Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 4*x^13 - 9*x^12 + 13*x^11 - 15*x^10 + 18*x^9 - 23*x^8 + 33*x^7 - 44*x^6 + 48*x^5 - 44*x^4 + 30*x^3 - 16*x^2 + 6*x - 1)
gp: K = bnfinit(y^15 - 2*y^14 + 4*y^13 - 9*y^12 + 13*y^11 - 15*y^10 + 18*y^9 - 23*y^8 + 33*y^7 - 44*y^6 + 48*y^5 - 44*y^4 + 30*y^3 - 16*y^2 + 6*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 2*x^14 + 4*x^13 - 9*x^12 + 13*x^11 - 15*x^10 + 18*x^9 - 23*x^8 + 33*x^7 - 44*x^6 + 48*x^5 - 44*x^4 + 30*x^3 - 16*x^2 + 6*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 2*x^14 + 4*x^13 - 9*x^12 + 13*x^11 - 15*x^10 + 18*x^9 - 23*x^8 + 33*x^7 - 44*x^6 + 48*x^5 - 44*x^4 + 30*x^3 - 16*x^2 + 6*x - 1)
\( x^{15} - 2 x^{14} + 4 x^{13} - 9 x^{12} + 13 x^{11} - 15 x^{10} + 18 x^{9} - 23 x^{8} + 33 x^{7} + \cdots - 1 \)
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: |
\(1788527348449625\)
\(\medspace = 5^{3}\cdot 7^{10}\cdot 37^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.40\) | 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: | $5^{1/2}7^{2/3}37^{1/2}\approx 49.77193869732466$ | ||
| Ramified primes: |
\(5\), \(7\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{185}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7561}a^{14}-\frac{3367}{7561}a^{13}+\frac{3581}{7561}a^{12}+\frac{2160}{7561}a^{11}-\frac{2266}{7561}a^{10}+\frac{3587}{7561}a^{9}-\frac{2881}{7561}a^{8}+\frac{1340}{7561}a^{7}-\frac{2711}{7561}a^{6}-\frac{3656}{7561}a^{5}+\frac{741}{7561}a^{4}+\frac{1621}{7561}a^{3}-\frac{3154}{7561}a^{2}-\frac{2450}{7561}a+\frac{2766}{7561}$
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: |
$\frac{12121}{7561}a^{14}-\frac{19812}{7561}a^{13}+\frac{42966}{7561}a^{12}-\frac{93115}{7561}a^{11}+\frac{123903}{7561}a^{10}-\frac{141382}{7561}a^{9}+\frac{162439}{7561}a^{8}-\frac{210596}{7561}a^{7}+\frac{325198}{7561}a^{6}-\frac{415210}{7561}a^{5}+\frac{430170}{7561}a^{4}-\frac{365826}{7561}a^{3}+\frac{202929}{7561}a^{2}-\frac{95135}{7561}a+\frac{16334}{7561}$, $\frac{49915}{7561}a^{14}-\frac{73507}{7561}a^{13}+\frac{154795}{7561}a^{12}-\frac{358827}{7561}a^{11}+\frac{443708}{7561}a^{10}-\frac{475718}{7561}a^{9}+\frac{602424}{7561}a^{8}-\frac{792411}{7561}a^{7}+\frac{1179168}{7561}a^{6}-\frac{1501583}{7561}a^{5}+\frac{1488120}{7561}a^{4}-\frac{1268294}{7561}a^{3}+\frac{713966}{7561}a^{2}-\frac{347942}{7561}a+\frac{91762}{7561}$, $\frac{220}{7561}a^{14}+\frac{238}{7561}a^{13}+\frac{1476}{7561}a^{12}-\frac{1143}{7561}a^{11}+\frac{506}{7561}a^{10}-\frac{4765}{7561}a^{9}+\frac{1304}{7561}a^{8}-\frac{79}{7561}a^{7}+\frac{899}{7561}a^{6}-\frac{2854}{7561}a^{5}+\frac{11800}{7561}a^{4}-\frac{6308}{7561}a^{3}-\frac{5829}{7561}a^{2}+\frac{5392}{7561}a-\frac{11482}{7561}$, $\frac{18511}{7561}a^{14}-\frac{23897}{7561}a^{13}+\frac{53531}{7561}a^{12}-\frac{127345}{7561}a^{11}+\frac{146161}{7561}a^{10}-\frac{160526}{7561}a^{9}+\frac{216811}{7561}a^{8}-\frac{275097}{7561}a^{7}+\frac{407330}{7561}a^{6}-\frac{511853}{7561}a^{5}+\frac{500023}{7561}a^{4}-\frac{441816}{7561}a^{3}+\frac{259422}{7561}a^{2}-\frac{137170}{7561}a+\frac{43700}{7561}$, $\frac{24301}{7561}a^{14}-\frac{34130}{7561}a^{13}+\frac{70381}{7561}a^{12}-\frac{164644}{7561}a^{11}+\frac{197283}{7561}a^{10}-\frac{199668}{7561}a^{9}+\frac{253192}{7561}a^{8}-\frac{349693}{7561}a^{7}+\frac{528252}{7561}a^{6}-\frac{652952}{7561}a^{5}+\frac{616741}{7561}a^{4}-\frac{499915}{7561}a^{3}+\frac{250016}{7561}a^{2}-\frac{107990}{7561}a+\frac{21959}{7561}$, $a$, $\frac{11682}{7561}a^{14}-\frac{16094}{7561}a^{13}+\frac{28473}{7561}a^{12}-\frac{73547}{7561}a^{11}+\frac{82820}{7561}a^{10}-\frac{67777}{7561}a^{9}+\frac{96462}{7561}a^{8}-\frac{148610}{7561}a^{7}+\frac{222396}{7561}a^{6}-\frac{261938}{7561}a^{5}+\frac{218286}{7561}a^{4}-\frac{162564}{7561}a^{3}+\frac{60213}{7561}a^{2}-\frac{17637}{7561}a+\frac{11820}{7561}$, $\frac{41303}{7561}a^{14}-\frac{65777}{7561}a^{13}+\frac{141420}{7561}a^{12}-\frac{315321}{7561}a^{11}+\frac{413315}{7561}a^{10}-\frac{465155}{7561}a^{9}+\frac{560589}{7561}a^{8}-\frac{726356}{7561}a^{7}+\frac{1087200}{7561}a^{6}-\frac{1394261}{7561}a^{5}+\frac{1442746}{7561}a^{4}-\frac{1263179}{7561}a^{3}+\frac{739785}{7561}a^{2}-\frac{373976}{7561}a+\frac{95681}{7561}$
| 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: | \( 26.5068203244 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 26.5068203244 \cdot 1}{2\cdot\sqrt{1788527348449625}}\cr\approx \mathstrut & 0.154258478351 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 4*x^13 - 9*x^12 + 13*x^11 - 15*x^10 + 18*x^9 - 23*x^8 + 33*x^7 - 44*x^6 + 48*x^5 - 44*x^4 + 30*x^3 - 16*x^2 + 6*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 - 2*x^14 + 4*x^13 - 9*x^12 + 13*x^11 - 15*x^10 + 18*x^9 - 23*x^8 + 33*x^7 - 44*x^6 + 48*x^5 - 44*x^4 + 30*x^3 - 16*x^2 + 6*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 - 2*x^14 + 4*x^13 - 9*x^12 + 13*x^11 - 15*x^10 + 18*x^9 - 23*x^8 + 33*x^7 - 44*x^6 + 48*x^5 - 44*x^4 + 30*x^3 - 16*x^2 + 6*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 - 2*x^14 + 4*x^13 - 9*x^12 + 13*x^11 - 15*x^10 + 18*x^9 - 23*x^8 + 33*x^7 - 44*x^6 + 48*x^5 - 44*x^4 + 30*x^3 - 16*x^2 + 6*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
$C_3\times S_5$ (as 15T24):
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 360 |
| The 21 conjugacy class representatives for $S_5 \times C_3$ |
| Character table for $S_5 \times C_3$ is not computed |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 5.1.9065.1 |
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 18 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 36 sibling: | data not computed |
| Degree 45 sibling: | 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 | $15$ | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.5.0.1}{5} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | $15$ | ${\href{/padicField/29.3.0.1}{3} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | R | ${\href{/padicField/41.5.0.1}{5} }^{3}$ | ${\href{/padicField/43.5.0.1}{5} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(7\)
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
|
\(37\)
| 37.3.0.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 37.6.3.1 | $x^{6} + 2775 x^{5} + 2566998 x^{4} + 791680745 x^{3} + 110476893 x^{2} + 4842384300 x + 27887445532$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 37.6.0.1 | $x^{6} + 35 x^{3} + 4 x^{2} + 30 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |