Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 2*x^13 + 3*x^12 + 11*x^11 - 5*x^10 - 14*x^9 - 5*x^8 - 10*x^7 + 22*x^6 + 14*x^5 + 2*x^4 - 13*x^3 - 5*x^2 + 1)
gp: K = bnfinit(y^15 - y^14 - 2*y^13 + 3*y^12 + 11*y^11 - 5*y^10 - 14*y^9 - 5*y^8 - 10*y^7 + 22*y^6 + 14*y^5 + 2*y^4 - 13*y^3 - 5*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 - 2*x^13 + 3*x^12 + 11*x^11 - 5*x^10 - 14*x^9 - 5*x^8 - 10*x^7 + 22*x^6 + 14*x^5 + 2*x^4 - 13*x^3 - 5*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - x^14 - 2*x^13 + 3*x^12 + 11*x^11 - 5*x^10 - 14*x^9 - 5*x^8 - 10*x^7 + 22*x^6 + 14*x^5 + 2*x^4 - 13*x^3 - 5*x^2 + 1)
\( x^{15} - x^{14} - 2 x^{13} + 3 x^{12} + 11 x^{11} - 5 x^{10} - 14 x^{9} - 5 x^{8} - 10 x^{7} + 22 x^{6} + \cdots + 1 \)
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: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(137631064560062464\)
\(\medspace = 2^{12}\cdot 23^{6}\cdot 61^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.89\) | 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: | $2^{4/5}23^{1/2}61^{1/2}\approx 65.21580088926439$ | ||
| Ramified primes: |
\(2\), \(23\), \(61\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{61}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{23}a^{11}-\frac{6}{23}a^{10}-\frac{5}{23}a^{9}+\frac{1}{23}a^{8}+\frac{2}{23}a^{7}+\frac{2}{23}a^{6}-\frac{11}{23}a^{5}-\frac{7}{23}a^{4}+\frac{5}{23}a^{3}+\frac{10}{23}a^{2}+\frac{5}{23}a+\frac{1}{23}$, $\frac{1}{23}a^{12}+\frac{5}{23}a^{10}-\frac{6}{23}a^{9}+\frac{8}{23}a^{8}-\frac{9}{23}a^{7}+\frac{1}{23}a^{6}-\frac{4}{23}a^{5}+\frac{9}{23}a^{4}-\frac{6}{23}a^{3}-\frac{4}{23}a^{2}+\frac{8}{23}a+\frac{6}{23}$, $\frac{1}{529}a^{13}+\frac{8}{529}a^{12}+\frac{2}{529}a^{11}+\frac{98}{529}a^{10}+\frac{136}{529}a^{9}-\frac{201}{529}a^{8}-\frac{54}{529}a^{7}+\frac{67}{529}a^{6}+\frac{79}{529}a^{5}-\frac{212}{529}a^{4}+\frac{255}{529}a^{3}+\frac{199}{529}a^{2}-\frac{175}{529}a-\frac{1}{529}$, $\frac{1}{529}a^{14}+\frac{7}{529}a^{12}-\frac{10}{529}a^{11}+\frac{249}{529}a^{10}-\frac{185}{529}a^{9}-\frac{102}{529}a^{8}+\frac{223}{529}a^{7}-\frac{43}{529}a^{6}-\frac{108}{529}a^{5}+\frac{42}{529}a^{4}-\frac{70}{529}a^{3}+\frac{211}{529}a^{2}-\frac{96}{529}a-\frac{199}{529}$
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: |
$\frac{162}{529}a^{14}-\frac{84}{529}a^{13}-\frac{504}{529}a^{12}+\frac{466}{529}a^{11}+\frac{2114}{529}a^{10}-\frac{316}{529}a^{9}-\frac{3527}{529}a^{8}-\frac{1152}{529}a^{7}-\frac{1646}{529}a^{6}+\frac{2548}{529}a^{5}+\frac{5430}{529}a^{4}-\frac{882}{529}a^{3}-\frac{1624}{529}a^{2}-\frac{1542}{529}a+\frac{12}{23}$, $\frac{144}{529}a^{14}-\frac{44}{529}a^{13}-\frac{379}{529}a^{12}+\frac{358}{529}a^{11}+\frac{1828}{529}a^{10}+\frac{128}{529}a^{9}-\frac{2187}{529}a^{8}-\frac{564}{529}a^{7}-\frac{1642}{529}a^{6}+\frac{867}{529}a^{5}+\frac{2910}{529}a^{4}-\frac{370}{529}a^{3}-\frac{866}{529}a^{2}-\frac{213}{529}a+\frac{17}{23}$, $\frac{12}{529}a^{14}+\frac{71}{529}a^{13}-\frac{130}{529}a^{12}-\frac{1}{529}a^{11}+\frac{355}{529}a^{10}+\frac{605}{529}a^{9}-\frac{614}{529}a^{8}+\frac{15}{529}a^{7}-\frac{290}{529}a^{6}-\frac{1299}{529}a^{5}+\frac{1322}{529}a^{4}-\frac{1434}{529}a^{3}+\frac{515}{529}a^{2}-\frac{375}{529}a+\frac{761}{529}$, $\frac{150}{529}a^{14}-\frac{155}{529}a^{13}-\frac{374}{529}a^{12}+\frac{467}{529}a^{11}+\frac{1759}{529}a^{10}-\frac{921}{529}a^{9}-\frac{2913}{529}a^{8}-\frac{1167}{529}a^{7}-\frac{1356}{529}a^{6}+\frac{3847}{529}a^{5}+\frac{4108}{529}a^{4}+\frac{24}{23}a^{3}-\frac{93}{23}a^{2}-\frac{1167}{529}a-\frac{485}{529}$, $\frac{128}{529}a^{14}+\frac{133}{529}a^{13}-\frac{547}{529}a^{12}-\frac{48}{529}a^{11}+\frac{2241}{529}a^{10}+\frac{1975}{529}a^{9}-\frac{3334}{529}a^{8}-\frac{3340}{529}a^{7}-\frac{1929}{529}a^{6}-\frac{741}{529}a^{5}+\frac{7103}{529}a^{4}+\frac{109}{23}a^{3}-\frac{39}{23}a^{2}-\frac{2650}{529}a-\frac{535}{529}$, $\frac{208}{529}a^{14}-\frac{197}{529}a^{13}-\frac{350}{529}a^{12}+\frac{539}{529}a^{11}+\frac{2149}{529}a^{10}-\frac{665}{529}a^{9}-\frac{2135}{529}a^{8}-\frac{1536}{529}a^{7}-\frac{3122}{529}a^{6}+\frac{4339}{529}a^{5}+\frac{1947}{529}a^{4}+\frac{1376}{529}a^{3}-\frac{1295}{529}a^{2}-\frac{1363}{529}a+\frac{113}{529}$, $\frac{68}{529}a^{14}+\frac{141}{529}a^{13}-\frac{213}{529}a^{12}-\frac{145}{529}a^{11}+\frac{1103}{529}a^{10}+\frac{1950}{529}a^{9}-\frac{363}{529}a^{8}-\frac{1512}{529}a^{7}-\frac{2194}{529}a^{6}-\frac{123}{23}a^{5}+\frac{1392}{529}a^{4}+\frac{1571}{529}a^{3}+\frac{1950}{529}a^{2}-\frac{38}{529}a+\frac{12}{529}$, $\frac{164}{529}a^{14}-\frac{234}{529}a^{13}-\frac{287}{529}a^{12}+\frac{721}{529}a^{11}+\frac{1528}{529}a^{10}-\frac{1697}{529}a^{9}-\frac{1986}{529}a^{8}+\frac{678}{529}a^{7}-\frac{1823}{529}a^{6}+\frac{4466}{529}a^{5}+\frac{1480}{529}a^{4}-\frac{2495}{529}a^{3}-\frac{1290}{529}a^{2}-\frac{2}{529}a+\frac{557}{529}$
| 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: | \( 381.090214491 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 381.090214491 \cdot 1}{2\cdot\sqrt{137631064560062464}}\cr\approx \mathstrut & 0.252818449870 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 2*x^13 + 3*x^12 + 11*x^11 - 5*x^10 - 14*x^9 - 5*x^8 - 10*x^7 + 22*x^6 + 14*x^5 + 2*x^4 - 13*x^3 - 5*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 - 2*x^13 + 3*x^12 + 11*x^11 - 5*x^10 - 14*x^9 - 5*x^8 - 10*x^7 + 22*x^6 + 14*x^5 + 2*x^4 - 13*x^3 - 5*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 - 2*x^13 + 3*x^12 + 11*x^11 - 5*x^10 - 14*x^9 - 5*x^8 - 10*x^7 + 22*x^6 + 14*x^5 + 2*x^4 - 13*x^3 - 5*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 - 2*x^13 + 3*x^12 + 11*x^11 - 5*x^10 - 14*x^9 - 5*x^8 - 10*x^7 + 22*x^6 + 14*x^5 + 2*x^4 - 13*x^3 - 5*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, 5.3.22448.1 |
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 | R | $15$ | ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ | $15$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.3.0.1}{3} }^{5}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.15.12.1 | $x^{15} + 5 x^{13} + 5 x^{12} + 10 x^{11} + 26 x^{10} + 20 x^{9} - 145 x^{7} + 70 x^{6} + 73 x^{5} + 315 x^{4} - 105 x^{3} + 200 x^{2} + 5 x + 1$ | $5$ | $3$ | $12$ | $F_5\times C_3$ | $[\ ]_{5}^{12}$ |
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(61\)
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |