Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + x^13 - x^12 - 5*x^11 - 4*x^10 - 5*x^9 - x^8 + 4*x^7 + 2*x^6 - 2*x^5 - 4*x^4 - 5*x^3 - 4*x^2 - 3*x - 1)
gp: K = bnfinit(y^15 + y^13 - y^12 - 5*y^11 - 4*y^10 - 5*y^9 - y^8 + 4*y^7 + 2*y^6 - 2*y^5 - 4*y^4 - 5*y^3 - 4*y^2 - 3*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 + x^13 - x^12 - 5*x^11 - 4*x^10 - 5*x^9 - x^8 + 4*x^7 + 2*x^6 - 2*x^5 - 4*x^4 - 5*x^3 - 4*x^2 - 3*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 + x^13 - x^12 - 5*x^11 - 4*x^10 - 5*x^9 - x^8 + 4*x^7 + 2*x^6 - 2*x^5 - 4*x^4 - 5*x^3 - 4*x^2 - 3*x - 1)
\( x^{15} + x^{13} - x^{12} - 5 x^{11} - 4 x^{10} - 5 x^{9} - x^{8} + 4 x^{7} + 2 x^{6} - 2 x^{5} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(31123779291971584\)
\(\medspace = 2^{18}\cdot 587^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.58\) | 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: | $2^{7/4}587^{1/2}\approx 81.49323216626063$ | ||
| Ramified primes: |
\(2\), \(587\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3501}a^{14}+\frac{436}{3501}a^{13}+\frac{1043}{3501}a^{12}-\frac{383}{3501}a^{11}+\frac{1055}{3501}a^{10}+\frac{1345}{3501}a^{9}+\frac{1748}{3501}a^{8}-\frac{1091}{3501}a^{7}+\frac{464}{3501}a^{6}-\frac{752}{3501}a^{5}+\frac{1220}{3501}a^{4}-\frac{236}{3501}a^{3}-\frac{1372}{3501}a^{2}+\frac{475}{3501}a+\frac{538}{3501}$
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
Trivial group, which has order $1$
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: | $8$ | 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: |
$a$, $\frac{431}{1167}a^{14}-\frac{749}{1167}a^{13}+\frac{1016}{1167}a^{12}-\frac{1304}{1167}a^{11}-\frac{814}{1167}a^{10}+\frac{1252}{1167}a^{9}-\frac{2050}{1167}a^{8}+\frac{1636}{1167}a^{7}+\frac{38}{1167}a^{6}-\frac{1631}{1167}a^{5}+\frac{281}{1167}a^{4}+\frac{1369}{1167}a^{3}-\frac{1219}{1167}a^{2}-\frac{278}{1167}a-\frac{248}{389}$, $\frac{248}{389}a^{14}-\frac{431}{1167}a^{13}+\frac{1493}{1167}a^{12}-\frac{1760}{1167}a^{11}-\frac{2416}{1167}a^{10}-\frac{2162}{1167}a^{9}-\frac{4972}{1167}a^{8}+\frac{1306}{1167}a^{7}+\frac{1340}{1167}a^{6}+\frac{1450}{1167}a^{5}+\frac{143}{1167}a^{4}-\frac{3257}{1167}a^{3}-\frac{3922}{1167}a^{2}-\frac{2924}{1167}a-\frac{1954}{1167}$, $\frac{1496}{3501}a^{14}-\frac{97}{3501}a^{13}+\frac{49}{3501}a^{12}+\frac{29}{3501}a^{11}-\frac{10007}{3501}a^{10}-\frac{2122}{3501}a^{9}-\frac{2573}{3501}a^{8}-\frac{1837}{3501}a^{7}+\frac{12616}{3501}a^{6}-\frac{2338}{3501}a^{5}-\frac{4736}{3501}a^{4}-\frac{4123}{3501}a^{3}-\frac{3260}{3501}a^{2}-\frac{1270}{3501}a-\frac{2716}{3501}$, $\frac{1054}{3501}a^{14}-\frac{254}{3501}a^{13}+\frac{1175}{3501}a^{12}-\frac{2234}{3501}a^{11}-\frac{3682}{3501}a^{10}-\frac{4943}{3501}a^{9}-\frac{1468}{3501}a^{8}+\frac{748}{3501}a^{7}+\frac{3584}{3501}a^{6}+\frac{4453}{3501}a^{5}-\frac{8323}{3501}a^{4}-\frac{1340}{3501}a^{3}-\frac{2509}{3501}a^{2}-\frac{1160}{3501}a-\frac{2444}{3501}$, $\frac{322}{3501}a^{14}+\frac{1519}{3501}a^{13}-\frac{1417}{3501}a^{12}+\frac{376}{3501}a^{11}-\frac{4555}{3501}a^{10}-\frac{6869}{3501}a^{9}+\frac{1529}{3501}a^{8}+\frac{3466}{3501}a^{7}+\frac{8201}{3501}a^{6}+\frac{7594}{3501}a^{5}-\frac{7441}{3501}a^{4}-\frac{11807}{3501}a^{3}-\frac{1825}{3501}a^{2}+\frac{3574}{3501}a+\frac{4021}{3501}$, $\frac{4682}{3501}a^{14}-\frac{898}{3501}a^{13}+\frac{4099}{3501}a^{12}-\frac{5362}{3501}a^{11}-\frac{23741}{3501}a^{10}-\frac{12679}{3501}a^{9}-\frac{17540}{3501}a^{8}+\frac{2231}{3501}a^{7}+\frac{24001}{3501}a^{6}+\frac{3476}{3501}a^{5}-\frac{14429}{3501}a^{4}-\frac{20809}{3501}a^{3}-\frac{19208}{3501}a^{2}-\frac{10855}{3501}a-\frac{4138}{3501}$, $\frac{218}{3501}a^{14}-\frac{646}{3501}a^{13}+\frac{976}{3501}a^{12}-\frac{637}{3501}a^{11}+\frac{91}{3501}a^{10}+\frac{1460}{3501}a^{9}-\frac{2879}{3501}a^{8}-\frac{937}{3501}a^{7}-\frac{2711}{3501}a^{6}-\frac{556}{3501}a^{5}+\frac{4552}{3501}a^{4}+\frac{3401}{3501}a^{3}+\frac{3157}{3501}a^{2}-\frac{2647}{3501}a-\frac{4084}{3501}$
| 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: | \( 152.23669177 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 152.23669177 \cdot 1}{2\cdot\sqrt{31123779291971584}}\cr\approx \mathstrut & 0.21237939329 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 + x^13 - x^12 - 5*x^11 - 4*x^10 - 5*x^9 - x^8 + 4*x^7 + 2*x^6 - 2*x^5 - 4*x^4 - 5*x^3 - 4*x^2 - 3*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^15 + x^13 - x^12 - 5*x^11 - 4*x^10 - 5*x^9 - x^8 + 4*x^7 + 2*x^6 - 2*x^5 - 4*x^4 - 5*x^3 - 4*x^2 - 3*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^15 + x^13 - x^12 - 5*x^11 - 4*x^10 - 5*x^9 - x^8 + 4*x^7 + 2*x^6 - 2*x^5 - 4*x^4 - 5*x^3 - 4*x^2 - 3*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^15 + x^13 - x^12 - 5*x^11 - 4*x^10 - 5*x^9 - x^8 + 4*x^7 + 2*x^6 - 2*x^5 - 4*x^4 - 5*x^3 - 4*x^2 - 3*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 non-solvable group of order 720 |
| The 11 conjugacy class representatives for $S_6$ |
| Character table for $S_6$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 6 siblings: | 6.2.37568.1, 6.2.51779072768.2 |
| Degree 10 sibling: | 10.2.828465164288.1 |
| Degree 12 siblings: | deg 12, deg 12 |
| Degree 15 sibling: | deg 15 |
| Degree 20 siblings: | deg 20, 20.4.2745418113754971322187776.1, deg 20 |
| Degree 30 siblings: | data not computed |
| Degree 36 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Degree 45 sibling: | data not computed |
| Minimal sibling: | 6.2.37568.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.5.0.1}{5} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }^{3}$ | ${\href{/padicField/23.5.0.1}{5} }^{3}$ | ${\href{/padicField/29.5.0.1}{5} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.5.0.1}{5} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.12.18.52 | $x^{12} + 10 x^{10} + 18 x^{8} - 8 x^{7} + 104 x^{6} + 48 x^{5} + 204 x^{4} + 216 x^{2} + 216$ | $4$ | $3$ | $18$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ | |
|
\(587\)
| $\Q_{587}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $8$ | $2$ | $4$ | $4$ |