Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - x^13 + 2*x^12 - 4*x^11 - x^10 + 3*x^9 + 3*x^8 + 12*x^7 - 2*x^6 - 14*x^5 + 7*x^4 + 11*x^3 - 6*x^2 + 1)
gp: K = bnfinit(y^15 - y^14 - y^13 + 2*y^12 - 4*y^11 - y^10 + 3*y^9 + 3*y^8 + 12*y^7 - 2*y^6 - 14*y^5 + 7*y^4 + 11*y^3 - 6*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 - x^13 + 2*x^12 - 4*x^11 - x^10 + 3*x^9 + 3*x^8 + 12*x^7 - 2*x^6 - 14*x^5 + 7*x^4 + 11*x^3 - 6*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - x^14 - x^13 + 2*x^12 - 4*x^11 - x^10 + 3*x^9 + 3*x^8 + 12*x^7 - 2*x^6 - 14*x^5 + 7*x^4 + 11*x^3 - 6*x^2 + 1)
\( x^{15} - x^{14} - x^{13} + 2 x^{12} - 4 x^{11} - x^{10} + 3 x^{9} + 3 x^{8} + 12 x^{7} - 2 x^{6} + \cdots + 1 \)
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: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-6401597515801839\)
\(\medspace = -\,3^{9}\cdot 13^{3}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.32\) | 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: | $3^{7/6}13^{1/2}23^{1/2}\approx 62.298423685493326$ | ||
| Ramified primes: |
\(3\), \(13\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-39}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{23}a^{13}-\frac{7}{23}a^{11}-\frac{5}{23}a^{10}+\frac{10}{23}a^{9}-\frac{7}{23}a^{8}+\frac{5}{23}a^{7}+\frac{4}{23}a^{6}+\frac{9}{23}a^{5}+\frac{6}{23}a^{4}+\frac{7}{23}a^{3}+\frac{1}{23}a^{2}-\frac{7}{23}a+\frac{4}{23}$, $\frac{1}{1917533}a^{14}+\frac{29448}{1917533}a^{13}-\frac{761330}{1917533}a^{12}-\frac{611332}{1917533}a^{11}-\frac{232445}{1917533}a^{10}+\frac{820230}{1917533}a^{9}+\frac{707153}{1917533}a^{8}-\frac{293390}{1917533}a^{7}-\frac{138626}{1917533}a^{6}+\frac{781020}{1917533}a^{5}-\frac{300256}{1917533}a^{4}-\frac{327532}{1917533}a^{3}+\frac{534843}{1917533}a^{2}+\frac{642407}{1917533}a+\frac{946988}{1917533}$
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: | $7$ | 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: |
$\frac{360497}{1917533}a^{14}+\frac{286226}{1917533}a^{13}-\frac{682720}{1917533}a^{12}-\frac{117120}{1917533}a^{11}-\frac{616888}{1917533}a^{10}-\frac{2471842}{1917533}a^{9}-\frac{639983}{1917533}a^{8}+\frac{2154207}{1917533}a^{7}+\frac{7491088}{1917533}a^{6}+\frac{8248471}{1917533}a^{5}-\frac{1484900}{1917533}a^{4}-\frac{5293656}{1917533}a^{3}+\frac{3104612}{1917533}a^{2}+\frac{4585530}{1917533}a-\frac{404054}{1917533}$, $\frac{329265}{1917533}a^{14}-\frac{101693}{1917533}a^{13}-\frac{233360}{1917533}a^{12}+\frac{44452}{1917533}a^{11}-\frac{1008070}{1917533}a^{10}-\frac{987354}{1917533}a^{9}-\frac{380756}{1917533}a^{8}+\frac{1753964}{1917533}a^{7}+\frac{4851047}{1917533}a^{6}+\frac{4367316}{1917533}a^{5}-\frac{1376217}{1917533}a^{4}-\frac{1934350}{1917533}a^{3}+\frac{1434176}{1917533}a^{2}+\frac{159448}{1917533}a+\frac{713029}{1917533}$, $\frac{37196}{1917533}a^{14}+\frac{353094}{1917533}a^{13}-\frac{303336}{1917533}a^{12}-\frac{415161}{1917533}a^{11}+\frac{548932}{1917533}a^{10}-\frac{1426193}{1917533}a^{9}-\frac{871109}{1917533}a^{8}+\frac{1246541}{1917533}a^{7}+\frac{1497590}{1917533}a^{6}+\frac{5197230}{1917533}a^{5}+\frac{35101}{83371}a^{4}-\frac{5211786}{1917533}a^{3}+\frac{3367048}{1917533}a^{2}+\frac{3093189}{1917533}a-\frac{1249046}{1917533}$, $\frac{197086}{1917533}a^{14}+\frac{166476}{1917533}a^{13}-\frac{527130}{1917533}a^{12}-\frac{127337}{1917533}a^{11}+\frac{209004}{1917533}a^{10}-\frac{1936527}{1917533}a^{9}+\frac{322878}{1917533}a^{8}+\frac{1980237}{1917533}a^{7}+\frac{2767704}{1917533}a^{6}+\frac{4899196}{1917533}a^{5}-\frac{4353734}{1917533}a^{4}-\frac{4476132}{1917533}a^{3}+\frac{5946360}{1917533}a^{2}+\frac{2892370}{1917533}a-\frac{1996231}{1917533}$, $\frac{159678}{1917533}a^{14}-\frac{260140}{1917533}a^{13}+\frac{105394}{1917533}a^{12}+\frac{415045}{1917533}a^{11}-\frac{1084188}{1917533}a^{10}+\frac{18691}{83371}a^{9}-\frac{155327}{1917533}a^{8}-\frac{179471}{1917533}a^{7}+\frac{1645718}{1917533}a^{6}-\frac{1049807}{1917533}a^{5}+\frac{1550822}{1917533}a^{4}+\frac{4059235}{1917533}a^{3}-\frac{1091168}{1917533}a^{2}-\frac{3164245}{1917533}a+\frac{1499744}{1917533}$, $\frac{69054}{1917533}a^{14}-\frac{333353}{1917533}a^{13}+\frac{120441}{1917533}a^{12}+\frac{652890}{1917533}a^{11}-\frac{1005594}{1917533}a^{10}+\frac{989747}{1917533}a^{9}+\frac{40509}{83371}a^{8}-\frac{1517141}{1917533}a^{7}+\frac{395271}{1917533}a^{6}-\frac{3563031}{1917533}a^{5}-\frac{1344286}{1917533}a^{4}+\frac{6493316}{1917533}a^{3}+\frac{112377}{1917533}a^{2}-\frac{4104220}{1917533}a+\frac{431792}{1917533}$, $\frac{487045}{1917533}a^{14}-\frac{62083}{1917533}a^{13}-\frac{943508}{1917533}a^{12}+\frac{410055}{1917533}a^{11}-\frac{44659}{83371}a^{10}-\frac{2151367}{1917533}a^{9}+\frac{1228043}{1917533}a^{8}+\frac{3344595}{1917533}a^{7}+\frac{7323747}{1917533}a^{6}+\frac{4611865}{1917533}a^{5}-\frac{9367957}{1917533}a^{4}-\frac{4920090}{1917533}a^{3}+\frac{5922147}{1917533}a^{2}+\frac{2760191}{1917533}a-\frac{1860241}{1917533}$
| 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: | \( 46.8775403574 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 46.8775403574 \cdot 1}{2\cdot\sqrt{6401597515801839}}\cr\approx \mathstrut & 0.226506000756 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - x^13 + 2*x^12 - 4*x^11 - x^10 + 3*x^9 + 3*x^8 + 12*x^7 - 2*x^6 - 14*x^5 + 7*x^4 + 11*x^3 - 6*x^2 + 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^14 - x^13 + 2*x^12 - 4*x^11 - x^10 + 3*x^9 + 3*x^8 + 12*x^7 - 2*x^6 - 14*x^5 + 7*x^4 + 11*x^3 - 6*x^2 + 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^14 - x^13 + 2*x^12 - 4*x^11 - x^10 + 3*x^9 + 3*x^8 + 12*x^7 - 2*x^6 - 14*x^5 + 7*x^4 + 11*x^3 - 6*x^2 + 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^14 - x^13 + 2*x^12 - 4*x^11 - x^10 + 3*x^9 + 3*x^8 + 12*x^7 - 2*x^6 - 14*x^5 + 7*x^4 + 11*x^3 - 6*x^2 + 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
$S_3\times S_5$ (as 15T29):
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 21 conjugacy class representatives for $S_5 \times S_3$ |
| Character table for $S_5 \times S_3$ |
Intermediate fields
| 3.1.23.1, 5.1.8073.1 |
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 siblings: | 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 | $15$ | R | ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | R | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.0.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 3.9.9.1 | $x^{9} + 90 x^{7} - 207 x^{6} + 540 x^{5} + 324 x^{4} + 243 x^{3} + 324 x^{2} + 162 x + 27$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $[3/2]_{2}^{3}$ | |
|
\(13\)
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.6.3.1 | $x^{6} + 390 x^{5} + 50743 x^{4} + 2208202 x^{3} + 765301 x^{2} + 5017316 x + 24555184$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(23\)
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.3.0.1 | $x^{3} + 2 x + 18$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |