - 30*x^28 - x^27 + 405*x^26 + 27*x^25 - 3250*x^24 - 324*x^23 + 17250*x^22 + 2278*x^21 - 63756*x^20 - 10416*x^19 + 168244*x^18 + 32508*x^17 - 319752*x^16 - 70720*x^15 + 435915*x^14 + 107592*x^13 - 419353*x^12 - 113139*x^11 + 275835*x^10 + 79936*x^9 - 117504*x^8 - 36027*x^7 + 29442*x^6 + 9423*x^5 - 3555*x^4 - 1173*x^3 + 108*x^2 + 36*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 - 30*x^28 - x^27 + 405*x^26 + 27*x^25 - 3250*x^24 - 324*x^23 + 17250*x^22 + 2278*x^21 - 63756*x^20 - 10416*x^19 + 168244*x^18 + 32508*x^17 - 319752*x^16 - 70720*x^15 + 435915*x^14 + 107592*x^13 - 419353*x^12 - 113139*x^11 + 275835*x^10 + 79936*x^9 - 117504*x^8 - 36027*x^7 + 29442*x^6 + 9423*x^5 - 3555*x^4 - 1173*x^3 + 108*x^2 + 36*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 - 30*x^28 - x^27 + 405*x^26 + 27*x^25 - 3250*x^24 - 324*x^23 + 17250*x^22 + 2278*x^21 - 63756*x^20 - 10416*x^19 + 168244*x^18 + 32508*x^17 - 319752*x^16 - 70720*x^15 + 435915*x^14 + 107592*x^13 - 419353*x^12 - 113139*x^11 + 275835*x^10 + 79936*x^9 - 117504*x^8 - 36027*x^7 + 29442*x^6 + 9423*x^5 - 3555*x^4 - 1173*x^3 + 108*x^2 + 36*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 - 30*x^28 - x^27 + 405*x^26 + 27*x^25 - 3250*x^24 - 324*x^23 + 17250*x^22 + 2278*x^21 - 63756*x^20 - 10416*x^19 + 168244*x^18 + 32508*x^17 - 319752*x^16 - 70720*x^15 + 435915*x^14 + 107592*x^13 - 419353*x^12 - 113139*x^11 + 275835*x^10 + 79936*x^9 - 117504*x^8 - 36027*x^7 + 29442*x^6 + 9423*x^5 - 3555*x^4 - 1173*x^3 + 108*x^2 + 36*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
| \(\Q(\sqrt{33}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\), 6.6.26198073.1, \(\Q(\zeta_{33})^+\), 15.15.10943023107606534329121.1 |
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 | $15^{2}$ | R | $30$ | $30$ | R | $30$ | ${\href{/padicField/17.5.0.1}{5} }^{6}$ | ${\href{/padicField/19.10.0.1}{10} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{5}$ | $15^{2}$ | $15^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{6}$ | $15^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{5}$ | $30$ | ${\href{/padicField/53.10.0.1}{10} }^{3}$ | $30$ |
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 $30$ | $6$ | $5$ | $45$ | |||
|
\(11\)
| Deg $30$ | $10$ | $3$ | $27$ |