Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 58*x^28 + 1512*x^26 + 23400*x^24 + 239200*x^22 + 1700160*x^20 + 8614144*x^18 + 31380096*x^16 + 81861120*x^14 + 150492160*x^12 + 189190144*x^10 + 154791936*x^8 + 76038144*x^6 + 19496960*x^4 + 1966080*x^2 + 32768)
gp: K = bnfinit(y^30 + 58*y^28 + 1512*y^26 + 23400*y^24 + 239200*y^22 + 1700160*y^20 + 8614144*y^18 + 31380096*y^16 + 81861120*y^14 + 150492160*y^12 + 189190144*y^10 + 154791936*y^8 + 76038144*y^6 + 19496960*y^4 + 1966080*y^2 + 32768, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 + 58*x^28 + 1512*x^26 + 23400*x^24 + 239200*x^22 + 1700160*x^20 + 8614144*x^18 + 31380096*x^16 + 81861120*x^14 + 150492160*x^12 + 189190144*x^10 + 154791936*x^8 + 76038144*x^6 + 19496960*x^4 + 1966080*x^2 + 32768);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 + 58*x^28 + 1512*x^26 + 23400*x^24 + 239200*x^22 + 1700160*x^20 + 8614144*x^18 + 31380096*x^16 + 81861120*x^14 + 150492160*x^12 + 189190144*x^10 + 154791936*x^8 + 76038144*x^6 + 19496960*x^4 + 1966080*x^2 + 32768)
\( x^{30} + 58 x^{28} + 1512 x^{26} + 23400 x^{24} + 239200 x^{22} + 1700160 x^{20} + 8614144 x^{18} + \cdots + 32768 \)
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: |
\(-20159382829191092591451779536401274948781988965475418112\)
\(\medspace = -\,2^{45}\cdot 31^{28}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(69.74\) | 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: | $2^{3/2}31^{14/15}\approx 69.74010138658973$ | ||
| Ramified primes: |
\(2\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $30$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(248=2^{3}\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{248}(1,·)$, $\chi_{248}(131,·)$, $\chi_{248}(227,·)$, $\chi_{248}(211,·)$, $\chi_{248}(129,·)$, $\chi_{248}(9,·)$, $\chi_{248}(81,·)$, $\chi_{248}(19,·)$, $\chi_{248}(171,·)$, $\chi_{248}(225,·)$, $\chi_{248}(25,·)$, $\chi_{248}(219,·)$, $\chi_{248}(107,·)$, $\chi_{248}(33,·)$, $\chi_{248}(67,·)$, $\chi_{248}(35,·)$, $\chi_{248}(113,·)$, $\chi_{248}(163,·)$, $\chi_{248}(195,·)$, $\chi_{248}(97,·)$, $\chi_{248}(41,·)$, $\chi_{248}(193,·)$, $\chi_{248}(235,·)$, $\chi_{248}(49,·)$, $\chi_{248}(51,·)$, $\chi_{248}(187,·)$, $\chi_{248}(233,·)$, $\chi_{248}(121,·)$, $\chi_{248}(59,·)$, $\chi_{248}(169,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{16384}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$, $\frac{1}{16384}a^{28}$, $\frac{1}{16384}a^{29}$
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
$C_{2}\times C_{116182}$, which has order $232364$ (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: |
$\frac{1}{1024}a^{20}+\frac{5}{128}a^{18}+\frac{85}{128}a^{16}+\frac{25}{4}a^{14}+\frac{2275}{64}a^{12}+\frac{1001}{8}a^{10}+\frac{2145}{8}a^{8}+330a^{6}+\frac{825}{4}a^{4}+50a^{2}+2$, $\frac{1}{32}a^{10}+\frac{5}{8}a^{8}+\frac{35}{8}a^{6}+\frac{25}{2}a^{4}+\frac{25}{2}a^{2}+2$, $\frac{1}{8192}a^{26}+\frac{13}{2048}a^{24}+\frac{299}{2048}a^{22}+\frac{1001}{512}a^{20}+\frac{8645}{512}a^{18}+\frac{12597}{128}a^{16}+\frac{12597}{32}a^{14}+\frac{8619}{8}a^{12}+\frac{31603}{16}a^{10}+\frac{9295}{4}a^{8}+\frac{13013}{8}a^{6}+\frac{1183}{2}a^{4}+\frac{169}{2}a^{2}+2$, $\frac{1}{2048}a^{22}+\frac{11}{512}a^{20}+\frac{209}{512}a^{18}+\frac{561}{128}a^{16}+\frac{935}{32}a^{14}+\frac{1001}{8}a^{12}+\frac{11011}{32}a^{10}+\frac{4719}{8}a^{8}+\frac{4719}{8}a^{6}+\frac{605}{2}a^{4}+\frac{121}{2}a^{2}+2$, $\frac{1}{2}a^{2}+2$, $\frac{1}{512}a^{18}+\frac{9}{128}a^{16}+\frac{135}{128}a^{14}+\frac{273}{32}a^{12}+\frac{1287}{32}a^{10}+\frac{891}{8}a^{8}+\frac{693}{4}a^{6}+135a^{4}+\frac{81}{2}a^{2}+1$, $\frac{1}{16384}a^{28}+\frac{27}{8192}a^{26}+\frac{325}{4096}a^{24}+\frac{575}{512}a^{22}+\frac{10625}{1024}a^{20}+\frac{16815}{256}a^{18}+\frac{18615}{64}a^{16}+\frac{115599}{128}a^{14}+\frac{124137}{64}a^{12}+\frac{89309}{32}a^{10}+\frac{40589}{16}a^{8}+\frac{10443}{8}a^{6}+296a^{4}+\frac{11}{2}a^{2}-3$, $\frac{1}{8}a^{6}+\frac{3}{2}a^{4}+\frac{9}{2}a^{2}+1$, $\frac{1}{4}a^{4}+2a^{2}+2$, $\frac{1}{2048}a^{22}+\frac{11}{512}a^{20}+\frac{105}{256}a^{18}+\frac{285}{64}a^{16}+\frac{3875}{128}a^{14}+\frac{4277}{32}a^{12}+\frac{6149}{16}a^{10}+\frac{2805}{4}a^{8}+\frac{6105}{8}a^{6}+\frac{1749}{4}a^{4}+99a^{2}+1$, $\frac{1}{4096}a^{24}+\frac{3}{256}a^{22}+\frac{253}{1024}a^{20}+\frac{385}{128}a^{18}+\frac{5985}{256}a^{16}+\frac{969}{8}a^{14}+\frac{27131}{64}a^{12}+\frac{7953}{8}a^{10}+\frac{24255}{16}a^{8}+\frac{11319}{8}a^{6}+\frac{2871}{4}a^{4}+149a^{2}+2$, $\frac{1}{2048}a^{22}+\frac{11}{512}a^{20}+\frac{105}{256}a^{18}+\frac{285}{64}a^{16}+\frac{3875}{128}a^{14}+\frac{4277}{32}a^{12}+\frac{6149}{16}a^{10}+\frac{2805}{4}a^{8}+\frac{6105}{8}a^{6}+\frac{875}{2}a^{4}+101a^{2}+3$, $\frac{1}{4096}a^{24}+\frac{3}{256}a^{22}+\frac{63}{256}a^{20}+\frac{95}{32}a^{18}+\frac{2907}{128}a^{16}+\frac{14687}{128}a^{14}+\frac{12369}{32}a^{12}+\frac{13689}{16}a^{10}+\frac{19085}{16}a^{8}+\frac{7679}{8}a^{6}+\frac{735}{2}a^{4}+35a^{2}-1$, $\frac{1}{16384}a^{28}+\frac{27}{8192}a^{26}+\frac{325}{4096}a^{24}+\frac{575}{512}a^{22}+\frac{5313}{512}a^{20}+\frac{33649}{512}a^{18}+\frac{74613}{256}a^{16}+\frac{14535}{16}a^{14}+\frac{62985}{32}a^{12}+\frac{46189}{16}a^{10}+\frac{21879}{8}a^{8}+1547a^{6}+455a^{4}+\frac{105}{2}a^{2}+1$
| 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: | \( 4316173757.895952 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 4316173757.895952 \cdot 232364}{2\cdot\sqrt{20159382829191092591451779536401274948781988965475418112}}\cr\approx \mathstrut & 0.104881086445514 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 + 58*x^28 + 1512*x^26 + 23400*x^24 + 239200*x^22 + 1700160*x^20 + 8614144*x^18 + 31380096*x^16 + 81861120*x^14 + 150492160*x^12 + 189190144*x^10 + 154791936*x^8 + 76038144*x^6 + 19496960*x^4 + 1966080*x^2 + 32768)
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 + 58*x^28 + 1512*x^26 + 23400*x^24 + 239200*x^22 + 1700160*x^20 + 8614144*x^18 + 31380096*x^16 + 81861120*x^14 + 150492160*x^12 + 189190144*x^10 + 154791936*x^8 + 76038144*x^6 + 19496960*x^4 + 1966080*x^2 + 32768, 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 + 58*x^28 + 1512*x^26 + 23400*x^24 + 239200*x^22 + 1700160*x^20 + 8614144*x^18 + 31380096*x^16 + 81861120*x^14 + 150492160*x^12 + 189190144*x^10 + 154791936*x^8 + 76038144*x^6 + 19496960*x^4 + 1966080*x^2 + 32768);
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 + 58*x^28 + 1512*x^26 + 23400*x^24 + 239200*x^22 + 1700160*x^20 + 8614144*x^18 + 31380096*x^16 + 81861120*x^14 + 150492160*x^12 + 189190144*x^10 + 154791936*x^8 + 76038144*x^6 + 19496960*x^4 + 1966080*x^2 + 32768);
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
| \(\Q(\sqrt{-2}) \), 3.3.961.1, 5.5.923521.1, 6.0.472842752.1, 10.0.27947533514866688.1, \(\Q(\zeta_{31})^+\) |
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 | R | $15^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{5}$ | $30$ | $15^{2}$ | $30$ | $15^{2}$ | $15^{2}$ | ${\href{/padicField/23.10.0.1}{10} }^{3}$ | ${\href{/padicField/29.10.0.1}{10} }^{3}$ | R | ${\href{/padicField/37.6.0.1}{6} }^{5}$ | $15^{2}$ | $15^{2}$ | ${\href{/padicField/47.10.0.1}{10} }^{3}$ | $30$ | $15^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.15.9 | $x^{10} + 20 x^{9} + 206 x^{8} + 1376 x^{7} + 6504 x^{6} + 22496 x^{5} + 57328 x^{4} + 105856 x^{3} + 135504 x^{2} + 108864 x + 9568$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
| 2.10.15.9 | $x^{10} + 20 x^{9} + 206 x^{8} + 1376 x^{7} + 6504 x^{6} + 22496 x^{5} + 57328 x^{4} + 105856 x^{3} + 135504 x^{2} + 108864 x + 9568$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ | |
| 2.10.15.9 | $x^{10} + 20 x^{9} + 206 x^{8} + 1376 x^{7} + 6504 x^{6} + 22496 x^{5} + 57328 x^{4} + 105856 x^{3} + 135504 x^{2} + 108864 x + 9568$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ | |
|
\(31\)
| Deg $30$ | $15$ | $2$ | $28$ |