Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 2)
gp: K = bnfinit(y^30 + 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 + 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 + 2)
\( x^{30} + 2 \)
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: |
\(-110536959860366678949888000000000000000000000000000000\)
\(\medspace = -\,2^{59}\cdot 3^{30}\cdot 5^{30}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(58.63\) | 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: | $2^{59/30}3^{7/6}5^{23/20}\approx 89.6360328228267$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| 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: | $14$ | 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: |
$a^{2}+1$, $a^{6}+1$, $a^{10}+1$, $a^{18}+a^{6}-1$, $a^{16}+a^{2}-1$, $a^{27}-a^{24}+a^{18}-a^{15}+a^{9}-a^{6}+1$, $a^{25}-a^{20}+a^{10}-a^{5}+1$, $a^{27}+a^{18}+a^{12}+a^{9}-a^{6}+a^{3}+1$, $a^{26}-a^{24}+a^{20}-a^{18}+a^{14}-a^{12}+a^{8}-a^{6}+1$, $a^{29}-a^{27}-a^{26}+a^{24}+a^{23}-a^{21}-a^{20}+a^{18}+a^{17}-a^{15}-a^{14}+a^{12}+a^{11}-a^{9}-a^{8}+a^{6}+a^{5}-a^{3}-2a^{2}-3a-3$, $a^{26}-a^{24}-a^{22}+a^{20}-a^{16}-a^{14}+a^{12}-a^{10}-a^{4}+1$, $3a^{29}+9a^{28}+2a^{27}-5a^{26}-3a^{25}-8a^{23}-a^{22}+8a^{21}+6a^{20}-a^{19}+2a^{18}+2a^{17}-12a^{16}-8a^{15}+2a^{14}+4a^{13}-2a^{12}+8a^{11}+9a^{10}-7a^{9}-11a^{8}-2a^{7}-3a^{6}-10a^{5}+6a^{4}+14a^{3}+3a^{2}-6a+3$, $35a^{29}+19a^{28}+6a^{27}-12a^{26}-32a^{25}-54a^{24}-71a^{23}-74a^{22}-67a^{21}-59a^{20}-48a^{19}-34a^{18}-11a^{17}+22a^{16}+51a^{15}+69a^{14}+79a^{13}+83a^{12}+87a^{11}+87a^{10}+68a^{9}+37a^{8}+a^{7}-32a^{6}-55a^{5}-74a^{4}-100a^{3}-118a^{2}-121a-103$, $2a^{29}+12a^{28}+4a^{27}+5a^{26}-15a^{25}+8a^{24}-3a^{23}+16a^{22}-8a^{21}-8a^{20}-10a^{19}-a^{18}+17a^{17}+7a^{15}-20a^{14}+12a^{13}+18a^{11}-7a^{10}-15a^{9}-10a^{8}-5a^{7}+23a^{6}-7a^{5}+8a^{4}-26a^{3}+15a^{2}+6a+19$
| 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: | \( 584910974292674.1 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 584910974292674.1 \cdot 1}{2\cdot\sqrt{110536959860366678949888000000000000000000000000000000}}\cr\approx \mathstrut & 0.826045636830044 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 + 2)
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, 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);
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);
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
$D_6\times F_5$ (as 30T51):
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 solvable group of order 240 |
| The 30 conjugacy class representatives for $D_6\times F_5$ |
| Character table for $D_6\times F_5$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 3.1.108.1, 5.1.50000.1, 6.0.1492992.1, 10.0.5120000000000.2, 15.1.7174453500000000000000.1 |
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]
Sibling fields
| Degree 30 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.12.0.1}{12} }^{2}{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.10.0.1}{10} }^{2}{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }^{2}{,}\,{\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.4.0.1}{4} }^{6}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{6}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{15}$ | ${\href{/padicField/31.10.0.1}{10} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.4.0.1}{4} }^{6}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{6}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{14}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $30$ | $30$ | $1$ | $59$ | |||
|
\(3\)
| 3.3.3.2 | $x^{3} + 3 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ |
| 3.3.3.2 | $x^{3} + 3 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
| 3.12.12.23 | $x^{12} + 6 x^{11} + 12 x^{10} + 12 x^{9} + 189 x^{8} + 108 x^{7} + 324 x^{6} + 648 x^{5} + 891 x^{4} + 918 x^{3} + 648 x^{2} + 324 x + 81$ | $3$ | $4$ | $12$ | $S_3 \times C_4$ | $[3/2]_{2}^{4}$ | |
| 3.12.12.23 | $x^{12} + 6 x^{11} + 12 x^{10} + 12 x^{9} + 189 x^{8} + 108 x^{7} + 324 x^{6} + 648 x^{5} + 891 x^{4} + 918 x^{3} + 648 x^{2} + 324 x + 81$ | $3$ | $4$ | $12$ | $S_3 \times C_4$ | $[3/2]_{2}^{4}$ | |
|
\(5\)
| 5.10.10.7 | $x^{10} - 50 x^{7} + 10 x^{6} + 10 x^{5} + 175 x^{4} - 250 x^{3} - 225 x^{2} + 50 x + 25$ | $5$ | $2$ | $10$ | $F_{5}\times C_2$ | $[5/4]_{4}^{2}$ |
| 5.10.10.7 | $x^{10} - 50 x^{7} + 10 x^{6} + 10 x^{5} + 175 x^{4} - 250 x^{3} - 225 x^{2} + 50 x + 25$ | $5$ | $2$ | $10$ | $F_{5}\times C_2$ | $[5/4]_{4}^{2}$ | |
| 5.10.10.7 | $x^{10} - 50 x^{7} + 10 x^{6} + 10 x^{5} + 175 x^{4} - 250 x^{3} - 225 x^{2} + 50 x + 25$ | $5$ | $2$ | $10$ | $F_{5}\times C_2$ | $[5/4]_{4}^{2}$ |